a characteristic of the APU bnd boundary, used to describe local SOC bulk ... APU power delivery (uni-directional). [W]. PL max maximum sustained load power.
BATTERY STATE ESTIMATION AND CONTROL FOR POWER BUFFERING APPLICATIONS by Herman L. N. Wiegman
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy Electrical and Computer Engineering
at the
UNIVERSITY OF WISCONSIN - MADISON
1999
© Copyright by Herman L.N. Wiegman, 1999 All rights reserved
i
Abstract A summary of aqueous solution, porous electrode cell modeling and performance evaluation is presented. Applications featuring electrochemical cells as load leveling devices are categorized according to battery state controllability and the relative impedance to power buffering break frequency ratio.
Special attention is given to charge-sustaining hybrid
electric vehicles. A battery state control algorithm based solely on terminal voltage is analyzed under switched mode loading conditions, and further developed for dynamic driving cycle loading conditions. Advanced models of parasitic effects in aqueous solution, porous electrode cells are developed for the purpose of battery state control and power capability prediction.
Direction sensitive series resistance, diffusion effect and mass transport
limitation are developed and evaluated for three battery chemistries. Non-linear impedance estimation methods which use the natural excitation of the load leveling application are outlined. Several short term effective source model structures for the battery are developed and evaluated. The short term power acceptance and delivery capabilities and effective efficiency characteristics are calculated to provide the battery state controller with control metrics.
ii
Acknowledgments I am grateful for the guidance and support of Prof. Robert D. Lorenz, without which this research effort would never have matured to its present state. My gratitude also goes to the staff and faculty of the Wisconsin Power Electronics Research Center (WisPERC) and the member companies of the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) for the great facilities, friendships, and research opportunities. Especially remembered are;
Ted Bohn, Ethan
Brodsky, Clark Hochgraf, and Michael Ryan, for the wonderful interactions via Team Paradigm, the most successful HEV team around. I am grateful to have received a fellowship from the Netherland-America Foundation during my Ph.D. studies. Via this fellowship I was able to begin exploring this exciting topic with Prof. A. Vandenput and the staff at the Technische Universiteit Eindhoven. I am grateful to my parents who have had a positive influence not only on my upbringing but also in my maturation. I am grateful to my good wife Holly, and my children, Nathan and Anna. I feel that I receive the recognition of this work, but they have paid the price. Holly's dedication to maintaining a solid foundation at home, her encouragement during trying times, and her valuable edits, rewrites, and clarifications have not gone unnoticed. Half van dit doctoraat is van jou.
iii
iv
Forward It was a cold, Wisconsin winter’s night, when I first thought of adding a power buffering battery to my bicycle lamp's generator. The rhythmic hum of the generator would often set my pace as I cut through the biting air by Lake Mendota. The head lamp’s glow would slowly surge with each pedal stoke and wobble of my rear wheel, lighting my way around ice patches and snow drifts. The security and friendship of the lamp would fade though, as I slowed for the steep, wooded hill leading to the orange glow of my apartment complex. Without the generator’s power, I was left alone, unguided. The solution to my dilemma came from a crackle of tree branches. If I added a diode and a set of rechargeable batteries to the system, I could avoid losing the lamp's light. This application of a small rechargeable battery as a load leveling device is just the tip of the iceberg. The proliferation of electrochemical cells as power buffering elements is accelerating. In order to maximize the capability of the cells and to provide more optimal control of the cell’s state, a sound method of estimating the cell's power processing capabilities and efficiency is presented. This thesis shows the direction of the author’s recent work to further the successful application of electrochemical cells as load-leveling devices. As it turned out, I removed the small rechargeable battery from my bicycle’s lighting system. It was an inappropriate application of technology to a rather bearable situation. With the battery, the surging of my lamp was eliminated, the hill no longer challenged me to pedal my best, and my friend - my guide became a characterless feature of my transport home. I missed the rhythmic feedback. The battery improved the lighting system's performance, but it removed a small piece of life in the process. Herman L.N. Wiegman Madison, WI 1999
v
Table of Contents Abstract ................................................................................................................. i Acknowledgments................................................................................................ ii Forward ............................................................................................................... iv Table of Contents................................................................................................. v Nomenclature ..................................................................................................... vii N.1 Acronyms and Abbreviations..................................................................................... vii N.2 Subscripts and Superscripts....................................................................................... viii N.3 Application Level Variables........................................................................................ ix N.4 Electrochemical and Electrical Circuit Modeling ....................................................... ix N.5 State Regulation and Power Prediction....................................................................... xi N.6 Parameter Estimation ................................................................................................. xii
Introduction.......................................................................................................... 1 I.1 Electrochemical Cells as Load Leveling Devices .......................................................... 2 I.2 Electrochemical Cell Performance Evaluation .............................................................. 6 I.3 Contributions ................................................................................................................. 7 I.4 Chapter Summary .......................................................................................................... 8
Chapter 1, State of the Art Assessment ............................................................. 10 1.0 Introduction ................................................................................................................. 11 1.1 Modeling of the Electrochemical Cell ........................................................................ 12 1.2 Battery Performance Evaluation ................................................................................. 35 1.3 Battery Management and Monitoring ......................................................................... 45 1.4 Application Specific Simulation of Cells.................................................................... 48 1.5 Battery State Control................................................................................................... 52 1.6 Recursive Parameter Estimation ................................................................................. 55 1.7 Summary ..................................................................................................................... 66
Chapter 2, Battery State Control via Terminal Voltage .................................... 69 2.1 Battery State Descriptors............................................................................................. 70 2.2 Voltage Control Analysis ............................................................................................ 72 2.3 Constant Load Experiments ........................................................................................ 91 2.4 Voltage Control for Dynamic Loading...................................................................... 100 2.5 Dynamic Load Experiments...................................................................................... 105 2.6 Properties of the Voltage Control Technique............................................................ 111 2.7 Summary ................................................................................................................... 112
Chapter 3, Advances in Modeling and Impedance Estimation ....................... 116
vi 3.1 Estimating Internal Voltages..................................................................................... 117 3.2 Impedance Modeling................................................................................................. 120 3.3 Filtering and Robust Estimation................................................................................ 135 3.4 Summary ................................................................................................................... 138
Chapter 4, Power Capability Prediction .......................................................... 141 4.1 Short Term Modeling................................................................................................ 142 4.2 Power Capability Calculations .................................................................................. 157 4.3 Power Capability Prediction Validation.................................................................... 160 4.4 Summary ................................................................................................................... 169
Chapter 5, Weighted Effective Efficiency....................................................... 171 5.1 Efficiency Calculations ............................................................................................. 171 5.2 Weighted Efficiency Evaluation ............................................................................... 179 5.3 Summary ................................................................................................................... 182
Chapter 6, Summary of Key Conclusions and Future Work........................... 183 6.1 Summary of Key Conclusions................................................................................... 183 6.2 Future Work .............................................................................................................. 186
Bibliography .................................................................................................... 189 Appendix.......................................................................................................... 196 A.1 System of Electrical Normalization's ....................................................................... 196 A.2 Example Battery Modules ........................................................................................ 197 A.3 Example Hybrid Electric Vehicle............................................................................. 201 A.4 Battery Test Bench Implementation ......................................................................... 204 A.5 Electrochemical Cells as Power Buffering Elements............................................... 210 A.6 Voltage Prediction Error Analysis............................................................................ 213
vii
Nomenclature N.1 Acronyms and Abbreviations..................................................................................... vii N.2 Subscripts and Superscripts ...................................................................................... viii N.3 Application Level Variables ....................................................................................... ix N.4 Electrochemical and Electrical Circuit Modeling ....................................................... ix N.5 State Regulation and Power Prediction....................................................................... xi N.6 Parameter Estimation ................................................................................................. xii Acronyms are listed first, with their relevant location in the document. Second, variables are categorized with Roman then Greek letters. Many of the variable's units are included even thought they may be realized in normalized or per unit [pu] form (unitless) in this document.
N.1 Acronyms and Abbreviations Section
APU CS-HEV CD-HEV FLOPS FRE FUDS LF-PBM LLD MF-PBM MT Ni-Cd Ni-MH OE PAI PDI PNGV
Alternate Power Unit, primary energy source Charge Sustaining Hybrid Electric Vehicle Charge Depleting Hybrid Electric Vehicle floating point operations (per sample) Fast Response Engine, typically in a low energy storage HEV Federal Urban Driving Schedule Low Frequency Power Buffering Model Load Leveling Device, power buffering element Medium Frequency Power Buffering Model Mass Transfer, the current rate limit of the cell Nickel-Cadmium chemistry Nickel-Metal Hydride chemistry Output Error, structure of the prediction model Power Acceptance Index, higher = better Power Delivery Index, higher = better Partnership for Next Generation Vehicles
I.1.2 1.5.1 1.5.1 4.1.3 2.3 2.3 3.1.2 2.1 3.1.2 1.1.5
1.6.1 3.2 3.2 I.1.2
PV RGN RLS SFUDS SOC SOV SRE USABC VRLA ZOH
viii Photovoltaic, solar energy collection system 1.1.1 Recursive Gauss-Newton, error reduction method 1.6.3 Recursive Least Squares, error reduction method 1.6.3 Simplified FUDS, simple power trace with discrete steps 2.3 State of Charge, the concentration of the electrolyte's reaction species, boundary and bulk descriptions are treated separately. 1.1.1 State of Voltage, descriptor of terminal voltage characteristic 2.1 Slow Response Engine, typically in a high energy storage HEV 2.3 United States Advanced Battery Consortium I.2.2 Valve Regulated Lead Acid, battery chemistry 1.2.3 Zero Order Hold 1.6.4
N.2 Subscripts and Superscripts apu bnd bulk chrg ct d dis dif dl e, ele g grid hi [k] L lo mt n p req s t
a characteristic of the APU boundary, used to describe local SOC bulk, used to describe average SOC charging direction dependent (current into cell, negative value) charge transfer descriptor discrete parameter descriptor (see p descriptor) discharging direction dependent (current out of cell, positive value) diffusion effect descriptor double layer descriptor electrolyte descriptors gassing descriptor electrode grid descriptor associated with an upper (high) control limit the kth sampled point, also [k-n] is the nth previous data sample the system's load descriptor, discharge is positive (also see veh) associated with a lower control limit mass transport descriptor normalizing value for model variables (nominal value) physical parameter descriptor (see d descriptor) required performance levels (PNGV standards) short term model descriptor a quantity measurable at the terminals of the cell
ix veh θ ' + -
a description of the vehicle (HEV) load, equivalent to "L" thermal model or OE model parameter descriptor weighted efficiency term based on energy accumulation function. a characteristic of the positive (cathode) electrode reaction a characteristic of the negative (anode) electrode reaction
N.3 Application Level Variables dapu Eapu EL ∆Es Icom It* PAPU PL max PL, Pveh ~ T Tapu ∆Qmax
APU duty cycle (percentage of time that it is operational) net energy supplied by the APU over driving cycle test net energy consumed by load over test net change in battery energy state over test current command to experimental cell or battery (current regulator) reference terminal current command (power regulation loop) APU power delivery (uni-directional) maximum sustained load power vehicle or "load" power demand (bi-directional) approximate model APU operating period (Sec 2.2.3) period of APU cycling maximum allowed variation in normalized SOC
Units
[%] [Wh] [Wh] [Wh] [A] [A] [W] [W] [W] [h] [h] [-]
N.4 Electrochemical and Electrical Circuit Modeling Ae Bg Cdif Cdl Co,Cr Cθ D Eo Eo Es F
equivalent area of the electrode surface [m2] Offset term for gassing model [A] diffusion capacitance (coupled to diffusion coefficient) [F] double layer capacitance (Udl dependent) [F] concentrations of the oxidized and reduced species [-] thermal capacity of the cell or battery [J/°K] -9 2 diffusion coefficient (on the order of ~10 [m /sec]) [m2/sec] standard potential of the reaction process (2.047 V for lead-acid) [V] initial energy state at t=0 (Sec 2.1.1) [Wh] stored energy in the cell or battery [Wh] Faraday’s charge constant = 96,487 [Coul./mole]
x ~ I Ict Ig Igo Imt Io It k1,3 k2,4 kct1 kct2
~ approximate model current level (function of Q, P, Rs) Faradaic, or charge transfer current gassing current, function of Temp and Ut normalizing gassing current gain parameter mass transport current limit (functions of D, δ, and current direction) exchange current constant (shown in Figure 1.2 c) terminal current, difference of Ig and Ict offset resistance fitting parameters (model for Rs, Sec 2.2.2) stored charge fitting parameters (model for Rs, Sec 2.2.2) current parameter (function of SOC, sign(Ict) and T) voltage parameter (function of SOC, sign(Ict) and T)
kdif
Impedance parameter ( function of D and Temp)
kdl1 kdl2 kgv kgθ ki kp ksh M Mct ne Ploss Q ~ Q Qdl Qeff Qo
capacitance parameter (function of SOC, sign(Ict) and T) voltage parameter (function of SOC, sign(Ict) and T) voltage dependence parameter in gassing relation thermal dependence parameter in gassing relation current rate penalty factor for Qeff (typically 1.2~1.4) linear polarization parameter (Sec 1.4.1, Shepherd's model) standard heterogeneous rate constant (app = apparent) battery, module or cell mass slope of Rct vs. Us number of electrons participating in the electrode process power loss map (Sec. 1.2.1, function of Q, It, and Temp) average electrolyte concentration (bulk SOC)
R
gas constant = 8.314
Rct Rdif Re Rele Rgrid
charge transfer resistance (function of current and local SOC) diffusion resistance (coupled to diffusion coefficient) electrolyte resistance in the equivalent circuit (Figure 1.7) equivalent series resistance (function of SOC and temperature ) grid resistance in the equivalent circuit (Figure 1.7)
approximate model SOC (Sec 2.2.3) amount of charge accumulated in the double layer effective remaining charge for end-of-use prediction (Sec.1.2.1) initial stored charge at t = 0
Ω
[Α] [A] [A] [A] [A] [A] [A] [Ω] [-] [A] [V-1]
[F] [V-1] [V] [°K] [-] [V] -1 -2 [sec m ] [kg] [Ω V-1] [moles] [W] [Ah] rad sec
[Ah] [Coul.] [Ah] [Ah] o J K mole [Ω] [Ω] [Ω] [Ω] [Ω]
xi Ro Rs ~ Rs T tn Udif Udl Uele Ug Umt Un Uo Uoc Us Ut Uz Zdif zo α δ τ τθ
offset electrolyte resistance (function of construction and temp.) effective short term series resistances (dis and chrg directions)
[Ω] [Ω]
mean values of short term series resistances (approximate model) average cell temperature (internal to cell) base quantity for time = 3600 [sec] = 1 [hour] diffusion voltage which modifies Uoc voltage across interfacial double layer (function of local SOC, current level, and temperature) ohmic voltage drop in electrolyte (function of SOC and temperature) driving voltage of the combined gassing reaction mass transport limited reaction diffusion voltage open circuit potential at rated acid concentration (100% SOC) open circuit potential at minimum acid concentration (0% SOC) open circuit potential based on bulk SOC (function of SOC, Uo ≤ Uoc ≤ Un ) effective source voltage for power prediction (Uoc - Udif) terminal voltage of cell or battery
[Ω] [oK] [h] [V] [V] [V] [V] [V] [V] [V] [V] [V] [V]
voltage across the cell's impedances (neglecting gassing circuit)
[V]
diffusion impedance (function of cell structure and current level) charge number of the oxidized species (zo - ne = zr) transfer coefficient fraction (function of Udl magnitude and direction) Nernst diffusion distance transition time to fully depleted electrolyte at the boundary thermal time constant (usually in air)
[Ω] [-] [-] [m] [h] [h]
N.5 State Regulation and Power Prediction Imax Pcap Pdis,chrg Pload Ploss Pmax Q*
the dc current limit for the energy storage device (or system) power capability to given set of voltage limits loading specific power flow out of and into the battery power drawn from the cell (positive = out of cell) power loss map (function of Q, It, and Temp) maximum power capability (function of direction) target value for the SOC regulator (mean(Qlo,Qhi))
[A] [W] [W] [W] [W] [W] [Ah]
xii [Ah] [Ah] [V] [V]
Qavg Qlo,hi Ulo,hi Umax,min
analytical average SOC level (function of Pveh, Uhi, Ulo, ) lower and upper limits for the SOC trajectory lower and upper limits for the trend voltage indicator the extreme terminal voltage limits for the system
ηcycle ηpow
net LLD energy efficiency (@ some rate)
[-]
power efficiency (function of current direction)
[-]
N.6 Parameter Estimation e(k) G(q,θ) H(q,θ) k nθ P q Ts u(k) y(k) ε(k) φ θ ψ ^
outside error injected into the system plant model (polynomial transfer function) error model (polynomial transfer function) index (k=1 at first sample) number of parameters to be fitted from model adaptive step size matrix (diagonal) shift operator (q-n a(k) = a(k-n)) sampling period input of the discrete sampled system output of the discrete sampled system prediction error ( y(k) - ^y(k) ) regressor vector (inputs & predicted outputs) parameter vector (coefficients) prediction gradient (dy/dθ) indicates an estimated quantity or variable
1
Introduction I.1 Electrochemical Cells as Load Leveling Devices...........................................................2 I.1.1 Hybrid Stand Alone Power Source ......................................................................3 I.1.2 Hybrid Electric Vehicles ......................................................................................4 I.2 Electrochemical Cell Performance Evaluation ...............................................................6 I.2.1 State of Charge Evaluator ....................................................................................6 I.2.2 EV and HEV Battery Evaluation..........................................................................7 I.3 Contributions ..................................................................................................................7 I.4 Chapter Summary ...........................................................................................................8 The use of electrochemical cells in power buffering applications has given rise to new challenges. One of these challenges is to provide appropriate battery state control over long periods with highly stressful operating conditions. The battery, acting as a temporary energy storage device, should provide trouble free service under these conditions.
A second
challenge is to identify the power processing capability and efficiency of the load-leveling battery, and then to use this information to make better system level decisions. The ultimate goal of this study is to develop advanced battery state control techniques which; •
provide battery state control techniques which are drift-free and robust.
•
provide battery parameter information for power capability prediction and diagnostic purposes.
•
reduce the battery's thermal stresses and increase the battery’s operating life time via lower power losses.
•
minimize the size, cost or mass of the battery for a given cell technology and application, i.e. minimize over-design and maximize battery utilization. Two specific applications are reviewed in this chapter which show these set of needs.
The generally accepted methods of battery state evaluation and control are also introduced,
2 thus revealing a need for new state descriptors and control methodologies. The challenge of appropriate state control and predicting power processing capability of the electrochemical cell is identified as the focus of the proposed research.
I.1 Electrochemical Cells as Load Leveling Devices Electrochemical energy storage devices are being used as temporary sources and sinks of power in a wide range of applications. This section introduces several applications which continually alter the flow of power into and out of batteries over short to moderate time periods. These applications can be grouped according to the application's ability to influence or "steer" the battery's state. The controllability of the battery state will be defined as the ability of the system to actively influence the battery state over a wide range of operating conditions. Certain systems can only marginally drive the state of the battery towards a more desirable level, hence these systems have low controllability. The specific requirements and design constraints of each application dictate how the battery is utilized and controlled. Five power buffering applications are summarized in Table I.1.
The last two
applications with high battery state controllability are discussed further in this section, while the other applications are explored in Appendix A.5.
3 Table I.1
Battery state controllability for several load-leveling applications.
Application Utility Tariff Optimizer Frequency Regulation Satellite / Space Station Stand Alone Power System Hybrid Electric Vehicle
Uncontrolled Generation
Controlled Generation
Load
Battery State Controllability
Weak Grid
None
Dynamic
Low
Weak Grid
None
Dynamic
Low
Fuel Cell
Steady
Moderate
Solar (eclipse cycles) Solar or Wind (cyclic power) Brake Regeneration
EngineDynamic Generator Heat Engine, Dynamic Fuel Cell (bi-directional)
High High
I.1.1 Hybrid Stand Alone Power Source Two common photovoltaic (PV) applications utilize batteries as load leveling devices; spacecraft and stand alone power sources. Spacecraft use PV arrays as primary source of power and batteries for buffering the array's output over the craft's eclipse cycle. Due to the regular and predictable nature of eclipse mode in space craft, the power buffering cycles are fairly regular and the PV system is mass optimized with respect to the power demands of the space craft. The battery state is only moderately controllable due to very little excess capacity of the PV arrays over the lifetime of the mission. A more challenging PV power buffering application is in stand alone power systems for remote locations. Electrification of remote sites is a natural application for the use of electrochemical systems as power buffers. Often the primary source of power is either wind, solar, or fossil fuel based. The battery’s responsibility is to provide energy storage such that a stable intermediate bus voltage is provided, while the other sources are manipulated to provide the time-average power required by the customer. Often the battery is oversized to
4 accommodate large variations in the quality and quantity of power from the primary sources. I.1.2 Hybrid Electric Vehicles Internationally, many governments have applied both lower exhaust gas emission and higher fuel economy standards on new vehicle fleets in order to lessen fuel consumption and improve air quality. These trends have forced manufactures to seek major improvements in vehicular energy efficiencies. The two main drive-train technology thrusts which show promise in attaining the goals of this new transportation paradigm are regenerative braking, and engine load-leveling. Hybridizing a vehicle's drive-train with an electrochemical storage system and electric drive-train is the present system of choice to meet these new goals. A general Hybrid Electric Vehicle (HEV) power flow diagram and example power trace are shown in Figure I.2. The battery and Alternate Power Source (APU) together must satisfy the driving cycle's axle power trace.
Alternate Power Unit Computer Controller
Axle Power
Battery
a) General power and control diagram Figure I.2
b) Example axle power trace from FUDS.
General HEV power flow diagram and typical axle power demand vs. time for a standard U.S. driving cycle.
The relative stresses on HEV batteries are much greater than the utility applications discussed in Appendix A.5.
The higher stresses are a result of battery mass and cost
reduction. The constraints themselves are imposed by consideration of vehicle size, packaging problems and consumer acceptance. The battery mass minimization results in very
5 high power pulses and moderate state of charge (SOC) variations experienced over a period of seconds. The battery state controller must steer the battery towards regions of high operating efficiency to avoid high fuel consumption and high thermal stresses. Long driving cycles accentuate the problems found in the presently used battery state control techniques that employ charge integrators. This is because the charge integrators themselves are prone to drift. The goal of maximizing system efficiency while reducing battery mass is presently making the problem of optimal battery state control difficult. “The key problem for hybrid vehicles is battery performance and additional work is needed to reduce its weight and size.”
[HYZEM 1997]
The general solution for most recently developed HEV’s is to over-design the battery capacity, or to significantly limit the power flow to the battery. This lessens the stresses on the battery, and it is more able to provide higher power cycle efficiencies over a wider set of operating conditions. There is a pressing need to develop techniques which provide HEV controllers with up-to-date information about the battery’s power capabilities and relative energy efficiency so measures can be taken to steer the battery state towards optimal performance. Addressing the challenges of more optimally controlling the battery state in demanding HEV application's is the primary focus of this work. The basic requirements for the power buffering battery are given in Table I.2 as defined by the Partnership for a New Generation of Vehicles (PNGV) work group on HEV batteries [PNGV 1997].
This
document will use these specifications as a guideline in developing advanced state controllers and on-line performance prediction algorithms.
6 Table I.2
PNGV Charge Sustaining HEV, Battery Specification
Description Energy Cycle Efficiency Power Processing Voltage Variation Battery System Mass Temperature Range Minimum Energy Storage
Requirement (slow response engine) 90~95 % +65 ~ +80 kW (18 sec) -70 ~ -150 kW (10 sec) +/- 15 % from nominal < 65 kilo grams (>1000W/kg) -40oC ~ 52oC >> 0.54 M Joules (150 Wh)
I.2 Electrochemical Cell Performance Evaluation Historically, batteries have been primarily used as power sources for consumptive loads. This has led to certain modes of thinking when evaluating and rating battery storage systems.
With the recent growth of power buffering applications, battery performance
evaluation has slowly taken on a new direction. State regulators and control algorithms for complex power buffering systems are not mature technologies. I.2.1 State of Charge Evaluator The battery’s charge content, or SOC, is often considered as the main indicator of the state of the battery. Much work has been done to empirically model and describe the useful remaining charge as a function of temperature, current loading, and historical use. Unfortunately, effective remaining charge is not related to the fundamental requirements of a typical load leveling application as shown in Table I.2. The charge state estimate of the battery will be shown to be a secondary state description, which will be used only to assist the capabilities of the primary state regulation technique.
7 I.2.2 EV and HEV Battery Evaluation The United States Advanced Battery Consortium (USABC), and the PNGV has recently addressed the special requirements of EV and HEV batteries.[USABC 1996, PNGV 1997] The PNGV's battery tests report on the battery technology's short term gravinametric [W/kg], and volumetric [W/liter] power delivery and acceptance capabilities (see Appendix A.2.1). These power capability results and estimated power cycle efficiency ("round trip efficiency") are reported as a function of relative stored charge. These test results place emphasis on the battery's power buffering capabilities rather than the effective energy or charge storage capabilities. The results of the PNGV testing gives a suggested stored charge operating range for the battery in order to provide the required levels of charge and discharge power performances. This return to battery charge control to provide the a priori tested performance of the battery will be challenged in this document. Instead a move towards terminal voltage and power capability will be shown as a more direct way to insure the high performance capabilities of the battery for HEV applications.
I.3 Contributions This thesis research has made the following technical contributions in the field of battery state control and utilization for power buffering applications; Cell Modeling •
Reviewed physically insightful equivalent circuit models for porous electrode cells and advanced the modeling of direction sensitive series resistance, diffusion effect and Mass Transport effect in these cells.
8 •
Condensed the complex cell model into an effective source voltage and resistance for short term power capability and efficiency prediction.
State Regulation •
Developed a drift-free battery state regulation technique based on terminal voltage information.
•
Formulated analysis methods for evaluating voltage-based battery state regulators under both steady state and dynamic loading conditions.
•
Presented a direct comparison of voltage and charge based state regulation techniques.
On-line Performance Prediction •
Applied on-line recursive parameter estimation techniques to an advanced cell model using only the excitation present in the power buffering application.
•
Formulated on-line power capability prediction algorithms and used the index to make better battery state regulation decisions. Special attention was given to Lead-Acid, Nickel-Cadmium, and Nickel-Metal
Hydride storage systems in charge sustaining HEV applications. The above contributions should also be applicable to other porous electrode battery chemistries as well as other power buffering applications with high battery state controllability.
I.4 Chapter Summary Chapter 1 assesses the state of the art in four fields which will be used to advance the field of battery state control; •
Modeling of porous electrode electrochemical cells,
•
Battery performance evaluation and simulation,
•
Battery system monitoring and control,
9 •
Recursive parameter estimation Chapter 2 introduces a drift free battery state control method based on terminal
voltage measurements.
The chapter presents analytical methods for evaluating the
performance characteristics of the terminal voltage control technique.
Experimental
comparisons between charge and voltage based state regulators are made for both steady state and dynamic loading conditions. Chapter 3 presents contributions made in modeling the excitation direction dependent characteristics of the cell, and the mass transport limited region of operation. The methods used to estimate the model's impedances are also discussed.
These modeling
advances are then used in the following chapters to improve the on-line performance prediction of porous electrode battery systems. Chapter 4 presents a framework for short term power capability prediction. Several methods to simplify the full equivalent electrical model are presented and compared for the three battery chemistries of interest. Comparative experimental results of the on-line power capability prediction algorithms are given. Chapter 5 presents new methods for evaluating the short term effective operating efficiency of a battery based on historical use and the present state. Chapter 6 summarizes the main contributions of the thesis and proposes several areas of future research which further advance the field of study. The Appendix includes breadth and background material which should be reviewed by readers unfamiliar with the corresponding aspects of the material.
10
Chapter 1, State of the Art Assessment 1.0 Introduction..................................................................................................................11 1.1 Modeling of the Electrochemical Cell .........................................................................12 1.1.1 Open Circuit Voltage.........................................................................................13 1.1.2 Ohmic potential .................................................................................................16 1.1.3 Activation Overpotential and the Double Layer Effect .....................................16 1.1.4 Diffusion Effects ...............................................................................................19 1.1.5 Mass Transport Limitations...............................................................................22 1.1.6 Gas Evolution and Self Discharge.....................................................................23 1.1.7 Equivalent Electrical Circuit Model..................................................................25 1.1.8 Impedance Modeling .........................................................................................28 1.1.9 Summary of Electrochemical Cell Modeling ....................................................34 1.2 Battery Performance Evaluation ..................................................................................35 1.2.1 Empirically Modified Charge and Energy Descriptions....................................35 1.2.2 Power Capability Testing ..................................................................................38 1.2.3 Off-line Impedance Measurements ...................................................................40 1.2.4 Summary of Battery Performance Evaluation ...................................................45 1.3 Battery Management and Monitoring ..........................................................................45 1.3.1 End of Use and Recharge Determination ..........................................................46 1.3.2 Equalization of Cells .........................................................................................47 1.3.3 Age Determination and Maintenance ................................................................47 1.3.4 Summary of Battery Management Systems.......................................................48 1.4 Application Specific Simulation of Cells ....................................................................48 1.4.1 Phenomenological Modeling.............................................................................49 1.4.2 Physical Law Modeling .....................................................................................50 1.4.3 Grey Box Modeling ...........................................................................................51 1.4.4 Summary of Application Specific Simulation of Cells .....................................51 1.5 Battery State Control....................................................................................................52 1.5.1 SOC Control in Charge Sustaining Hybrid Electric Vehicles...........................52 1.5.2 Summary of Battery State Control ....................................................................54 1.6 Recursive Parameter Estimation ..................................................................................55 1.6.1 Model Set Selection...........................................................................................55 1.6.2 Black Box Model Structures .............................................................................57 1.6.3 Recursive Least Squares and Gauss-Newton Algorithms .................................61 1.6.4 Indirect Estimation of Model Parameters..........................................................63 1.6.5 Direct Estimation of Physical Model Parameters..............................................65 1.6.6 Non-linear Modeling .........................................................................................65 1.7 Summary ......................................................................................................................66 1.7.1 State of the Art Review .....................................................................................66 1.7.2 Areas of Opportunity .........................................................................................67
11
1.0 Introduction To contribute meaningfully to the advancement of battery state evaluation and control for power buffering applications, an understanding of several technologies is required. The focus of this chapter is to review the relevant portions of the state of the art in the fields listed below, and based on the review, to identify areas of research opportunity. The relevant fields are: •
Modeling of electrochemical cells.
•
Characterization and evaluation of cell impedance.
•
Evaluation of cell performance.
•
Battery monitoring systems.
•
Simulation of cells in various applications.
•
Battery state control techniques for various applications.
•
Recursive parameter estimation for Output Error systems. The above fields of study are interwoven and this research seeks to more closely join
the fields of on-line performance evaluation with that of battery loading control. One goal of this work is to integrate the relevant dimensions of this technology and apply them to the control of batteries in load-leveling applications. A diagram showing the areas of study is shown in Figure 1.1. The off-line fields of study are more closely meshed, and advanced analysis and modeling techniques are usually developed in this arena. The on-line fields of study often lag behind, and are hampered by non-ideal, or unpredictable operating conditions.
12 physical laws
empirical data
parameter extraction
concentration distributions
Cell Design
specific tests
Cell Parameter Determination
voltage trajectory
Off Line load leveling
On Line end of recharge
Opportunity
Control of Cell Loading
end of use charge equalization
parameter estimation
Performance Evaluation Cell Monitoring
SOC determination
Figure 1.1
gassing
Simulation
starting power
Ah & Energy aging maintenance
Summary diagram of the fields involved in battery design, evaluation and state control.
1.1 Modeling of the Electrochemical Cell The fundamental characteristics of aqueous, porous electrode electrochemical cells are outlined in this section. Six main phenomena in the cells are discussed to provide a foundation in the physical law modeling of electrode processes. These relationships are then transferred to an equivalent electrical circuit. This phenomenological impedance model is then used in later sections for state control and performance prediction. The modeling techniques presented in the following sections are applicable to several common battery chemistries; Lead-Acid (Pb-acid), Nickel-Cadmium (NiCd), & Nickel-Metal Hydride (NiMH) in use for power buffering applications. Other theory and modeling techniques are
13 necessary if alternate chemistries (Lithium-ion, Lithium-polymer, Zinc-air, high temperature batteries, etc.) are investigated. A purely physical law (or "white box") description of the reactions potentials and acid concentration gradients in the cell is often preferred when considering cell design and the effect of design parameters on the resulting cell’s performance [Newman 1975, Gu 1987, Karden 1997]. These macrohomogeneous and single pore models are not discussed in detail in this document because the emphasis of the research is on the estimation and evaluation of a cell’s externally measurable electrical performance, rather than the cell’s internally adjustable design optimization. To that end, a model based on physical law descriptions of the equivalent electrical components is the focus of the review in Chapter 1, Section 1. Definition of Terms The bulk State-of-Charge (SOC) of the cell is defined as the average reaction species concentration in the electrolyte. The normalized bulk SOC will be labeled as Q and has a value between 0 & 1. The boundary SOC is defined as the concentration of the reaction species at the electrodeelectrolyte interfacial region. Diffusion effects often cause differences between the bulk and boundary SOC values. Positive current flow (It > 0) in the cell is defined as being the opposite direction of the net electron flow while the electrolyte's SOC is being depleted by an external load. The voltage potentials present in the cell due to electrode reactions are considered positive voltage drops, thus acting as sinks of power. 1.1.1 Open Circuit Voltage An electrochemical cell is fundamentally a device that converts stored chemical energy directly into electrical energy via an oxidation-reduction (redox) process. A sample
14 aqueous cell with porous electrodes is shown in Figure 1.2.a. The porous anode, cathode, and aqueous electrolyte are shown. Other elements such as the separator (insulator) or electrolyte carrier (matting material) found in practical cells are not shown. A current, It, is shown flowing out of the cathode due to a hypothetical external load (power sink). The terminal voltage, Ut , and the center reference potential, φ½ = 0 [V], are defined in order to distinguish cathode and anode electrode reaction potentials. Other reference potentials in the region of the electrode-electrolyte interface are not shown due to simplicity. The nominal, no-load potential of each electrode’s reaction with the electrolyte is defined by the Nernst relation in (1.1). This describes the potential when no current is flowing across the interfacial region. This “open-circuit” or “rest” potential, Uoc , (often labeled "E") is a function of the boundary SOC (local to the surface of the electrodes) and is graphically shown in Figure 1.2.b vs. the normalized bulk SOC, Q, thus assuming there are no SOC gradients present when the cell is truly at rest. The actual relationship for Uoc in practical cells tends to have a weak temperature dependence, and often some non-linear effects vs. bulk SOC. Generally, it can be considered to be a positive slope relation with slight temperature dependent offset [Bode 1977, Kahlen 1994].
15 It
Ut +
Uoc Un cathode
φ1/2
Uo
electrolyte anode
a) Example porous electrode aqueous cell with center reference potential.
Figure 1.2
1
b) Example Nernst relation.
Basic electrochemical cell layout and open circuit potential relationship.
RT Co Uoc = E° + n F ln Cr e where Uoc Eo R T ne F Co,Cr
Q
open circuit potential as function of SOC standard potential of the electrode’s process
[V]
(1.1)
[V] [V]
o J K mole temperature of the reaction [oK] charge number participating in the process (small integer) [±] Faraday’s charge constant, 96,487 [Coul./mole] concentrations of oxidized and reduced species [mole/m3] gas constant, 8.314
Special cases defined include: Un potential at rated concentration (100% SOC) Uo potential at minimum concentration (0% SOC)
[V] [V]
The terminal voltage, Ut, deviates from the open-circuit potential, Uoc, due to several effects. These voltage drops (often labeled “overpotentials”) under loaded conditions occur due to four main effects; Ohmic potential drops in the bulk electrolyte and other conduction paths. Charge Transfer or activation potential drops of the electrolyte-electrode interfacial region. Diffusion or concentration potential drops from the temporal and spatial SOC gradients in the electrolyte. Mass Transport limitations from the rate limited operation of the cell.
16 1.1.2 Ohmic potential The effective series ohmic potential drop in the cell, Uele, is due to the combined series conduction drops in the bulk electrolyte, separator, grid and interconnections. This combined ionic and electrical voltage has a weak temperature and sometimes strong SOC dependence [Vinod 1994, Kahlen 1994]. The effective series resistance of the electrolyte, Rele, will be modeled in more detail in Section 1.1.8. Uele = It⋅Rele
[V]
Rele
[Ω]
(1.2)
where resistance of electrolyte and interconnections, f(Q,T)
1.1.3 Activation Overpotential and the Double Layer Effect The activation of the electrode-electrolyte boundary during current conduction leads to a charge transfer potential drop which behaves logarithmically with current magnitude and boundary SOC levels.
Alternatively, the current resulting from an external load is an
exponential function of the charge transfer potential, Udl, as shown in the Butler-Volmer relation of (1.3) [Linden 1984]. (1-α) ne F Udl α ne F Udl Ict = ne F Ae ksh Co exp - RT − Cr exp RT
[A]
(1.3)
Io = ne F Ae ksh Co(1-α)Cr(α)
[A]
(1.4)
where Ict Ae ksh α Udl Io
charge transfer (Faradaic) current effective area of the electrode
[A] [m2] m standard heterogeneous rate constant (at Eo, Udl=0) sec transfer coefficient fraction (typically 0.3~0.7) [-] (function of the reaction potential, Udl, and current direction) approximate activation potential (often labeled ηs) [V] exchange current constant (at Eo, Udl=0) [A]
17 An “electrical double layer” results from the attraction of opposite polarity charges at the electrode-electrolyte boundary, and from the orientation of solvent dipoles (commonly H20) [Grahame 1947].
This effect causes charge accumulation and results in a large
capacitance in the interface region. This phenomena effects the boundary SOC and the charge transfer potential, Udl. The modeling of this capacitive effect is explored further in section 1.1.8. Both the capacitive double layer and the charge transfer effects will use a common potential, Udl, in this document. This assumption is reasonable for many chemistries, and it allows for the convenient reduction of the equivalent electrical model in section 1.1.8. The double layer potential, Udl, will be used in place of other specialized potentials which are defined with different reference electrodes, resulting in small modeling errors.
This
document tries to minimize the number of potentials used to sufficiently, and simply describe the main reactions present in aqueous, porous electrode cells. The double layer (charge transfer) voltage-current relationship, Udl-Ict, and the exchange current, Io, are shown graphically in Figure 1.3.a. The resulting terminal voltage vs. charge transfer current plane, Ut vs. Ict , for two normalized bulk SOC levels (high and low Q) are shown in Figure 1.3.b [Schöner 1988]. No differences between the boundary and bulk SOC are assumed in this example, hence the terminal voltage, Ut, is equal to the open circuit potential, Uoc, minus the charge transfer potential drop, Udl.
18 Ut Udl
Uoc+
Ict Io combined anodic and cathodic
a) Udl-Ict plane with exchange current, Io. Figure 1.3
high Q low Q
φ1/2 = 0 V
cathode anode
Ict
Uoc-
b) Ut, vs. Ict, with respect to φ1/2.
Double layer and terminal voltage planes vs. charge transfer current.
The Butler-Volmer relationship of (1.3) is often simplified and rearranged to describe the double layer potential, Udl , as a function of the charge transfer (Faradaic) current, Ict , as shown in (1.5). This potential is graphically shown in Figure 1.3.a for a reaction with balanced charge and discharge transfer coefficients (αchrg = αdis = ½). This assumption makes the example simple, but leads to errors when predicting the double layer potentials in practical cells. Udl =
RT |Ict| ln α ne F |Io|
[V]
(1.5)
where |Ict| >> Io, valid only for large reaction rates Equations (1.3) and (1.4) assume fixed values for α, ksh and Io. The exchange current constant, Io, and the standard heterogeneous rate constant, ksh, should be modified for (1.3) and (1.4) to be valid over a wide range of charge transfer potentials. The new “apparent” terms are a function of Udl, and the charge number of the oxidized species. Often these nonlinearity's are neglected in order to simplify the modeling of the cell, but this results in modeling errors [Linden 1987].
19 (α ne-zo )F Udl ksh app = ksh exp RT (α ne-zo )F Udl Io app = Io exp RT where zo
charge number of the oxidized species (zo - ne = zr)
m sec
(1.6)
[A]
(1.7)
[-]
The corrections to the exchange current and rate constant are not insignificant. These phenomena do vary with the boundary SOC, and the reaction direction as the role of oxidized and reduced species switch, thus showing how complex it is to model the charge transfer potential. The charge transfer descriptions should utilize the acid concentrations at the boundary of the electrode-electrolyte, and the above figures assumed that the boundary and bulk SOC's were identical such that there was no ambiguity in the description of the potentials. Unfortunately, gradients in the electrolyte do exist and need to be described in order to ascertain the effect on the relations governing the main reactions in the cell. The following section addresses the modeling of electrolyte charge gradients. 1.1.4 Diffusion Effects Electrolyte concentration gradients exist in the cell due to the history of the applied current, and spatial anomalies (pores, separators, and electrolyte carriers). The transport of the electrolyte reactants to the electrodes occurs via three mechanisms; electrical migration due to an electric potential gradient diffusion due to a concentration gradient. convection, due to the active stirring of the electrolyte The first two methods of transport work simultaneously with each other, and are
20 inseparable. The third method, which actively encourages uniform concentrations, is not typically applied to common battery systems, although some large systems do use this method in flooded cells to improve voltage stiffness performance. Only the combined effects from electrical migration and diffusion will be addressed in this document. The transport of electrolyte from the bulk solution to the electrode boundary via diffusion and electrical migration is the focus of modeling and cell performance improvement. An example of the diffusion effect for the application of a constant load (discharge) current is shown in Figure 1.4, both the normalized electrolyte concentration as a function of distance from the electrode, and terminal voltage as a function of time are shown. The concentration of the electrolyte near the boundary of the electrode reduces rapidly and the distribution into the electrolyte has a characteristic diffusion distance.
This Nernst
diffusion distance, δ, elongates with the square root of time for constant currents and is defined in (1.8) [Bode 1977, Gileadi 1993]. The diffusion transition time, τ, is the length of
Qbulk
Ut
to
Uoc(Qbulk) Voltage
Norm. SOC
time necessary to deplete the boundary SOC for a given applied current.
t1 τ 0 0
Qboundary 25), charge equalization between battery cells (or modules) is desirable to maintain the battery over long periods of time [Kutkut 1998]. Charge equalization may be implemented after imbalance is detected, or simply on a regular basis. Many equalizers use a simple charge shunting method or a non dissipative charge shuffler.
The BMS may or may not be active in the actual charge
equalization process, but it should be able to warn the user of imbalances, faulty cells or modules. 1.3.3 Age Determination and Maintenance Due to the critical nature of many applications with battery back-up systems, maintenance of the battery is vital to high power availability and reliability. Presently, few BMS's have an integrated age or cell diagnostic function. These functions are traditionally reserved for specialized instrumentation [Markle 1992, Kanya 1994]. The instrumentation usually utilizes either injected or natural system noise during live operation of the battery to determine the relative age via AC impedance information [Kato 1995, Robinson 1994]. Most researchers have focused on measuring the electrolyte resistance to indicate relative capacity,
48 and age of the cells or modules. The general consensus for identifying aged modules or poor interconnections is to allow for a 20% increase (from the cell's historical average) in the electrolyte resistance before considering the module faulty or imbalanced. Other pitfalls of the diagnostic method are; the repeatability of measurements, ability to probe individual live modules, accounting for temperature variation in data sets [Hawkins 1994, Heron 1994]. The pitfalls of applying the specialized instruments for purposes of maintenance, can be avoided with data from built in BMU's and via a BMS with impedance analysis capabilities. 1.3.4 Summary of Battery Management Systems Complex battery systems require a multi-function BMS to insure, power availability, quick recharge, efficient operation, and high reliability through age determination of the battery's cells. The BMS is strategically positioned in the system, and can incorporate the proposed advanced state estimation and power capability prediction routines. The power prediction techniques will make on-line impedance parameter estimates available for the diagnostic, and age determination functions, thus furthering the value of the proposed algorithms.
1.4 Application Specific Simulation of Cells For many applications it is vital to simulate battery performance in order to properly evaluate and design the system. A review of simulation methods and modeling techniques is presented in this section.
Two general methods for the simulation of cells are
phenomenological and physical law modeling ("black" and "white-box" modeling). A third
49 approach is to use several physical law relations to help describe the elements of a phenomenological model. This results in a "gray-box" model, which is often very successful in describing and simulating the desired attributes quickly and accurately. 1.4.1 Phenomenological Modeling Common phenomenological models use a simple set of parameters to describe specific cell characteristics over a limited range of operating conditions.
The Peukert
modified charge integrator of (1.27) was one such model. Often, these models can not be extended to describe more complex cell behavior.
Phenomenological models also lack
physical insight into the mechanisms present in the cells, and do not provide a means of design improvement, as do the physical law models. Many phenomenological simulations borrow heavily from Peukert's [1897] modified charge relation. Shepherd [1965] included Peukert's capacity relation into a terminal voltage prediction equation for constant current charge and discharge conditions as described in (1.31).
The Shepherd relations for terminal voltage used either logarithmic or linear
descriptions of the potential drops as a function of current to describe the trajectories of Figure 1.12. Qn Ut = Uoc - It Rs - kp Qn - ∫ I dt
[V]
(1.31)
where Rs kp Qn
average series resistance parameter (from empirical data) [Ω] linear polarization parameter [V] rated charge capacity of cell (at some nominal rate) [Ah]
This basic terminal voltage modeling technique was extended by Copetti [1994], and Duval [1995] to include temperature effects for photovoltaic and automotive battery
50 applications. Moore [1996] added dynamic (double layer time constant) effects to the model, and provided 2-dimensional maps to more accurately describe the Rs parameter for dynamic HEV simulations. The Shepherd based simulation results are often adequate (~5% error) over short periods, but the parameter sets are abstract and have not been extended to include mass transport and charge-discharge disparities in the battery. 1.4.2 Physical Law Modeling Physical law modeling requires an in-depth knowledge of electrochemistry, but it provides an excellent means of simulating cell performances with respect to actual design parameters and construction techniques. Newman and Tiedeman [1975] outlined the basis for porous electrode mathematical modeling.
The 1-dimensional (1-D), macroscopic
description of the reactions tried to model unbalanced transfer coefficients (α variances, αchrg ≠ αdis) and non-uniform reaction rates. The differential equations needed to solve for the reaction potentials were complex. Gu et. al. [1987] further extended the 1-D model to include charging currents, resting potentials and diffusion effects for a single transfer coefficient (α=0.5).
Temperature sensitivity was also added to several of the main
parameters. This model provided 1-D descriptions of the acid concentrations for a wide range of applied currents and temperature. A simpler version of the Gu model was presented by Karden [1997]. This model did not include double layer or diffusion gradient effects, and hence provided poor dynamic simulation capabilities, but still offered a sound simulation basis.
51 1.4.3 Gray Box Modeling The combination of the two above modeling techniques forms a "gray-box" type model which can vary in complexity, and flexibility, depending upon the simulation application. An early electrical equivalent model by Appelbaum [1982] used a RandlesErshler type RC network to describe the terminal voltage characteristics of the battery. Several parameter sets (RC values) were used to extend the performance of the simulations to a wide range of SOC values and charging conditions. Others have presented similar work with similar curve fitted RC values [Owen 1991, Salameh 1992]. Often the fitted parameters follow arbitrary and convenient mathematical descriptions, rather than using the physical law expressions. A noteworthy gray box modeling technique was undertaken by Mauracher and Karden [1996]. The study introduced non-linear electrical circuit elements for which the mathematical forms were dictated by electrochemical physical laws. The coefficients of the equations were then constant over a particular range of SOC and temperature. Impedance spectroscopy was used to extract the small signal characteristics of a cell as a function of discharge current level. The model successfully simulated electrolyte, double layer, and diffusion effects to produce 99.5% accurate terminal voltage predictions for dynamic EV driving cycles. The modeling technique was not extended to wide ranges of SOC, mass transport effects, or temperature, hence accuracy degraded at extremes. 1.4.4 Summary of Application Specific Simulation of Cells Different simulation techniques were discussed. The most accurate technique, which uses physically meaningful descriptions of an equivalent electrical model, was able to
52 describe battery behavior over a wide range of operation. This modeling technique will serve as the foundation in the proposed battery state control, and power capability prediction algorithms. The previous sections discussed the modeling, performance evaluation and simulation of cells. The next section reviews the methods presently used in actively controlling the battery state for power buffering applications.
1.5 Battery State Control Battery state control techniques for load-leveling applications depend upon the requirements of the specific application. The applications which have low battery state controllability (Table I.1) have many constraints which dictate how the battery is operated. The state regulation techniques for these applications are discussed in Appendix A.5. Those applications which have high controllability of the battery state can actively drive the battery towards a more desirable state. The control methods are discussed below. 1.5.1 SOC Control in Charge Sustaining Hybrid Electric Vehicles Many of the present battery state regulators for HEV's borrow heavily from EV type battery monitors. Charge Depleting HEV's (CD-HEV's) have less stringent battery state regulation requirements than Charge Sustaining HEV's (CS-HEV's), and often directly use EV battery state estimation equipment. Development of EV's preceded that of CS-HEV's due to pressure from government agencies to produce "zero-emission" vehicles.
These EV
battery state estimators are focused on predicting end-of-use. These effective remaining charge monitors have been adapted to CS-HEV's via different current compensation gains
53 and several proprietary SOC estimate reset mechanisms [Hughes 1997, Visteon 1997, Panasonic 1997, Hiramatsu 1998]. A block diagram of a typical charge estimate control technique and the actual charge state vs. time for a 1999 CS-HEV is shown in Figure 1.19
Qo
∫
It
^ Q Q*
Temp.
APU control
reset mech.
Ut
a) Charge control block diagram
b) Actual charge vs. time [Hiramatsu 1998]
Figure 1.19 CS-HEV battery state control based on a modified current integrator with SOC estimate reset mechanism and reference SOC value, Q*. ^ where Q Q*
estimate of normalized bulk SOC, f(|It| and T) target Q value during charge sustaining operation
[-] [-]
The SOC of the EV or HEV battery is predominantly estimated via integration of the cell's current. Other methods of SOC estimation (measurement of electrolyte specific gravity or correlation of SOC to impedance characteristics) are generally not attempted in vehicle applications which used inexpensive, sealed battery modules. The integration of the gain ^ . These errors may compensated current tends to result in drift errors in the SOC estimate, Q come from parasitic gassing current effects, current measurement errors, integrator errors, or poor compensation gains. Several proprietary error reset mechanisms are in use to assist the CS-HEV battery state controller performing well over a wide range of operating conditions.
54 APU Operation The battery state is driven towards higher SOC values by initiating the APU, or by increasing the power output relative to the load's power demand. Generally the battery state controller tries to maintain the SOC estimate close to a target value, Q*, or within a recommended SOC range (Figure 1.19.b) . This target value and deviation are usually predetermined from laboratory testing, and they may be temperature, age, or driving cycle dependent.
The battery state control loop may be slow, or fast, depending upon the
configuration, ratings and characteristics of the battery and APU in the CS-HEV. The previously discussed PNGV battery testing extracts the relative power capability performance estimates into SOC dependent quantities. This information then caters to and perpetuates the use of SOC as a control variable for CS-HEV batteries. The PNGV testing adequately predicts power capability in the absence of diffusion effects, but it does not guarantee performance as the battery ages. The control of a battery via SOC is an indirect method for controlling the battery's power capability. Aging, temperature, imbalances in the battery string, and diffusion effects will significantly effect the performance of SOC controlled batteries. 1.5.2 Summary of Battery State Control There is a need to advance the present state of the art in CS-HEV battery state regulation which should more directly measure and control the terminal voltage variations and power capability of the cell during HEV operation as outlined in the battery specification of Table I.2. To develop state regulation techniques which can directly estimate the power
55 capability of the battery, the parameters of the battery impedance model must be estimated on-line. The next section reviews methods of recursive parameter estimation for use in online discrete control systems.
1.6 Recursive Parameter Estimation This section introduces the elements used in recursive Output Error model parameter estimation techniques.
The objective is to review the technique and some of the
simplifications as well as discuss potential pitfalls of applying the technique to a non-linear battery model. The proposed power capability prediction algorithms require knowledge of the impedance model. Once the model parameters are estimated, the efficiency, power capability, and aging/diagnostic phenomena of the cell can be exposed. The gassing current, Ig, and Nernst open circuit voltage, Uoc, are assumed to be age independent in this study, and are modeled once via an a priori test sequence, leaving only the impedances of the RandlesErshler model to be estimated via on-line data. 1.6.1 Model Set Selection The field of system identification is used in determining the structure and order of the system model. In this document, the model structure and order is given by the RandlesErshler impedance model. The exact order can be user defined, because of the ambiguity of how many RC-circuits are needed to adequately model the complex diffusion impedance. The focus of the remainder of this chapter is on the parameter estimation of the given impedance model, from regularly sampled, input-output data. This chapter will also be limited to recursive identification methods and Output Error (OE) modeling, due to the
56 interest in implementing the algorithms on finite memory computers, and the modeling the battery system rather than the noise model. Bayesian or statistical methods for recursive parameter estimation [Kulhavý 1997] and neural nets [Chen & Billings 1990] can also be used to estimate non-linear model parameters. This document will focus on the general methods of OE models for discrete sampled data. Appropriate application of the OE parameter estimation techniques has given good results for other non-linear systems [Ljung 1987]. Nomenclature The basic nomenclature used to describe the discrete modeling methods is discussed. The traditional variable "k" represents the discrete sample index, Ts is the sampling period, fs and ωs are the sampling frequencies [Hz and rad/sec], q is the shift operator, and θ is the set of parameters.. The nomenclature for describing variables as a function of k, q, or θ are described below. (k) - being a function of the sampled data index (or time). [k] - specific value at a particular sampled data index (or time). (q) - being a function of the shift operator, indicates the function or filter has a finite set of distinct coefficients relative to the present sample data point. [ q·a(k) = a(k+1) and more typically, q-n·a(k) = a(k-n)] (θ) - being a function of the parameter set, θ, indicates that the variable's values are sensitive to the values of the actual model's parameters or of the algorithm's estimated parameters. ^ ) = estimated or An estimated variable is shown by a circumflex, or "hat," e.g. ^y(k,θ ^. predicted output, ^y, as a function of the sampled data index, k, and estimated parameter set, θ
57 1.6.2 Black Box Model Structures General discrete parameter estimation routines assume discrete sampled data at regular intervals. A block diagram of the black box model structure set is shown in Figure 1.20. The input, u(k), and the external noise, e(k), are inputs to the plant and error models. The output, y(k), is effected by both systems. e(k) H(q,θ) υ(k) +
u(k) G(q,θ)
y(k)
+
Figure 1.20 Standard discrete black-box model set for both plant and output noise dynamics. The output, y(k), as a function of the inputs, u(k) and e(k), requires knowledge of the plant and noise transfer functions, G(q,θ) and H(q,θ), as shown in (1.32) [Ljung 1987]. Prediction of the output with out knowledge of the error input is shown in (1.33). y(k) = G(q,θ)u(k) + [H(q,θ)-1]e(k)
[-]
(1.32)
^y(k) = H-1(q,θ)G(q,θ)u(k) + [1-H-1(q,θ)]y(k)
[-]
(1.33)
Note: H-1(0,θ) = 1, thus reducing the predicted output to only a function of previous inputs. The objective of parameter estimation is to tune the model parameters based on the minimization of the predicted output error, ε(k), as defined in (1.34). Note that only previous inputs and outputs need to be measured. ε(k) = y(k) - ^y(k) = H-1(q,θ) [y(k) - G(q,θ) u(k)]
[-]
(1.34)
58 Model and Data Constraints Constraints on the model and on the data exist in order to achieve an asymptotic convergence of the estimated parameters towards a minimum output prediction error [Ljung 1987]. The output of the model must be differentiable with respect to each of its parameters, and the data must obey the following rules as the sample size becomes large. The injected noise, e(k), must be independent of the input excitation, u(k). The injected noise, e(k), must have zero mean. The input, u(k), must has sufficient variance to excite the modeled dynamics, G(q,θ), and/or the input must have at least N appropriately spaced non-zero frequency components which excite the N, modeled parameters in the system. Modeling Forms Many modeling forms exist, each one characterizing the plant and noise systems differently. Single Input Single Output (SISO) black box modeling techniques use rational functions whose polynomial coefficients become the model parameter set. The general polynomial model form is shown in (1.35) and the polynomials are defined below in (1.36)(1.40). c(q) b(q) a(q) y(k) = f(q) u(k) + d(q) e(k) where a(q) = 1+ a1q-1 + ⋅⋅⋅ + anaq-na b(q) = b0 + b1q-1 + ⋅⋅⋅ + bnbq-nb c(q) = 1+ c1q-1 + ⋅⋅⋅ + cncq-nc d(q) = 1+ d1q-1 + ⋅⋅⋅ + dndq-nd f(q) = 1+ f1q-1 + ⋅⋅⋅ + fnfq-nf
[-]
(1.35)
[na] [nb+1] [nc] [nd] [nf]
(1.36) (1.37) (1.38) (1.39) (1.40)
Note that the input polynomial b(q) includes a constant term, b0, for the most recent sampled data, u[k]. The order of the polynomials (dimension of the vectors) are shown in
59 brackets in place of the units. The resulting parameter vector for the general polynomial model of (1.35) is given in (1.41). The length of this parameter vector is nθ, the total number of parameters to be describe the dynamics of the plant and error models. θ = [a1, ..., ana, b0, ..., bnb, c1, ..., cnc, d1, ..., dnd, f1, ..., fnf]T
[nθx1]
(1.41)
Common black box models which describe different aspects of the plant and error systems are listed in Table 1.1 [Box 1978, Ljung 1987, Gorter 1997]. This research study will limit itself to OE modeling due to it's applicability to linear and non-linear, real time systems with unknown noise models. Table 1.1
Common black box models which describe plant and error systems. Model
Polynomials Used
FIR (Finite Impulse Response)
b(q)
ARX (Auto Regres. w/ eXogenous input)
a(q), b(q)
ARMAX (ARX w/ Moving Avg. )
a(q), b(q), c(q)
OE (Output Error)
b(q), f(q)
BJ (Box-Jenkins)
b(q), c(q), d(q), f(q)
Output Error Models The OE model structure is reviewed, and two discrete modeling methods are given. The OE model estimation block diagram is shown in Figure 1.21. The b and f polynomials are used to estimate the output of the physical plant. The estimation of the output by the OE model is given in (1.42) in both general and specific discrete sample forms, where φ(k,θ) is the regressor vector of lagged input/output data as defined in (1.43).
60 υ(t) u(t)
+
G(s,θ)
Plant
y(t)
+
Ts Output Error Model
u(k)
^ b(q,θ) ^ f(q,θ)
^ ^y(k,θ)
-
+
y(k) ^ ε(k,θ)
Figure 1.21 Discrete OE model structure which utilizes an estimate of the plant model to predict the output. ^ ^y(k,θ ^ ) = b(q,θ) u(k) ^) f(q,θ
[-]
(1.42.a)
^y = b^ ⋅u + ... + b^ ^ ^ ^ ^ [k] 0[k] [k] nb[k]⋅u[k-nb] - f 1[k]⋅y[k-1] - ... - f nf [k]⋅y[k-nf]
[-]
(1.42.b)
^y = φ θ ^ [k] [k] [k]
[-]
(1.42.c)
φ[k] = [u[k],...,u[k-nb],-y^[k-1],...,-y^[k-nf]]
[1 x nθ]
(1.43)
where [k-n] indicates the value of the discrete variable at the nth previously sampled point. ^ θ estimate of the plant's polynomial coefficients, [b0, ..., bnb, f1, ..., fnf]T . φ regressor vector of past inputs and predicted outputs. More specifically, for OE prediction methods, the regressor vector, φ, uses the estimate of the output, ^y(k), and not the measured output, y(k) as in ARX, ARMAX and BJ model structures. The definition of the prediction error is shown in both general and sample specific discrete forms for OE models in (1.44). ^ ) = y(k) - ^y(k,θ ^ ) = y(k) - φ(k,θ ^) θ ^ ε(k,θ [k-1] [k-1] [k-1]
[-]
(1.44.a)
^ ε[k] = y[k] - ^y[k]= y[k] - φ[k] θ [k-1]
[-]
(1.44.b)
^ The predicted error at sample, k, uses the previous sample's parameter estimate,θ [k-1], and the most recent regressor vector, φ[k]. This slight lag in the updating of the error is a
61 characteristic of recursive routines. The parameters will still asymptotically approach stable values for long data sets (k → ∞) [Ljung 1981, 1987]. 1.6.3 Recursive Least Squares and Gauss-Newton Algorithms Updating the parameter estimate is generally addressed with two error minimization routines; the Recursive Least-Squares (RLS) and Recursive Gauss-Newton (RGN) algorithms.
These can be successfully used to converge the prediction errors towards
minimal values. This document will explore both methods for flexibility issues. The RLS method is simplest to implement, while the RGN method is more applicable to direct estimation techniques of the system's physical system parameters, rather than the discrete parameters. Recursive Least Squares The celebrated RLS methods is based on the minimization of the square of the error by the incremental update of the parameter set based on the relative sensitivity of the error to each parameter. The incremental update can either be based on fixed step sizes, µ, as shown in Figure 1.22, or by an adaptive step value which speeds convergence upon minimum prediction error. The update of the parameter estimates, and the adaptive step size matrix P (diagonal matrix) are defined below for the RLS algorithm as outlined in (1.45) and (1.46).[Ljung 1987]
62 ε2 δε2 δθ
µ
δε2 δθ
θ
Figure 1.22 Minimization of prediction error via a corrective step in the parameter based on the minimization of the error squared. P[k-1] φ[k]T φ[k] P[k-1] P [k] = P [k-1] 1+φ[k] P[k-1]φ[k]T
[nθ x nθ]
(1.45.a)
φmaxT φmax
[nθ x nθ]
(1.45.b)
δε ^ ^ =θ ^ θ = θ[k-1] - P[k] φ[k]T ε[k] [k] [k-1] - P[k] ^ δθ
[nθx1]
P max = µ max «
2
2
(1.46)
Typically, the diagonal values (eigen values) of P are artificially limited to only vary over a 10:1 range [Parkum 1992]. Large step sizes offer quick response, but greater noise sensitivity. If the input excitation of a parameter approaches zero, the parameter's eigen values in P tends toward infinity. Hence the user must supply a set of limits to each eigen value of P. This is often referred to as selective forgetting and many specialized methods for the step size selection exist [Box 1978, Ljung 1987] Recursive Gauss-Newton Algorithm A different algorithm, but with similar performance in the discrete domain, is the RGN gradient descent algorithm. The sensitivity of the predicted output, ^y, to the model parameters is equivalent to the negative sensitivity of the prediction error, ε. This sensitivity is used by the RGN algorithm in order to quickly converge upon minimum error. The
63 definition of the prediction gradient, ψ, for output error models is defined below in both general and specific sample discrete forms in (1.47) ^) δ ^y δε φ(k,θ ψ(k,θ) = (k,θ) = (k,θ) = ^f (q) δθ δθ
[1 x nθ]
(1.47.a)
φ[k] ψ[k] = ^ f [k-1]
[1 x nθ]
(1.47.b)
P[k-1] ψ[k]T ψ[k] P[k-1] 1 + ψ[k] P[k-1] ψ[k]T
[nθ x nθ]
(1.48)
φ[k]T T ^θ = θ ^ ^ ε[k] [k] [k-1] - P[k] ψ[k] ε[k] = θ[k-1] - P[k] ^f [k-1]
[nθx1]
(1.49)
P [k] = P [k-1] -
1.6.4 Indirect Estimation of Model Parameters The estimation of a system's model parameters in the discrete domain must be done with care to insure good prediction results. The Eulerian based discrete domain parameter ^ , is generated with the assumption that the input (and output) of the system behave with set, θ d discrete steps. This Zero Order Hold (ZOH) assumption is obviously false for the continuous time system, so inaccuracies in the parameter set estimation is inevitable. The error does reduce with increased sampling rates, but care must be used when selecting too high a sampling rate in order to avoid loss of parameter resolution [Borgard 1994]. The trapezoidal rule, or Tustin model assumes a ramp function between samples and generally gives better prediction results for the same sample frequency. Additionally, to convert the discrete parameter set, θd, to the physical domain, a transformation is required. This transformation assumes that the physical parameter set, θp, be differentiable over the operating space, and that it's dimension be equal to that of the
64 discrete parameter set. Example RC System Figure 1.22 shows an example RC electrical circuit (physical system) and the nomenclature for the OE modeled system. The input-output variables, (u, y), are regularly sampled and an estimate of the discrete domain parameters, (f^, b^), are made. These are then ^ ,C ^ ). used to find an estimate of the physical system's parameters (R υ
U I
+
C
u Ts
discrete est.
R u=I y=U
f^ ^ θd = ^ b
y
RC
^ R ^ θp = ^ C
a) input-output descriptors for an RC system.
f -1
^ θd ^ θp
b) indirect parameter estimation method.
Figure 1.22. Indirect estimation of physical parameters from discrete data. Two common ways to model the example R-C system in the discrete domain are given in the Table 1.2. The traditional Eulerian model does not offer the accuracy of the trapezoidal rule, Tustin model when relatively slow sampling is used, hence this document will use the Tustin model. Tustin Model The Tustin model of the first order R-C system traditionally incorporates two b(q) coefficients, but the resulting model would then have three coefficients, (b0, b1, f1), to the physical system's two, (R, C). To insure one-to-one mapping between the discrete and physical parameter spaces, the order of the Tustin model is reduced by combining the two b(q) coefficients into a single term, (b0) . The resulting prediction of the output in the
65 reduced Tustin model is shown in (1.50). Table 1.2.
Discrete and physical parameter transformations for Eulerian, and Tustin models of a first order, linear, parallel arranged R-C system.
Parameter [units]
Eulerian Model
Reduced Tustin Model
y[k] [V]
-f1 y[k-1] + b1 u[k-1]
-f1 y[k-1] + b0 (u[k] + u[k-1])
f1 [-]
-exp RC
b1 & b0 [Ω]
R(1-exp RC)
T sR Ts+2RC
R [Ω]
b1 1+f1
2 b0 1+f1
C [F]
-Ts (1+f1) b1 ln(-f1)
Ts 1-f1 4 b0
τ [sec]
-Ts ln(-f1)
Ts 1-f1 2 1+f1
-Ts
-Ts
^y = -f^ ^ [k] 1 [k-1] y[k-1] + b0 [k-1] (u[k] + u[k-1])
Ts-2RC Ts+2RC
[V]
(1.50)
1.6.5 Direct Estimation of Physical Model Parameters Methods exist for the direct estimation of the physical parameters without the use of a model transformation. The method is based on Recursive Gauss Newton algorithms and integration routines.
Direct estimation techniques require a computationally expensive
integration step, but the parameter estimate, θp, will be free of ZOH model deficiencies as found in the discrete time models.[Gorter 1997] 1.6.6 Non-linear Modeling Several basic non-linear modeling techniques have been developed.
The basic
approach is to rewrite the regressor vector, φ, with the non-linear description of the input/output data, and the parameter set, θ, then contains the coefficients of the non-linear
66 relations. This method will be explored further in Proposed Research chapter (Chapter 6). The basic premise will be to model the quickly modulating, nonlinear components of the battery's impedance, while the slow nonlinear effects will be treated as time variance tracking in the recursive algorithms.
1.7 Summary A review of the state of the art is given and several key areas are outlined as areas for research opportunities. 1.7.1 State of the Art Review Electrochemical laws describing the main reactions in porous electrode, aqueous cells were reviewed. An equivalent electrical model for these cells based on physical laws was outlined. This Randles-Ershler impedance model, when fitted with physical law based nonlinear descriptions, gives very accurate simulation and performance prediction results, thus indicating that it is an appropriate structure for modeling demanding load-leveling applications. The conventional methods for battery performance evaluation were reviewed. Simple off-line methods for power and efficiency performance evaluation reported results as a function of bulk SOC.
The bulk SOC descriptor of the battery was shown to be the
traditional control variable for battery state control, even though boundary SOC is more directly responsible for defining the power and efficiency performance. The many functions found in battery management systems were reviewed. reviewed functions do not directly address power capability nor cell aging.
The
67 Recursive parameter estimation for Output Error prediction models was reviewed. The methods were shown to estimate parameters of linear, uniformly sampled systems. Parameters of nonlinear systems can be estimated using these methods if they can be decomposed into nonlinear basis functions with coefficients which may be slowly time varying. 1.7.2 Areas of Opportunity The assessment of the state of the art reveals several areas of opportunity which should be addressed in order to advance the field of battery state control for load leveling applications. The areas of opportunity which this document addresses are categorized below; Modeling and Simulation •
Introduce models of porous electrode, aqueous cells which are appropriate for the fundamental analysis of battery state regulation methods and power capability prediction algorithms.
•
Develop non-linear description of the cells main reactions. The model must be valid over a wide range of operating conditions which include high charge and discharge currents, diffusion, and mass transfer limitations.
State Control •
Develop battery state control techniques for charge sustaining load-leveling applications which do not require complicated reset mechanisms. High cycle efficiency and good power capability performance are secondary goals of the new control methods.
•
Develop battery state control techniques which more directly regulate power cycle efficiency and power capability over a wide range of operating conditions.
68 On-line Power Capability Prediction •
Develop a non-linear double layer, and diffusion impedance models which account for mass transport limitations and arbitrary current histories.
•
Apply recursive parameter estimation techniques to a battery which is excited only by the load-leveling power flow and maintain good estimates of the mass transport limit.
•
Include diffusion and mass transport effects into the off-line PNGV power prediction model to make it applicable for on-line performance evaluation.
•
Evaluate on-line cell impedance estimates for power capability prediction, efficiency optimization, and cell aging diagnostics.
On-line Battery Efficiency Maximization •
Evaluate the effective power cycle efficiency of the battery based on the previous trend in power processing.
•
Modify the battery state controller to drive the effective cycle efficiency towards higher levels.
69
Chapter 2, Battery State Control via Terminal Voltage 2.1 Battery State Descriptors..............................................................................................70 2.1.1 Primary Descriptors...........................................................................................71 2.1.2 Secondary Descriptors.......................................................................................72 2.2 Voltage Control Analysis.............................................................................................72 2.2.1 Example Application Framework......................................................................73 2.2.2 Steady State Analysis ........................................................................................75 SOC Behavior Under Terminal Voltage Control.............................................77 Trajectory on the Q-Ut Plane ...........................................................................78 2.2.3 Approximate Model and Solution .....................................................................80 2.2.4 Effects of Terminal Voltage Control Limit Variances ......................................81 ~ Effects of Voltage Limit Shifts on Qavg ...........................................................82 ~ Effect of Voltage Limit Compression on ∆Q...................................................83 Effect of Voltage Limit Compression on APU Operating Period....................84 2.2.5 Numerical Evaluation of the Non-linear Model................................................85 SOC Behavior under Terminal Voltage Control..............................................87 2.3 Constant Load Experiments.........................................................................................91 2.3.1 Experimental Set-up ..........................................................................................92 2.3.2 State of Charge Control .....................................................................................94 2.3.3 State of Voltage Control....................................................................................96 Efficiency Evaluation.......................................................................................99 2.4 Voltage Control for Dynamic Loading ......................................................................100 2.4.1 2-D Voltage Control ........................................................................................101 LPF Corner Frequency ...................................................................................103 Trend and Dynamic Control Limits ...............................................................104 2.5 Dynamic Load Experiments.......................................................................................105 2.5.1 Experimental Set-up ........................................................................................105 2.5.2 State of Charge Control ...................................................................................106 2.5.3 Voltage Control ...............................................................................................108 2.5.4 Cold Temperature Operation...........................................................................110 2.6 Properties of the Voltage Control Technique ............................................................111 2.7 Summary ....................................................................................................................112 2.7.1 Charge Control ................................................................................................114 2.7.2 Voltage Control ...............................................................................................114
70 This chapter discusses new work in charge sustaining control of power buffering batteries. Battery state description categories are introduced. These are based on the relative importance of the variable to the battery specification for power buffering applications (see Table I.2). These categories challenge state controllers to utilize different descriptors as control variables. A method of state control based on terminal voltage is presented and analyzed within the framework of a simplified power flow model and an equivalent short term cell model. The sensitivity of many operating parameters to the terminal voltage control method is studied. A numerical analysis of the more complex non-linear short term model description is also applied to better compare the efficiency performances of charge and voltage based control techniques. The terminal voltage control technique is extended to dynamic loading conditions as experienced in CS-HEV's. Experimental comparisons of charge and voltage control techniques are provided for three battery chemistries.
2.1 Battery State Descriptors This section categorizes battery state descriptors based on importance to the Load Leveling Device (LLD) specification. An example LLD specification was given in the Introduction for a CS-HEV (Table I.2). The different ways to describe the battery are grouped according to the nature of the descriptions; some are slowly changing, accumulative descriptors, and others are quickly changing, excitation dependent, descriptors. A battery's “state” is often, and traditionally, described by the relative amount of active reaction species in the electrolyte (SOC). Other ways to describe the battery are via; the amount of stored energy, or "state-of-energy" (SOE)
71 internal temperature distribution terminal voltage behavior bi-directional short term power capability efficiency characteristic vs. loading internal gas pressure, etc.. These state descriptions can be divided into two groups based on the importance of the descriptor to the function of the electrochemical cell as an LLD. A proposed formulation of these state descriptors is summarized in Table 2.1. Table 2.1
Proposed descriptors for batteries in LLD applications. Primary Descriptors
Secondary Descriptors
Power Capability
Stored Charge (SOC)
Efficiency Characterisitc
Stored Energy (SOE)
Terminal Voltage Behavior
Temperature Distribution
2.1.1 Primary Descriptors Real-time control of LLD’s require knowledge about the power processing capability and power efficiency performance. Indirectly, terminal voltage variation under load gives power capability and efficiency information, as determined by the I⋅R voltage drop. Efficiency, power capability, and terminal voltage are fundamental to LLD operation as seen in the basic specification for the battery in a CS-HEV. These descriptors are classified as Primary Descriptors of the battery. These state descriptors are highly dependent upon the series impedance of the cell and must continually comply with the LLD specification. If these customer tangible characteristics do not comply with the specifications, then the LLD (battery) fails in it's objectives.
72 2.1.2 Secondary Descriptors Battery properties of less concern to the overall system (the LLD specification) are those that are not outwardly tangible; e.g. the actual stored charge, stored energy, and temperature distribution. As long as the battery is performing well with respect to the immediate needs of the specification, stored charge, stored energy and cell temperature are free to slowly change. These battery state descriptors are defined as Secondary States for LLD devices. These descriptors influence the series impedance of the battery, hence also the primary descriptors, and will be loosely bounded by a primary descriptor control scheme. It is interesting to note the above LLD descriptor groupings also apply to the batteries of EV’s and other charge depleting applications. If the battery can not meet the short term power demands (or voltage limits) set forth by the application, then the secondary goal of predicting effective stored charge is irrelevant. End-of-use is defined as the point where the battery can no longer meet the load's power and voltage requirements. It is the goal of many battery monitors to predict when a primary descriptor of the battery falters (terminal voltage), by monitoring an empirically modified secondary descriptor (Peukert modified SOC).
2.2 Voltage Control Analysis Control of the battery via terminal voltage is analyzed for a simple hysteresis controlled APU example (CS-HEV as outlined in Appendix A.3) . First the framework of the example is defined and the characteristics of the proposed cell model are explored. Next, the resistance description is simplified to obtain a closed form solution for the battery current and SOC variables as a function of loading. The effect of different voltage control limits on the
73 approximate model is then explored. Finally a numerical analysis of the more complete nonlinear resistance description is presented to accurately compare the efficiency performances for both charge and terminal voltage control techniques. 2.2.1 Example Application Framework The presented analysis of terminal voltage control uses the equivalent short term cell model of Figure 1.14 (PNGV model).
This model includes an open circuit voltage, a
direction sensitive equivalent series resistance, but no diffusion effects.
Although the
excitation of the battery typically extends into the diffusion region, beyond the model's valid range, it still represents a good place to begin the analysis of control techniques. The following analysis will ignore all diffusion effects to avoid temporal variations in the impedances. The SOV analysis is applied to an APU which features a constant output power during operation. The APU is cycled on and off by a hysteresis band regulator applied to either the charge content or the terminal voltage. An example APU power trace and resulting Q and Ut traces are shown in Figure 2.1. In this example, either the charge content, or terminal voltage could be used as a control variable. The control of the APU is assumed to be based on a hysteresis band regulator with a single SOC target, Q*, or a set of voltage limits, (Ulo, Uhi) as shown in Figure 2.2.
74 Papu
Pchrg
Pveh Pdis (1-d)Tapu
dTapu
Qhi
Q
Q* ∆Q
Qlo Uhi
Ut
Ulo
Uoc time
Figure 2.1
Example battery state regulation traces vs. time with hysteresis controlled APU operation.
APU on
APU on
off
off Qlo
Q*
Qhi
Q
a) Control via SOC target, Q*, or limits. Figure 2.2
Ulo
Uhi
Ut
b) Control via terminal voltage limits.
Two methods of APU hysteresis control for charge sustaining control.
The above example with steady vehicle demand, Pveh, is based on an CS-HEV driving at a steady state speed (see Appendix A.3). The battery (LLD) must always satisfy the power difference between the APU and vehicle demands, hence power flow is the foundation of the control analysis. The vehicle power demand, Pveh, and APU power level, Papu, dictate the charge and discharge power rates applied to the battery, as shown in (2.1)-(2.2). Note that Pchrg is a negative value, as is the case with the charging currents. Pdis = Pveh
(Pdis > 0)
[W]
(2.1)
Pchrg = Pveh - Papu
(Pchrg < 0)
[W]
(2.2)
75 where Papu Pveh
predetermined APU power level, (Papu > Pveh) steady vehicle power demand, (0 < Pveh < Papu)
[W] [W]
The APU power level is assumed to be a predetermined constant value which is higher than the range of load demands. 2.2.2 Steady State Analysis The analysis of battery state control algorithms begins with descriptions of the components in the simplified electrical model with diffusion effects ignored (Figure 1.14.b). Descriptions of the open circuit voltage, Uoc, and the short term series resistance, Rs, are given in (2.3) and (2.4). The open circuit voltage, Uoc, is described here as a linear function of normalized bulk SOC, Q, which is sufficient for typical lead-acid storage systems [Bode 1977]. The mathematical model for the effective high rate series resistance terms, Rs dis and Rs
chrg,
was chosen due to good empirical fit for a typical VRLA systems [Vinod 1994].
Other mathematical relations may be used to describe either Uoc or Rs as functions of Q for different cell technologies. Uoc = Q⋅(Un - Uo) + Uo Rs dis = k1
Rs chrg = k3 where Q k1,3 k2,4
1 Q+k2 1 1+k4-Q
normalized bulk SOC, [0≥Q≥1] offset resistance fitting parameters stored charge fitting parameters
[V]
(2.3)
[Ω]
(2.4.a)
[Ω]
(2.4.a)
[-] [Ω] [-]
The power flow relations of (2.5) are used to determine the battery current, It, as a function of Pveh, and Q. Once the terminal current, It, has been described for different
76 operating conditions, the direction dependent power efficiency of the battery, ηdis & ηchrg, can be described as a function of It using the resistance and open circuit voltage characteristics as shown in (2.6). The power cycle efficiency of the battery, η cycle, is defined as the LLD's net power efficiency for arbitrary steady vehicle loading. Pdis = It⋅Uoc - It2⋅Rs dis
(It > 0)
[W]
(2.5.a)
Pchrg = It⋅Uoc - It2⋅Rs chrg
(It < 0)
[W]
(2.5.b)
(It > 0)
[-]
(2.6.a)
(It < 0)
[-]
(2.6.b)
[-]
(2.7)
I ⋅R
ηdis = 1 - t U s dis oc
I ⋅R
-1
ηchrg = 1 - t Us chrg oc ηcycle = ηdis⋅ηchrg where It
η
battery current, f(Pveh and Q) see inverse of (2.5) power and net cycle efficiency, f(Pveh and Q)
[A] [-]
A description of the battery's efficiency enables the calculation of the APU duty cycle and operating period via the application of both an energy balance and charge balance relation to the ideal storage element, Uoc. The definitions for the variance in normalized stored charge, ∆Q, and the APU operating period, Tapu, are given in (2.8)-(2.11) for the storage element under charge sustaining operation of Figure 2.1.a.
∆Q = Qhi - Qlo ∆Q Tdis = mean(I ) = (1-d)⋅Tapu dis
[Ah] (APU=off)
[h]
(2.8) (2.9.a)
77 ∆Q Tchrg = mean(I
[h]
(2.9.b)
[h]
(2.10)
dapu⋅Tapu⋅Pchrg⋅η chrg = (1-d)⋅Tapu⋅Pdis⋅(η dis)
[Wh]
(2.11)
∆Q the variance in the cell's Q during APU operation Qhi,lo the high and low values of the SOC in Figure 2.1.a It dis,chrg the mean values of currents applied to the cell Tdis period of APU inactivity (off mode) Tchrg period of APU operation (on mode) Tapu net period of APU operation (on and off modes) d duty cycle of APU operation
[Ah] [Ah] [A] [h] [h] [h] [-]
chrg)
= d⋅Tapu
(APU=on)
Tapu = Tchrg + Tdis -1
where
SOC Behavior Under Terminal Voltage Control The above descriptions for the terminal current, efficiency, and APU duty cycle, all are functions of the load power level, Pveh, and the normalized bulk SOC of the battery, Q. This sub-section describes how the terminal voltage control technique naturally varies the Q of the battery when it is operated under different power flow conditions. The resulting Q level is then used to evaluate the battery's efficiency performance over a wide range of load power demands. The terminal voltage equations of the equivalent short term cell model (Figure 1.14.b) are used to determine the Q levels which result for a given load demand. The voltage control limits, (Ulo, Uhi), and the load's power level, Pveh, are the independent variables, and the resulting Q levels, (Qlo, Qhi), are the dependent variables. The relations used to solve for the Q levels are given in (2.12). The resulting average Q is then calculated via the mean value of the high and low Q values. The parentheses show at which value of Q the variables must be evaluated at in order to find a proper solution.
78 Ulo = Uoc(Qlo) - It(Pveh,Qlo)⋅Rs dis(Qlo)
(It > 0)
[V]
(2.12.a)
Uhi = Uoc(Qhi) - It(Pveh,Qhi)⋅Rs chrg(Qhi)
(It < 0)
[V]
(2.12.b)
Qavg = mean(Qhi, Qlo)
[-]
where Qlo Qhi Qavg
(2.13)
normalized SOC when Ut reaches Ulo (It > 0, APU turn on) [-] normalized SOC when Ut reaches Uhi (It < 0, APU turn off) [-] average normalized SOC level (function of Pveh, Uhi, Ulo) [-]
Trajectory on the Q-Ut Plane As the terminal voltage, and SOC vary in time with the control of the APU, a trajectory on the Q-Ut plane occurs. This trajectory is instrumental in visualizing how the terminal voltage control algorithm operates. The charge and voltage trajectories of Figure 2.1 are redrawn with key markers in Figure 2.3. A counter clockwise trajectory results when Q is plotted against Ut as in Figure 2.4. The trapezoidal trajectory is bounded by horizontally by the Ulo and Uhi control limits, where the APU is turned on/off at the A/B points respectively. The curvature in the trajectory is due to the non-linear nature of Rs and current It with respect to Q, hence these operating points are solved via numerical techniques in Section 2.2.3 B
B
Q
Q avg
∆Q
A
Qlo
A
B
Ut A
Qhi
Uhi D
Uoc(Q)
C
Ulo
time
Figure 2.3
Charge and voltage trajectories with event markers (A=APU on, B=APU off, C & D = charge and discharge voltage drops).
79 SOC
D
C
Qhi
high P veh
B
low Pveh
Qavg
Qavg
Qlo
C
SOC
A D Uoc(Q) Ulo
Uhi
Ut
a) Example Q-Ut plane with event markers.
Ulo
Uhi
Ut
b) Three Qavg levels resulting from different Pveh loading.
Figure 2.4. Charge vs. Voltage plane for a cell under terminal voltage control using the equivalent short term cell model (ignoring diffusion effects). The charging over potential, C , is determined from the charging current and the short term effective series resistance, Rs chrg as a function of Q. The discharging voltage droop, D, is determined from the discharge current, and Rs dis as a function of Q.(Equations 2.13-14) "D" = It(Pveh, Qlo)⋅Rs dis(Qlo)
[V]
(2.14.a)
"C" = - It(Pveh, Qhi)⋅Rs chrg(Qhi)
[V]
(2.14.b)
In the example of Figure 2.4.a, the discharge voltage droop, D, is greater than the charging overvoltage, C (thus indicating a low charge to discharge power ratio, assuming Rs dis
≈ Rs chrg). Figure 2.4.b shows how varying the loading (charge to discharge power ratio)
changes the relative C and D voltage drops, and the average charge state, Qavg. High values of vehicle load power relative to the APU power level (Pdis>>Pchrg), tend to produce higher Qavg values, due to smaller charging currents and charging voltage rises, C. The trapezoidal trajectory then "slides" to the upper right along the diagonal Q-Ut relation given in (2.3).
80 2.2.3 Approximate Model and Solution An approximate closed form solution for the average Q level is possible if the values for Rs are reduced to singular values. The approximate solution for Qavg for a given set of terminal voltage control limits is given in Equation (2.18). This solution assumes constant ~ values for the series resistances, Rs
dis
~ & Rs
chrg
(the tilde overstrike is used to indicate an
approximate description or solution). The approximate solution also uses the coefficients of ~ the linear form of Uoc from (2.3). Qavg is derived from the simplified form of (2.12) and is ~ shown in (2.16)-(2.17). Qavg shows a linearly increasing value with rising values of load power (rising Pdis & falling Pchrg) and the relationship is shown for the following non-linear numerical example in Figure 2.5.b.
Uoc(Q) ~ I dis =
~ Uoc(Q)2 - 4⋅Pdis⋅Rs dis ~ 2⋅Rs dis
~ ( I dis > 0)
[A]
(2.15.a)
~ ( I chrg > 0)
[A]
(2.15.b)
~ Ulo2 - Ulo⋅Uo + Rs dis⋅Pdis ~ Qlo = Ulo⋅(Un-Uo)
~ (0 < Qlo < 1)
[-]
(2.16.a)
~ Uhi2 - Uhi⋅Uo + Rs chrg⋅Pchrg ~ Qhi = Uhi⋅(Un-Uo)
~ (0 < Qhi < 1)
[-]
(2.16.b)
[-]
(2.17)
Uoc(Q) ~ I chrg =
~ Uoc(Q)2 - 4⋅Pchrg⋅Rs chrg ~ 2⋅Rs chrg
~ ~ ~ Qavg = mean(Qlo , Qhi)
81 ~ where Rs ~ I ~ Qlo ~ Qhi ~ Qavg
mean values for the short term series resistances ~ ~ discharge and charge currents, f(Q, Pveh, Rs)
[Ω]
approximate Q when Ut reaches Ulo
[-]
approximate Q when Ut reaches Uhi
[-]
~ approximate average Q level, f(Ulo,hi, Pveh, Rs)
[-]
[Α]
~ ~ The approximate Qavg solution is based on simple values of series resistance, (Rs dis,Rs chrg)
which results in an almost purely trapezoidal trajectory in the Q-Ut plane. Using the
non-linear forms for Rs (highly Q dependent) results in pronounced curvature of the trapezoidal trajectory as shown in Figure 2.4. The approximate Qavg solution has a limited accuracy for most conventional aqueous cell technologies with strongly non-linear Rs characteristics (and diffusion effects).
The approximate solution is more applicable to
emerging, high power density cell technologies which feature less Q dependence in the Rs dis, and Rs chrg characteristics. The approximate solution is still valuable in studying the operating trends of the terminal voltage control technique, which is further explored in the following section. 2.2.4 Effects of Terminal Voltage Control Limit Variances ~ The effects of shifting and compressing the terminal voltage control limits on Qavg, ~ ~ the peak to peak variation in Q, ∆Q, and the approximate operating period of the APU, Tapu, are explored in this section. In order to visualize the effects with numerical graphs, some typical per unit values for a battery system are assumed. The values for the open circuit voltage characteristic, nominal terminal voltage control limits, and the charge/discharge power and resistance levels are shown in Table 2.2.
82 Table 2.2
List of per unit values used for the control limit sensitivity study.
Uoc description & control limits Un Uo nominal Uhi nominal Ulo
Value
Loading and Resistance variables Papu Pveh Rs chrg Rs dis
1.0 [pu] 0.9 [pu] 1.0 [pu] 0.9 [pu]
Value 1 [pu] 0.15 - 0.85 [pu] 0.07 [pu] 0.05 [pu]
~ Effects of Voltage Limit Shifts on Qavg The nominal terminal voltage control limits are assumed to be equal to the min and max values of the open circuit voltage relationship (graphically shown in Figure 2.8.a). Shifting the terminal voltage control limits to higher or lower values has a pronounced effect on the average Q level. The resulting diagonal sliding effect is graphically shown on the QUt plane in Figure 2.5.a. Using Equations (2.16)-(2.17), the sensitivities are calculated and numerically shown in Figure 2.5.b. for ±3% shifts in the control limits. The large changes in Qavg result from the shallow slope of the Uoc vs. Q relationship. Manipulating the control limits can steer the SOC operating level up or down as desired. ~ Qavg Sensitivity to SOV Limit Shifts
SOC 1
Approx. SOC [pu]
increase in Q avg Qavg
shifted SOV limits Ulo
+3%
0.5
nom.
0.25
-3%
0 0
Uhi
Ut
a) Graphical example of shifting limits.
Figure 2.5
0.75
0.25
0.5
0.75
Load Power [pu]
b) Effects of ±3 % shifts in limits.
Effects of shifts in the terminal voltage limits on the average Q level.
1
83 ~ Effect of Voltage Limit Compression on ∆ Q The peak to peak variance in Q (∆Q in Figure 2.3) can be calculated from the approximate model and is shown in (2.18). This variance can be manipulated via the control limits. A graphical example of compressing the voltage limits on the Q-Ut plane is shown in Figure 2.6.a. The sensitivity is numerically shown in Figure 2.6.b for ±15% variation in the relative width of the control limits. Adjusting the relative width of the control limits has the same effect as adjusting the width of the hysteresis bands for the charge based controller of Figure 2.2.a. SOC
~
∆Q
decrease in ∆Q Approx. ∆ Q [pu]
0.75
∆Q
Ulo
compressed SOV limits
Uhi
a) Graphical example of width changes.
Figure 2.6
Sensitivity to SOV Limit Widths
+15% 0.5
nom. 0.25
-15% 0
Ut
0
0.25
0.5
0.75
1
Load Power [pu]
b) Effects of ±15 % changes in limit widths.
Effects of width changes in the control limits on the SOC variance.
~ ~ ~ ∆Q = Qhi - Qlo
~ (0 < ∆Q < 1)
[pu]
(2.18)
The terminal voltage control limits have a pronounced effect on the variances in SOC. This sensitivity comes from the shallow slope of the Uoc relationship (Figure 2.8.a) which is also responsible for the average SOC sensitivity.
Adjusting the relative width of the
hysteresis control limits of the terminal voltage controller (Ulo, Uhi), the variation in the battery's SOC can be easily manipulated for constant loading applications.
84 Effect of Voltage Limit Compression on APU Operating Period The approximate analysis can be extended to include APU's operating period (Tapu in ~ Figure 2.1). The approximate solution, Tapu, is found from the estimated charge variance to the applied mean current as shown in (2.19). The approximate APU operating period is shown vs. load power in Figure 2.7. ~ Tapu dis =
~ ∆Q ~ mean( I dis)
~ Tapu chrg = -
~ ∆Q ~ mean( I chrg)
(APU = off)
[h]
(2.19.a)
(APU = on)
[h]
(2.19.b)
[h]
(2.20)
~ ~ ~ Tapu = Tapu dis + Tapu chrg
A 15% variance in the relative width of the terminal voltage control limits has a pronounced impact on the operating time of the APU. ~ sensitivity of ∆Q to the width of the control limits.
This is directly related to the
85 ~ Tapu Sensitivity to SOV Limit Widths
3
2
+15%
1
0
APU always on
APU always off
Approx. APU Period [pu]
4
-15%
0
0.25
0.5
0.75
1
Load Power [pu]
Figure 2.7
Approximate APU period vs. load power for broadened and compressed control limits. (dash = 15% broader, solid = nominal, dots = 15% compressed).
2.2.5 Numerical Evaluation of the Non-linear Model To better evaluate and compare the operation of both charge and terminal voltage control techniques, a numerical analysis of the non-linear model is attempted. The example Rs coefficients are shown in per unit form in Table 2.3, and were derived from basic tests applied to the VRLA system described in Appendix A.2. The following results for power cycle efficiency do not account for diffusion effects, hence these studies are valuable in a comparative manner for trends. Table 2.3 Variable Un Uo Papu Pveh
List of parameter values used for the numerical comparison of control techniques. Value 1.0 [pu] 0.9 [pu] 1 [pu] 0.15 - 0.85 [pu]
Variable k1 k2 k3 k4
Value 0.04 [pu] 0.2 [-] 0.06 [pu] 0.3 [-]
86 0.12
Rs [pu]
Uoc [pu]
1.0
R s chrg 0.06
R s dis
0.9
0
0 0
0.5
Q [pu]
a) Open circuit voltage vs. Q
Figure 2.8
1
0
0.5
1
Q [pu]
b) Short Term Series Resistance vs. Q
Example characteristics for a VRLA cell outlined in the Appendix.
The open circuit voltage, Uoc, and the short term resistance, Rs, characteristics enable the calculation of first the terminal current, It, and then the power cycle efficiency, ηcycle, as a function of load power and Q. The power cycle efficiency results vs. load power and Q are shown in surface plot format in Figure 2.9. The plot indicates that the example battery should be operated over a range of 0.3 < Q < 0.8 , in order to obtain the best power cycle efficiency performance for a wide range of loading (0.15 < Pveh < 0.85 [pu]). The shape of the cycle efficiency contour will tend to be a "ridge" shape which is diagonally positioned in the 3-D space.
87 Cycle Efficiency vs. Q and Pveh
Cycle Efficiency
0.95 0.94 0.93 0.92 0.85 z
0.91 y 0.1
Pveh Q
0.8
x Figure 2.9
0.15
Pdis> Pchrg
Pdis< Pchrg
Estimated battery cycle efficiency vs. Q and load power; solid line indicates optimal trajectory (Papu = 1 [pu], Pveh = 0.15-0.85 [pu]).
SOC Behavior under Terminal Voltage Control This sub section shows how Qavg varies with both cell loading levels, and the terminal voltage control limits when using the non-linear description of the series resistance. The steady state value of Qavg in turn determines the cycle efficiency. The average Q level as a function of load power level is shown in Figure 2.10.a. The generally increasing relationship of the approximate solution as first seen in Figure 2.5.b is also witnessed here, but with less sensitivity to shifts in the terminal voltage limits. This is due to the rising and falling values of the short term series resistance model (Figure 2.8.b). The nonlinear relations for the Rs values tend to limit the excursions in the Qavg as the terminal voltage control limits are shifted. The charge and discharge voltage drops (C & D) become larger under extremes of operation thus reducing the typical range of Qavg.
88 Cycle Efficiency vs. Load Power
Average Q vs. Load Power Cycle Efficiency [pu]
Average Q [pu]
1
0.75
0.5
0.25
0.95
0.94
0.93
0.92
0 0
Pdis< Pchrg
0.25
0.5
0.75
Load Power [pu]
1
Pdis> Pchrg
a) Qavg variation with shifted voltage limits
0
Pdis< Pchrg
0.25
0.5
1
0.75
Load Power [pu]
Pdis> Pchrg
b) Cycle efficiency for shifted voltage limits
Figure 2.10 Qavg and ηcycle variations for shifts in the control limits. (nominal limits Ulo = Uo and Uhi =Un , dash = +3%, solid = nominal, dot = -3%) The resulting cycle efficiency performance for the cell under shifted terminal voltage control limits is shown in Figure 2.10.b. The plot shows how terminal voltage control achieves good efficiency performance over a wide range of operating conditions. As the control limits are shifted, a slightly different efficiency trajectory is realized.
These
trajectories are paths which mimic the optimal trajectory on the ridge shaped efficiency surface of Figure 2.9. Selecting an arbitrary set of upper and lower terminal voltage control limits (Ulo, Uhi) can be used to explore how the efficiency performance can be optimized for a given power flow condition. The cycle efficiency performance for arbitrary upper and lower voltage control limits is shown in the surface plot of Figure 2.11. The plot assumed a load power level of 0.5 [pu], and an APU power level of 1.0 [pu], resulting in a balanced charge/discharge power flow. For this operating point and given Rs characteristic, slightly higher terminal voltage limits resulted in the best performance.
For other power flow
conditions, different values for Ulo, and Uhi would offer optimal efficiency performance.
89 In general, a single set of terminal voltage control limits will give adequate efficiency performance over a wide range of steady state loading due to the relatively flat nature of these efficiency surfaces.
Cycle Efficiency vs. Arbitrary Control Limits
Cycle Efficiency
0.94
0.935
+4% 0.93
z
Uhi
y -4% x
Ulo
+4% Pdis = 0.5 [pu] , P chrg = 0.5 [pu]
-4%
Figure 2.11 Cycle efficiency vs. arbitrary terminal voltage control limits (Ulo, Uhi). (Pdis = Pchrg = 0.5 [pu]) To explore how shifts in the control limits (fixed width between Ulo and Uhi) effect the efficiency performance for arbitrary loading, a final surface plot is presented in Figure 2.12. The surface plot shows that the nominal terminal voltage control limit values result in near optimal power cycle efficiency performances for nominal loading conditions. The control limits can be shifted slightly as a function of loading conditions in order to maximize efficiency performance.
90 Efficiency vs. Load Power and Control Limit Shifts
Cycle Efficiency
0.95
0.94
0.85 Pveh Pdis> Pchrg
0.93 z
-4% y x
V limits
+4%
Nominal Voltage Limits: Ulo= Uo , Uhi = Un
0.15 Pdis< Pchrg
Figure 2.12 Cycle efficiency vs. shifts in control limits and load power, optimal trajectory indicated with bold line. A set of overlaid plots comparing terminal voltage and charge control techniques is shown in Figure 2.13. The voltage control techniques generally gives very good power cycle efficiency vs. a traditional charge based regulator. The graph in Figure 2.13 shows the efficiency performances of both a set of voltage and charge control limits. The two sets of control limits were tuned to give near optimal results at either low or high discharge to charge power ratios. The relative differences between the voltage controllers is minimal compared with the charge regulation controllers. For best results, the SOC control technique should make adjustments to the target value of Q* (Figure 2.2.a) in order to obtain optimal broad band efficiency.
91 Cycle Efficiency vs. Load Power
im Vl
im Vl
2
SO C= 0.3 0
=
0. 80
0.94
1
SO C
Cycle Efficiency [pu]
0.95
0.93 0
0.25
Pdis < P chrg
0.5
Load Power [pu]
0.75
1
Pdis > P chrg
Figure 2.13 Cycle efficiency vs. load power for both charge and voltage control techniques. (Vlim1 = [0.95, 0.99], Vlim2 = [0.93, 0.97]) The efficiency plots presented in Figures 2.10 - 2.13 should be used as guidelines for adjusting control limits for constant loading applications and not as accurate predictors of performances. The simplified short term model which ignores diffusion effects does not account for elevated values of charge transfer resistance, and hence the model predictions are optimistic. The general trends in efficiency performance indicate that shifting the voltage control limits can effectively steer the efficiency profile to favor different charge/discharge power flow conditions.
2.3 Constant Load Experiments This section experimentally compares charge and voltage control techniques in a CSHEV example with hysteresis controlled APU operation. Figure 2.14 shows the general power flow diagram for the CS-HEV example. The APU is assumed to operate with a
92 constant power output when enabled, and is cycled on-off to maintain the battery state within hysteresis band limits. This type of control was used on spark ignited, throttle controlled engine-generator set, as found in the example CS-HEV of Appendix A.3. Generally, the APU can be controlled either in a "power tracking" mode, or in a fully "load leveled" mode. The operation of the APU will significantly effect the power flow in the battery, and the overall fuel / battery efficiency of the system. APU APU Control
Vehicle Load dynamic
PAPU PL PB
Battery State Feedback
constant
Battery
Figure 2.14 General HEV power flow diagram with either dynamic or constant loading. 2.3.1 Experimental Set-up The experiments were accomplished in the laboratory with a bi-directional power source and a relatively high bandwidth (~50Hz) switching current regulator. Figure 2.15 shows the layout of the small-scale (36V, 12A) battery test stand assembled for the battery state control experiments. The computer controller observed the device under test's (DUT) voltage and current at a sufficiently high 4 Hz sample frequency, and received vehicle power trace information from an external file (1 Hz). The vehicle load trace, PL, was either a constant command or a scaled version of a power trace from a typical federal driving cycle. The controller regulated the battery power flow based on the difference between the vehicle
93 power trace and the APU power level (when enabled). Appendix A.2 and A.4 discuss the sample batteries under test and the control block diagram used to regulate power flow. The following scaled laboratory experiments applied true power control to the battery, unlike the simplified current control often found in literature [Kahlen 1994, Brandt 1992]. DUT 3φ net emf
AC & DC machines
Switching Current Regulator It Ut
Icom Current Command
PL Computer Controller
Figure 2.15 Connection diagram of the small scale battery test set-up. The energy efficiency of the battery in Figure 2.14 is defined in (2.25). This power sink centric description is further discussed in Chapter 5. For the purpose of comparing battery control techniques, the vehicle load, PL, and APU power, Papu, are assumed to be measurable before the summing junction, and there are no power processing (or losses) between the summing junction and battery terminals. EL = ∫ PL dt
[Wh]
(2.21)
Eapu = ∫ Papu dt
[Wh]
(2.22)
Es(Q) = ∫ Uoc(Q) dQ
[Wh]
(2.23)
∆Es = Es init - Es final
[Wh]
(2.24)
[-]
(2.25)
ηcycle =
EL EL+∆Es = E Eapu-∆Es apu
94 where EL EAPU ∆Es
net energy consumed by load, PL net energy supplied by the APU net change in battery energy storage, Es
[Wh] [Wh] [Wh]
A constant vehicle loading example is used to compare the charge and voltage based battery state controllers. A constant vehicle load profile with three distinc levels (PL = 1.5, 3, 4.5 [pu]) was applied to the system shown in Figure 2.14. The APU, capable of producing 6.0 per unit power, was turned on and off to maintain the state of the battery within a given set of voltage or charge hysteresis band limits. 2.3.2 State of Charge Control The performance of the gassing current compensated SOC regulator is presented. The terminal voltage characteristic of the battery under SOC control is examined, while the normalized SOC was regulated within an arbitrarily selected window of +/- 5% around a target SOC level. Three test with different SOC levels (0.30 , 0.50, 0.80 [pu]) were taken. Figure 2.16 shows how the battery terminal quantities, Q estimate, and APU reacted over the 0.75 hours of the constant load test. The Q level began the test at an initial value of 0.8 pu and the Q estimate was successfully regulated. The charge state, Q, was estimated using the Tafel gassing compensated method outlined in Section 1.1.5 and Appendix A.2.2.
95 Voltage, Current, SOC, APU and Pveh vs. Time
Current
Voltage [xVn] Current [xIn] SOC [xQn]
Charge Power
Power [xPn]
Voltage
1.1 1 0.9
5 0 5 0. 9 0. 8 0. 7 6 4 2 0 0
500
1000
1500
2000
2500
Time [sec.]
Figure 2.16 Terminal voltage, current, Q estimate and power vs. time for the SOC controlled example. (Qlim = [0.75, 0.85], Qo = 0.80 pu) Closer examination of the experiment emphasizes some of the disadvantages of charge based state control on long driving cycles. The initial Q level, Qo, for the control algorithm was determined from a battery which was discharged to this level from a known 100% SOC state. The battery was then allowed to rest for12 hours. Initializing the SOC value will always be a challenge. If a stable, diffusion free, open circuit voltage reading is not available, the algorithm may need to begin with the last estimated SOC value which may harbor previously accumulated errors. Drift in the SOC estimate due to current integration errors can also become significant over longer driving cycles. Figure 2.17 shows the terminal voltage vs. Q estimate for the constant load experiment in Figure 2.16. As the test progresses, a clock-wise trapezoidal trajectory is revealed. The voltage trace is seen to shift downward as the test progressed showing how the
terminal voltage reacted to the shift in discharge to charge loading.
96 The fundamental
characteristic is that the terminal voltage is not directly observed or bounded by the charge based state controller. Rather a secondary variable (charge content) is controlled based on a previously determined set of control limits which will give acceptable voltage or efficiency performance for a set of known power processing traces. Voltage vs. SOC 1.15
1.05 Terminal Voltage [xVn]
Terminal Voltage [xUn]
1.1
1
0.95
0.9
0.85
0.8 0.7
0.72 0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88 0.9
bulk State of Charge [xQn]
Figure 2.17 Terminal voltage vs. Q estimate for the constant load experiment under SOC control. The optimal operating range for SOC will vary as a function of the ratio between charging and discharging currents and resistances. To obtain good power cycle efficiency over a wide range of events, the controller must continually adjust the control limits applied to the Q estimate based on a priori experiments. 2.3.3 State of Voltage Control A terminal voltage hysteresis regulator example is now presented.
The SOC
97 trajectory is examined, while the terminal voltage is bounded. Figure 2.18 shows how the terminal quantities (Ut , It), Q estimate and APU reacted during the 0.75 hours of the constant load example under voltage control. The terminal voltage control limits were intentionally set to mimic central portion of the SOC control example in Figure 2.16. The Q estimate is shown to slowly approach a steady state value while the directly measured terminal voltage is well bounded.
Voltage [xVn] Current [xIn] Power [xPn]
Current Charge Power
SOC [xQn]
Voltage
Voltage, Current, SOC, APU and Pveh vs. Time 1.1 1 0.9
5 0 -5
0.8 0.7 0.6 0.5 6 4 2 0
0
500
1000
1500
2000
2500
Time [sec]
Figure 2.18 Terminal voltage, current, Q estimate and APU power vs. time for the voltage controlled example. (Vlim = [0.86, 1.06], Qo = 0.60 pu)
98 SOC vs. Voltage 1 0.9
0.7 0.6 SOC [xQn]
bulk State of Charge [xQn]
0.8
0.5 0.4 0.3 0.2 Q = f(Uoc)
0.1 0 0.8
0.85
0.9
0.95
1
1.05
1.1
Terminal Voltage [xUn]
Voltage [xUn]
Figure 2.19 Q estimate vs. terminal voltage for the voltage controlled constant load experiment Figure 2.19 shows how the Q estimate responded relative to controlled terminal voltage. The voltage controlled battery allows the SOC to drift to three different levels as the discharge to charging loading changed. Voltage control is stable, because the ideal Q-Ut relation is a diagonal line.
This diagonal relationship guarantees that the actual (not
estimated) bulk SOC is bounded within a nominal range, which is set by the voltage control limits. The voltage control limits do need to be spaced reasonably close to the upper and lower bounds of the Uoc relationship. Further discussions of setting the control limits is discussed in section 2.4. The voltage control variable is not an estimated variable. The terminal voltage is directly measurable, and no integration is used to estimate the value. This provides drift free
99 control of the terminal voltage, and when coupled with the diagonal, Q-U relationship found in Figures 2.17 & 2.19, it also bounds the range of SOC operation over wide loading conditions and temperature. Charge based control often requires compensation and reset mechanisms to insure stable operation over long term loading and temperature ranges. Efficiency Evaluation The net cycle efficiencies of the experiments were evaluated for each constant load power levels. The net cycle efficiency vs. the load power is shown in Figure 2.20. The efficiency results were understandably similar due to the similar operating conditions, but a fundamental difference is seen in how the battery state is controlled. The voltage controlled experiments maintained higher voltage stiffness (less variation under load), thus causing better overall efficiency performances. Efficiency vs. Load Power 0.98
Efficiency
Efficiency [-]
0.96
0.94
SOV1 Q=80 Q=50 Q=30
0.92
0.9
0.88 1
2
Pchrg > Pdis
3 Load Power [xPn]
4
5
Pdis > Pchrg
Load Power [xPn]
Figure 2.20 Net cycle efficiency vs. load power for the switched mode controlled APU experiments under voltage and charge control. (Papu = 6[pu])
100 The above constant vehicle load examples do not address many of the challenges present in typical driving cycles. The following section addresses the issues of controlling a dynamically loaded battery via terminal voltage.
2.4 Voltage Control for Dynamic Loading Typical driving cycles exhibit dynamic power demands with both positive and negative values as introduced in Figure I.2.b. The battery is required to accept or deliver very high power flows, and the terminal voltage often experiences wide variations.
The
previously outlined method of battery state control via a set of terminal voltage limits can not be directly applied to these dynamic loading conditions. There is a wide range of CS-HEV control strategies. Each dictates under which conditions the battery and APU supply the vehicle's power demands, but the spectral content of the power demand can still be divided into two general parts; that which the APU generally satisfies, and that which the battery generally satisfies. The average, and low frequency components of the vehicle's power spectrum is assumed to be supplied by the APU, and mid to high frequency power components are assumed to be supplied by the battery. The division between APU and battery responsibilities depends upon the response time and control of the APU (fast = load power tracking, slow = load leveling). The PNGV has categorized APU's as either Slow or Fast Response Engines (SRE & FRE) with typical response times, of 30 and 2 seconds respectively. These categories for APU's will be used here to help quantify the power buffering spectral demands on the battery. An example 1590 kg CS-HEV was simulated in several driving cycles and the axle
101 power spectrums were obtained. The resulting power spectrums [10log(PL / Pavg) vs. log frequency] were all similar. The Simplified Federal Urban Driving Schedule (SFUDS) did not reproduce the broadband spectral content as found in the realistic cycles (HFEDS & FTP75), hence it was not investigated further. Figure 2.21 shows the power spectrum for the first 505 seconds of the FUDS cycle [see Figure I.2.b and Appendix A-3]. Three U.S. driving cycles featured a corner frequency of approximately 0.02 Hz, and natural attenuation on the
Power Magnitude [dB]
order of ~30 dB/decade.
log Frequency [Hz]
Figure 2.21 Power spectrum of the FUDS driving cycle for a typical 1590 kg vehicle including slow and fast response engine regimes. 2.4.1 2-D Voltage Control The high frequency power spectrum which the battery must process causes dynamic swings in the terminal voltage. Terminal voltage control techniques as used in the constant load example are still applicable, but the control limits must be broadened to avoid premature
102 APU state changes. Practically, the terminal voltage control limits should be extended to the maximum allowed by the LLD specification (Table I.2). These broad limits generally satisfy APU control requirements for dynamic loads but have drawbacks; •
Broad terminal voltage control limits do not sufficiently constrain SOC for nondynamic loads. The SOC will be allowed to drift to one extreme or the other before a wide terminal voltage limit is reached.
•
Broad voltage control limits do not encourage small impedance drops necessary for good power efficiency during nominal loading conditions. To overcome these two deficiencies, a 2-dimensional voltage control algorithm is
introduced. Besides the dynamic terminal voltage events, the trend in terminal voltage (or moving average) is also monitored and constrained. The voltage trend is obtained from a low-pass-filter applied to the terminal voltage. The control limits applied to the trend in terminal voltage are much tighter than those applied to the dynamic terminal voltage. With the addition of the voltage trend control, the two above drawbacks are eliminated. Two possible implementations of APU controllers based on the trend and dynamic terminal voltage information are shown in Figure 2.22. Umax
APU Set
Reset Ut
Ulo Set Umin
LPF(Ut)
Uhi
Reset Umax
Ut
a) Fuzzy logic APU operation determination
Umin LPF
Uhi
Ulo
+ + + + -
Reset APU R S
APU on/off
Set APU
b) Conventional logic APU operation determination
Figure 2.22 APU operation determination via trend and direct terminal voltage information for both fuzzy and conventional logic systems.
103 LPF Corner Frequency The trend voltage's LPF corner frequency and control limits are derived from the spectrum of the battery's power flow and the ideal Q-Uoc relationship respectively. The LPF corner frequency should be placed below the most dynamic power buffering frequency of the battery. The frequency in the battery's power spectrum which has the largest magnitude is considered the characteristic buffering frequency, ωbuf [rad/sec], or fbuf [Hz]. For CS-HEV's with SRE's it is approximately 0.01 [Hz], and for FRE's fbuf is closer to 0.08 [Hz] as shown in Figure 2.19. The LPF will then attenuate the high frequency content in the dynamic terminal voltage, resulting in a low noise trend voltage battery control signal. The LPF corner frequency should not be set too low, thus causing significant delay in the trend indicator. Hochgraf [1996] introduced a practical lower limit on the LPF, based on worse case loading and acceptable deviations in the battery's SOC. The equations defining the trend voltage and LPF corner frequency are shown in (2.26) and (2.27). Utrend = LPF(Ut)
[V]
(2.26)
fLPF < fbuf
[Hz]
(2.27.a)
[Hz]
(2.27.b)
fLPF >
where ∆Qmax PB max En fbuf
PB max 2π⋅3600⋅En⋅∆Qmax maximum allowed variation in normalized SOC maximum sustained battery power (charge or discharge) nominal energy storage of battery dominant frequency of battery power spectrum
[-] [W] [Wh] [Hz]
The battery in the example CS-HEV (Appendix A.3) has a lower LPF limit of 0.001 Hz (based on ∆Qmax= 0.12, En = 7kWh, PB max=20kW). The upper limit for the LPF was 0.02Hz (based on a maximum spectral component at fbuf= 0.02 [Hz]). A first order LPF with
104 a corner frequency of 0.0026 Hz (60 second first order time constant) was successfully used and is also used in the below experiments. Trend and Dynamic Control Limits The control limits for the trend voltage indicator should be set close to the limits of the diagonal Uoc-Q relationship (values of Un and Uo in Figure 2.8.a). This will insure that the SOC always finds a steady state operating region before drifting too close to an extreme (0 or 100% SOC). The dynamic terminal voltage limits are set to the max allowed by the LLD specification (Table I.2) which are typically 1.15 [pu] and 0.8 [pu]. Figure 2.21 shows an example of the two voltage signals vs. time. The terminal voltage behaves dynamically within wide control limits, and the filtered terminal voltage behaves slowly within tighter control limits. The two upper control limits dictate when the APU should be turned off in order to reduce the voltage characteristic, and the two lower limits determine when the APU should be turned on in order to increase the voltage behavior of the battery. Either of the two voltage signals can influence the APU state to change. A 2-dimensional hysteresis control box is presented which shows how the two voltage indicators influence the APU state. Figure 2.23.b shows the terminal voltage vs. trend voltage plane. The steady state relation between the two axes is a unity slope line. The diagonal nature of the unity slope line insures that the voltage trajectory will generally move between APU off and on bounds. The 2-D voltage control plane can show which boundaries commonly trigger the changes in the APU state.
105 1.2
LPF(Ut)
Ut
0.9
Ut Limits LPF(Ut) Limits APU
1.0
APU off
Ut
LPF(Ut)
1.1 APU on
Event Marker
Ut
APU off
Umax
Umin
0.8
APU on
on off time
0.85
0.9
0.95 1.0
Ulo
Uhi
1.15
LPF(U t )
a) Dynamic and trend voltages vs. time.
b) 2-D voltage bounding box
Figure 2.23 Two dimensional voltage signal control example with event markers.
2.5 Dynamic Load Experiments Typical driving cycles exhibit broadband power spectrums. The average, and low frequency power demand in a CS-HEV is supplied by the APU, hence the battery must supply the mid- to high frequency power components. The exact characteristic of the spectral content of the battery power is dependent upon the brake regeneration events, APU response time, and specifics of the CS-HEV control algorithm. Special attention must be given to highly dynamic events in the power flow and terminal voltage if it is used as a control variable. 2.5.1 Experimental Set-up The first 505 seconds of the FUDS cycle was selected as the seed for the dynamic load experiments. The power demand for the example HEV was scaled to +/- 3 pu peak power stresses and used as the input load power vector, PL, for the scaled test bench. The APU power level was then set to 1.0 pu in order to mimic the stresses present in the example
106 high energy storage CS-HEV. Twelve driving cycles were repeated sequentially, extending the test to 6060 seconds. The below experiments were carried out in a temperature controlled oven at 30°C. To view the typical driving cycle load power traces for the example CS-HEV, see to Appendix A.3. 2.5.2 State of Charge Control The battery traces from the SOC controlled dynamic load example are shown in Figure 2.24.
The SOC was regulated within the range of 0.40~0.45 pu.
(empirically
determined to produce acceptable voltage results). The terminal voltage responded within the min and max allowed limits of 0.83-1.15 pu (+/- 12% variation). The net battery energy efficiency over the test was 84.0% and an 8°C temperature rise was witnessed.
Once
initiated, the APU operated with an approximate duty cycle of ~39% and with a period of ~640 seconds.
APU Power Charge
Voltage
107
Time [sec]
Figure 2.24 Terminal voltage, Q (estimate) and APU power vs. time for the dynamic load example under SOC control. The APU operating period when under charge control is determined by the average power demand, SOC control limits, and energy content of the battery. The period is inversely proportional to the right hand side of Equation 2.27.b. The SOC estimate is based on the time integral of the dynamic current, thus resulting in a slowly moving control variable which predictably moves between the hysteresis control limits.
Terminal Voltage [xUn]
108
bulk State of Charge [xQn]
Figure 2.25 Terminal voltage vs. Q (estimate) for the dynamic load example under SOC control. Figure 2.25 shows the terminal voltage vs. SOC (estimate) plane for the experiment in Figure 2.24. The SOC (estimate) is seen to be well regulated, and the voltage is well behaved. The target of ~43% SOC was selected after some trial and error. The V-Q trajectory of Figure 2.25 provided a way to evaluate an appropriate range for the SOC level; terminal voltage responds in an acceptable manner (within min/max limits). 2.5.3 Voltage Control The dynamic load experiment was reapplied under 2-dimensional voltage control as outlined in section 2.4.1. The limits were; (Ulo, Uhi) = (0.9, 1.0) [pu], (Umin, Umax) = (0.83, 1.15) [pu]. The voltage control technique achieved a power cycle efficiency of 85.1% and the battery experienced only a 7°C temperature rise. The APU operated with an approximate duty cycle of ~38% and with an average period of ~500 seconds.
109 The APU operating period is a function of the voltage control limits, internal resistance, the magnitude of the dynamic load events in the driving cycle and the LPF of the trend voltage. Slower trend filters and less dynamic events tend to increase APU operating periods. The battery quantities under voltage control are shown in Figure 2.24 and the events which changed the APU state are shown with circles. Figure 2.26 shows how voltage control allowed the SOC to vary asymptotically to a
Papu
SOC
LPF(Ut)
Ut
steady state operating level. Figure 2.25 shows the test’s 2-D voltage control box.
Time [sec]
Figure 2.26 Terminal voltage, filtered voltage, Q (estimate) and APU power vs. time for the dynamic load example under 2D voltage control.
Terminal Voltage [xUn]
110
Filtered Voltage [xUn]
Figure 2.27 Voltage control trajectory (Ut vs. LPF(Ut)) for the dynamic load experiment of Figure 2.24.. The trajectory in Figure 2.27 shows how both trend and instantaneous voltage signals are well contained inside the control limits. Under this test, the trend voltage dictated the majority of the APU control events. Once, at t= 4600 sec, the terminal voltage reached the lower bound, Umin, and initiated the APU. This example shows how both voltage signals work to insure that the battery specification is met (Table I.2). 2.5.4 Cold Temperature Operation The voltage control technique works well under cold operating conditions. The same control limits can be used in VRLA systems, independent of temperature.
The APU
operating periods are generally shorter for cold operation. The trend and dynamic voltage control limits are easily reached due to elevated series impedance's in the cold battery, but the battery terminal voltage is always well constrained (SOC control does not control terminal
111 voltage under cold operation). An example of low temperature (0 C) operation is shown in o
Figure 2.28. The battery began the test at 0oC (ice water bath) and experienced a 15oC rise internal to the modules. The net energy efficiency of the battery over the test was 83.1%. The SOC eventually drifted to the same range as Figure 2.26, but was initially at ~47% while the battery was still cold. Allowing the SOC to find an operating range which satisfies the
Papu
SOC
LPF(Ut)
Ut
primary voltage constraints is a great benefit to the voltage based state controller.
Time [sec]
Figure 2.28 Terminal voltage, filtered voltage, Q (estimate) and APU power vs. time for low temperature test.
2.6 Properties of the Voltage Control Technique The voltage control technique directly measures and constrains the terminal voltage of the battery. The terminal voltage is an important part of the battery's system specification (Table I.2). Constraining it between reasonable bounds through the control of the APU guarantees that the battery will not operate outside the specification. Charge based APU
112 control, has inherent drift due to integration errors and it can not, by itself, insure that the battery will not fail the voltage specifications. Some form of drift reset mechanisms are applied to all long-term SOC controlled batteries. Secondly, the voltage control technique does not require an estimate of the initial state. The initial terminal voltage is simply read into the control algorithm, and no historical data is needed, unlike the SOC method which requires an estimate for the initial charge, Qo. Thirdly, the voltage control technique naturally monitors the relative voltage stiffness of the battery when under load. The battery's impedance changes with temperature, age, and diffusion effects, and a single set of voltage control limits can still successfully insure that the battery does not violate the system specifications. Thus avoiding under/over voltages, and poor power cycle efficiencies. SOC control can mimic voltage control if the target level, Q*, is adjusted dynamically when different loading conditions, diffusion effects, or aging are experienced. But it is less complex to use voltage control with a single set of control limits which are estimated from the analysis presented in Section 2.2.4.
2.7 Summary Chapter 2 introduced a method for categorizing battery descriptions for load leveling applications into primary and secondary descriptors.
These categories highlighted the
opportunity for terminal voltage and power capability based regulators to outperform the traditional charge based battery regulation techniques. The terminal voltage control technique was analyzed for a simple hysteresis
113 controlled APU application. The equivalent short term model of the cell which ignored diffusion effects was used to evaluate the behavior of the voltage regulation technique. A simplified form of the model was used to determine an approximate closed form solution for the Q characteristic of a cell under voltage control. The effect of varying voltage control limits was shown to have predictable results on the cell's operation. This approximate solution technique also lead to a description of the power cycle efficiency and APU operating characteristic. Numerical analysis of the short term electrical model with non-linear series resistances was used to more ably describe the efficiency characteristics of the cell under voltage control. The voltage regulation technique was shown to naturally adjust the cell's Q level as the loading on the cell changed. This load sensitive adjustment in Q resulted in near optimal power cycle efficiency performance over a wide range of load conditions. A 2-dimensional voltage control technique for dynamic loading was presented. The technique uses both the dynamic terminal voltage and its trend to determine the operating state of the APU. This technique requires the use of a low pass filter with known bounds on the cut-off frequency. This chapter presents a sampling of experiments which evaluated the fundamental differences in operating characteristics between charge and voltage based state regulation techniques. Both steady state and dynamic loading profiles were explored.
The voltage
based methods are shown to be robust and provide excellent operating characteristics. Several conclusions can be made from the discussions and experiments made in this chapter. The material specially focused on a high energy storage CS-HEV with hysteresis
114 controlled APU and VRLA batteries. Some caution must be applied when extending the below statements beyond these contexts. 2.7.1 Charge Control The charge control method was difficult to initialize due to the ambiguity of the initial charge state, Qo.
Additionally, the presence of integrator drift and gassing currents will
cause further problems during extended driving cycles. The process of selecting the target charge value, Q*, and/or the hysteresis control bands, (Qlo, Qhi), is not simple. Due to the high dynamic loads present in the driving cycle, the best range of SOC was determined via experimentation. For cold temperature operation, no single set of SOC control limits resulted in acceptable operation of the battery. The battery was continually violating the lower and upper terminal voltage limits. SOC based control of the battery state does not directly control terminal voltage, which is effected by diffusion effects, dynamic loading and temperature. 2.7.2 Voltage Control Voltage based control techniques constrain terminal voltage and stably bound the SOC level during both dynamic and cold temperature operation.
The relative voltage
stiffness of the battery determined the APU operating points which avoided long periods of APU rest or operation. Thus deep diffusion effects are avoided and the terminal voltage stiffness is kept high. Terminal voltage based battery state control have no initialization problems, and are simple to implement. Voltage control provides well behaved performance even over a wide range of temperature and age conditions.
115
116
Chapter 3, Advances in Modeling and Impedance Estimation 3.1 Estimating Internal Voltages......................................................................................117 3.1.1 Charge Based Estimation of Uoc ......................................................................117 3.1.2 Voltage Based Estimation of Uoc.....................................................................118 3.1.3 Impedance Voltage ..........................................................................................119 3.2 Impedance Modeling..................................................................................................120 3.2.1 Electrolyte Resistance......................................................................................120 3.2.2 Charge Transfer Resistance Tracking..............................................................123 Updating Estimates in Absence of Excitation ...............................................124 3.2.3 Mass Transport Limitation ..............................................................................126 Observing Mass Transport Current Limit......................................................126 Tracking Mass Transport Current Limit ........................................................129 Mass Transport Diffusion Effect....................................................................130 3.2.4 Diffusion Effect ...............................................................................................133 3.3 Filtering and Robust Estimation ................................................................................135 3.3.1 Filtering of the Estimation Signals ..................................................................136 3.3.2 Ignoring Outliers..............................................................................................136 3.3.3 Sufficient Excitation........................................................................................137 3.3.4 Time variances.................................................................................................137 3.4 Summary ....................................................................................................................138 3.4.1 Electrolyte Circuit ...........................................................................................139 3.4.2 Charge Transfer Resistance .............................................................................139 3.4.3 Mass Transport Effect .....................................................................................139 3.4.4 Diffusion Effect ...............................................................................................140
The basic theory and equivalent electrical model of porous electrode cells were introduced in Chapter 1. Further modeling is necessary to account for the inherent practical limitations of realizable cells. Several phenomena related to the current magnitude and direction dependent structure of cells are introduced and examples are given for three cell chemistries. A recursive estimation techniques, as outlined in Section 1.7, is applied to the modified lumped Randles-Ershler circuit model to obtain parameter estimates during both
117 specialized tests and nominal CS-HEV operation.
3.1 Estimating Internal Voltages Two methods for estimating the open circuit voltage, Uoc, of the cell are presented. The charge and voltage based estimates for Uoc have strengths and weaknesses which will determine which is used in varying applications. The gassing current component present in the cell, as outlined in Figure1.6 (section 1.1.6), is assumed to be known a priori. See Appendix A.2.2 for the different Tafel gassing relationships for VRLA, NiCd and NiMH chemistries. Once the open circuit voltage is estimated, the impedance voltage Uz can be estimated and used in the following impedance estimation routines. 3.1.1 Charge Based Estimation of Uoc One method of estimating the open circuit voltage is via a charge state estimate and an open circuit voltage relationship. The open circuit voltage relation, as outlined in (1.1) and (2.3), can be mapped as a function of bulk SOC with a priori tests (Uoc = f(Q)). The bulk SOC is then updated during operation with a current integrator which compensates for the parasitic gassing current as shown in (1.13). Figure 3.1.b shows how a gassing current model and the open circuit voltage relation are used to estimate Uoc. In the estimation of Uoc, no Peukert style modifications are applied to the current integrator. This insures an accurate estimation of the true bulk SOC level of the cell. Decoupling the gassing current from the terminal current results in accurate bulk SOC estimates, and hence Uoc estimates. This method necessitates an initial state, Qo, for the charge integrator, and is prone to integrator drift. Occasional reset methods may be required for prolonged operation.
118 Uz Ict
Rele +
Uoc +
It Ig
- Qo
Uele
Tafel
Ut-Uele
T
Ut -
a) Equivalent circuit structure Figure 3.1
Uz
Ut Ln(Ig)
+
Ig It
Ict
∫
Q
Uoc
Qo
^ and U ^ estimation b) Block diagram for U co z
Estimation of open circuit and impedance voltages from terminal quantities, gassing model, initial charge state, and open circuit voltage relationship.
3.1.2 Voltage Based Estimation of Uoc An alternate method of estimating the open circuit voltage exists and is based on terminal voltage measurements. The estimation of Uoc can be based solely on the low frequency voltage trajectory of the battery, hence it does not necessitate an estimate of the bulk SOC. The best estimate of Uoc, via the model's diffusion, double layer, and electrolyte resistances, is subjected to a low pass filter. The time constant of the low pass filter should be below the nominally experienced diffusion time constant. Figure 3.2 shows the block diagram of the voltage based estimation of Uoc. ^ (t=0) = ⋅U (t=0) U oc t
[V]
(3.1.a)
^ U oc max = ⋅Un
[V]
(3.1.b)
^ U oc min = ⋅Uo
[V]
(3.1.c)
^ = ⋅LPF( U +U ^ +U ^ +U ^ ) U oc t dif dl ele
[V]
(3.1.d)
^ is obtained. If U ^ Equation (3.2) shows how the bulk SOC can be estimated once U oc oc
119 ^ will be bounded between 0 and100%. is bounded according to (3.1.b) and (3.1.c), then Q ^ ^ = ⋅ Uoc -Uo Q Un-Uo
[Ah]
(3.2)
Udif Udl Uele Ut
LPF
^ U oc
τLPF > τdif Figure 3.2
Block diagram for estimating Uoc without charge integration stage.
This Uoc estimation method as outlined in (3.1) produces a well bounded estimate and does not exhibit drift or initialization errors as does the charge based estimation method. This method does harbors a low pass filter stage which introduces a phase lag which will also ^ . effect the estimation of the diffusion voltage, U dif 3.1.3 Impedance Voltage ^ , is the next step in the process of impedance The impedance voltage estimate, U z ^ is calculated from the difference between, U ^ and the terminal voltage, U as estimation. U z oc t shown in (3.3) and Figure 3.1.a. This impedance voltage drop, Uz, represents the series combination of the diffusion, mass transport limitations, double layer, and electrolyte voltage drops (Udif + UMT +Udl + Uele).
Once Uoc and Uz estimates are known, the process of
estimating the components of the impedance model can begin. ^ = ⋅U ^ -U U z oc t
[V]
(3.3)
120
3.2 Impedance Modeling The variations in the terminal voltage from both discharge and charge excitation are due to the intrinsic properties and limitations of the electrochemical reactions and structure. The basic model elements introduced in Section 1.1 are further developed in this section. Example waveforms and phenomena from specialized tests and typical CS-HEV operation are used to elucidate the new model forms. 3.2.1 Electrolyte Resistance The series resistance, Rele, which represents the combined series resistance of the grid, interconnect and electrolyte, should ideally be a single value for constant temperature and bulk SOC.
Experimental testing shows a different paradigm where the effective series
resistance is dependent upon the direction of the current excitation.
This variation is
probably due to the equivalent electrical model's crisp separation between the electrolyte resistance and the double layer circuit (Rct, Cdl). The charge transfer resistance, Rct, has a strong current direction dependency, and some of this effect is indistinguishable from the electrolyte resistance. The lack of extension of the double layer capacitance effect into the electrolyte would suggest that portions of Rct are not being paralleled by Cdl, thus merging with the Rele term. This document lumps those portions of Rct which are outside the shunting effects of the double layer capacitance into the now modified electrolyte resistance structure of Figure 3.3.a. Experimentally, extreme SOC operation (Q>0.8 and Q predefined level } Imt = ⋅It[k] , if { R ct
[A]
(3.8.a)
^ dR ct Imt = ⋅It[k] , if {abs[ dt [k]] > predefined level }
[A]
(3.8.b)
129 Tracking Mass Transport Current Limit The example in Figure 3.8 showed that the charge region MT current limit moved from a higher magnitude (-2.5 [pu]) to a lower magnitude (-1.5 [pu]) over a short period of time. Predicting the onset of this effect is critical to accurate performance prediction. The MT current limit is fundamentally a function of boundary SOC of the cell.
The effective
source voltage, Us, which accounts for the diffusion modified acid concentration, is an indicator of the boundary SOC.
Figure 3.9 shows how the identified MT current limit
changes with the effective source voltage, and how this correlation can be used to predict the onset of the Imt . Imt (charge region) vs. Eff. Source Voltage
VRLA, Q=0.7, T=298*K
-0.5
δImt / δUs = 80 [pu]
-1.5
-2
-2.5
-3 0.955
w/o Us comp
-1
-1.5 Imt [xIn]
MT Current Limit [xIn]
-1
Imt (charge region) vs. Time
0
-0.5
Imt [xIn]
Mass Trans Current Limit [xIn]
0
-2
-2.5
w/ Us comp
-3
-3.5 0.96
0.965
0.97
0.975
0.98
50
100
150
200
250
300
350
400
450
500
Time [sec]
a)
Voltage, Us [xUn]
Figure 3.9
b)
Time [sec]
Charge region mass transport current limit, Imt chrg, in a VRLA battery over a portion of a driving cycle; a) Imt vs. Us, b) Imt vs. time with and without Us compensation.
The MT current limit varies linearly with the local acid concentration, hence a simple slope relation is seen with respect to Us in Figure 3.9.a. The definition of the slope and how the estimate of the MT current limit is updated is defined in (3.9) and (3.10). Figure 3.9.b
130 shows how the estimate is smoothly updated with Us during a typical driving cycle with a VRLA battery at a high bulk SOC. During the driving cycle, occasional regeneration events caused the battery to enter charge MT limit region. These events led to the update of the MT current limit, and these events were predicted via the Us updated Imt chrg estimate. d Imt lim Mmt = d U s
[Ah]
(3.9)
^I mt lim [k+n] = Mmt ⋅ (Us[k+n] - Us[k] ) + I mt lim [k]
[Ah]
(3.10)
where: sample [k] represents the last update (occurrence) of a MT current limit. Note: charge and discharge current limits and slopes must be maintained separately. Mass Transport Diffusion Effect During MT limited operation, the boundary SOC becomes either locally depleted (discharge limit) or locally saturated (charge limit). This effect is experienced over a very shallow portion of the electrode-electrolyte interface region.
This localized charge
differential between the interface and the electrolyte is similar to a diffusion effect, but has a characteristic depth, δmt, which is much less than the diffusion depth, δdif. Figure 3.10 shows an hypothesis of the distribution of charge extending out from the interfacial region for a cell experiencing a charge limited MT effect. This charge accumulation effect introduces an additional R-C circuit in series with the normal diffusion effects which occur in the electrolyte.
131
Norm. SOC
Zmt
Zdif
Uoc
Umt
Qmax I
II
III
Cmt Rmt
Zdif
Qbulk
Uoc 0
δmt
δdif
MT
dist.
a) Charge vs. distance from boundary
b) Equivalent circuit
Figure 3.10 Example charge region mass transport diffusion effect. Regions:
I - Boundary layer accumulation region, (δmt < δdif , τmt < τdif) II - Diffusion region (extending out of pours into electrolyte) III - Electrolyte region (in equilibrium)
Terminal Current, Uz, Udif, & Udifmt vs. Time
Current [xIn]
Current, It [xIn]
3 2 1 0 -1 -2
80
90
100 110 120
Uz, Udif [xUn]
Voltage [xVn]
0.05
130 140 150 160 170
180
Udif
0 -0.05
Udif +Umt
-0.1 -0.15
Uz
-0.2 -0.25
80
90
100 110 120
130 140 150 160 170 180
Time [sec] Figure 3.11 Mass Transport diffusion effect in a VRLA system exposed to a driving cycle test (Q=0.70, T=298°K, Rmt=0.15, Cmt=0.052, τmt = 28 [sec]).
132 The MT diffusion effect is often witnessed after the onset of MT limited operation. Figure 3.11 shows an example of how the modeling of the MT diffusion effect improves the terminal voltage prediction performance of the equivalent circuit model. A VRLA battery was exposed to a high charging current at t=105 seconds which forced the battery into a charge region MT limit. Later, between 127 and 150 seconds, no excitation was present, and the impedance voltage remained low, contrary to a conventional model prediction which accounts only for nominal diffusion effects. The MT diffusion circuit becomes inactive after the localized charge differential has been cleared. The exclusion of the Rmt-Cmt circuit from the model is done via a virtual shorting switch which is opened by onset of the MT effect, and closed by the clearing of accumulated charge on the circuit. The control rules for the virtual switch are given in (3.11) and shown in Figure 3.10.b. MT = iff ( Umt ≅ 0 and |It| < |Imt| )
(switch closed)
[-]
(3.11.a)
MT = iff ( |It| > |Imt| )
(switch opened)
[-]
(3.11.b)
Table 3.3 quantifies the improvement experienced when MT diffusion effects are included in the equivalent circuit model of the cell. Dynamic driving cycle tests were applied to a VRLA battery with a moderately high bulk SOC (Q=0.85) such that the charge region MT limit was reached several times over the 505 seconds of the test. The comparative results show that the terminal voltage prediction errors and the effective source voltage prediction errors were reduced by ~70% for this extreme example of a battery operated in the MT limited charge region . Note: the effective source voltage Us can also include the MT diffusion voltage and it's estimate also improves as shown in Figure 3.3.
133 Table 3.3
Voltage prediction errors relative to Mass Transport modeling.
Voltage prediction
VRLA
errors absolute rel. to Uz♠
Terminal Voltage, Ut
Eff. Source Voltage, Us
non MT model
MT model
non MT model
MT model
0.0104 [pu]
0.0028 [pu]
0.0133 [pu]
0.0033 [pu]
26 %
7.0 %
33 %
8.2 %
♠ rms Ut = 0.955 [pu], rms Uz =0.040 [pu], Q=0.85, T=298°K, UsPb50/MT
3.2.4 Diffusion Effect Modeling of nominal diffusion effects present in a cell was introduced in Section 1.1.4. The diffusion impedance is a complex circuit which tries to model the effects of acid concentration gradients from the interfacial region out through the pours and separator, and into the electrolyte. This complex impedance is not fully witnessed in charge sustaining applications which achieve only moderate charge gradients. A more challenging aspect is the movement of smaller charge differentials between positive and negative gradients. This section explores the diffusion direction dependent nature of the diffusion circuit and provides methodologies of properly modeling these effects. Researchers have noted a "hysteresis" effect in the voltage relaxation characteristic of NiCd and NiMH cells. The trajectory of the terminal voltage back to equilibrium from a diffusion disturbance is different for charge and discharge perturbations. This document addresses this effect with a modified diffusion circuit model. The diffusion voltage behavior under charge and discharge operation would indicate separate diffusion impedances with different resistive and capacitive elements. Figure 3.12 shows the proposed diffusion model with singular R-C circuit for the description of the first diffusion time constant. More R-C circuits can be added in series with
134 the first in order to describe deeper diffusion effects. The diffusion parameter estimation method is shown in Figure 3.12.b. The best estimate of the diffusion voltage is filtered and sub-sampled (to maintain good parameter resolution relative to the rapidly sampled Ut, It data). The remaining recursive parameter estimation routines then run at the slower, subsampled rate.
Uoc UeleUdl Tsub
Cdif dis
RLS
It
Rdif dis
Ig
mode
a)
bdif
µ4 µ 5 sign
Rdif f -1
Cdif Udif
dis
Rdif chrg Cdif chrg
fdif
LPF
Ut
Ts,Tsub
chr
b)
Figure 3.12 Estimation of the diffusion circuit parameters; a) equivalent circuit, b) estimation block diagram with sub-sampling. The determination of weather to update the charge or discharge region diffusion circuit is determined by the presence of any pre-existing diffusion effect. Changing modes between the regions can only be done once the previous circuit has been depleted (Udif = 0). The mode change rules are shown in (3.12). mode = ⋅charge iff (It0 and Udif chrg ≅ 0 )
[-]
(3.12.b)
Figure 3.13 shows a specialized pulse test for a NiMH system. The test specifically drove the battery into a discharge diffusion state, then later into a charge diffusion state. Two long periods of rest (no excitation) reveals the relaxation voltage trajectory (diffusion voltage). The figure also shows how a singular diffusion circuit compares with that of a
135 direction dependent circuit. The direction dependent diffusion circuit is able to accurately describe the disparity in the relaxation voltage trajectories, reducing the prediction error by an order of magnitude. This effect is most prominent in NiMH cells, and may be attributed to the asymmetrical way that hydrogen is accumulated in the metal matrix in either the charge or discharge modes.
Current [xIn]
Current, It [xIn]
Diffusion Test; dual time constants and regions 5
It = 0
0
-5 0
50
Voltage [xVn]
Uz, Udif [xUn]
0.05
100
150
200
250
Uz Udif dis
0
Udif chrg
mode -0.05 0
50
100
150
200
250
Time [sec] Figure 3.13 Discharge and charge region diffusion effects in a NiMH system exposed to a pulsed power test (Q=0.65, T=298°K, Cdif dis=1.30, τdif dis=100 [sec], Cdif chrg=0.78, τdif chrg = 170 [sec], DifMH66).
3.3 Filtering and Robust Estimation Several steps were taken to obtain adequate physical parameter estimates from the sampled data.
136 3.3.1 Filtering of the Estimation Signals The recursive estimation techniques from section 3.2 applied filters to the input signals of the three estimation blocks.
These filters improved noise immunity and
encouraged quicker convergence of the estimated parameters. The 2nd order filters used in the parameter estimation algorithms are shown in Figure 3.15. Additionally, sub-sampling was used to estimate the discrete domain diffusion parameters in order to avoid high sampling rates relative to the characteristic diffusion frequency. Sub-sampling from 10 Hz to 0.025 Hz helped the discrete diffusion parameters to avoid the unit circle in the discrete domain (lack of resolution in the discrete, and hence, physical parameters).
ωsamp
ωnyq
ωbp
ωlp
dB
Electrolyte ωhp
Double Layer ωSubSamp
Diffusion
0
20
10
5.0
2.0
1.0
0.5
0.2
0.1
0.05
0.02
0.01
-40
ω [r/s]
Figure 3.15 Second order filters applied to the signals for more robust estimation. (fs = 10[Hz], sub-sampling at (fs/400), τdl = 1 sec.) 3.3.2 Ignoring Outliers Additional steps were taken to insure that the recursive algorithms were well behaved. The sample data included some events which produced obviously false parameter estimates. These outliers were ignored by the RLS routines which were designed to converge upon a reasonable value for the non-linear circuit elements. The parameter estimates were limited over a 20:1 range in order to bound any overshoot in the estimates and to encourage quicker
137 convergence. 3.3.3 Sufficient Excitation The excitation level for each of the parameter estimation blocks was monitored. Low excitation levels were ignored and the algorithms temporarily stopped updating their parameter estimates. With this technique the parameter estimates are only influenced by the less noise prone, large signal excitations, which were of interest for the prediction of high power capabilities. 3.3.4 Time variances The direction sensitive descriptions of the elements in the impedance model described the main effects present in the model. The descriptions of the model elements did not describe the non-linear effects which are slowly changing with either time, temperature, crystallization or bulk SOC. The step sizes of the RLS algorithms were selected to give acceptably quick convergence's. The time variant parameter set is trackable if it moves more slowly than the convergence rate of the recursive estimation routines. Table 3.4 summarizes the fast and slow effects present in the lumped Randles-Ershler impedance model elements.
138 Table 3.4
Impedance parameter sensitivities and time variances.
Parameter
Fast Dynamics (nonlinearity)
Slow Dynamics (time variance)
Rele
sign(It)
bulk SOC, temp., age
Rct
magnitude and sign(Ict)
boundary SOC, temp., age
Cdl
magnitude and sign(Udl)
boundary SOC, temp., age
Zdif MT
Ilim, boundary SOC
temp., age
Zdif
sign(It)
current history, temp.
Imt lim
boundary SOC
temp.
3.4 Summary The resulting cell model structure which includes the modeling advances introduced in this chapter is shown in Figure 3.16. The modes of the switches are not dependent upon the current direction (as is the case with the diodes in the electrolyte circuit), but rather the direction of accumulation of stored charge in the respective regions. The MT diffusion circuit (Rmt, Cmt) is not shown as a direction dependent effect, but there may well be differences in the way cells behave under MT limited charge and discharge regions. Udif Cdif dis Ict
Rdif dis Rdif chrg Cdif chrg
Umt
Udl Cmt Rmt
Uele Cdl dis Rct dis
Rele dis
Rct chrg Cdl chrg
It
Rele chrg
Uoc
Ut Ig
Figure 3.16 Resulting equivalent electrical model of the cell with direction dependent effects.
139 Several contributions with respect to on-line modeling of aqueous solution porous electrode cells were outlined in this chapter. 3.4.1 Electrolyte Circuit The electrolyte resistance was modeled as a current direction dependent component, primarily due to the inclusion of a portion of the charge transfer resistance effect. This new modeling form for the series resistance tends to reduce the voltage prediction errors by 20% in typical driving cycle tests. 3.4.2 Charge Transfer Resistance The methods of tracking both the high rate charge and discharge values of Rct were outlined. Additionally, a method of updating either the charge or discharge estimates for Rct during a lack of sufficient excitation in the respective regions was outlined. This method took advantage of a correlation between the effective source voltage, Us, and the relative magnitude of Rct for both charge and discharge regions. 3.4.3 Mass Transport Effect Two methods for observing the MT effect were outlined. One method was based on the rapid change of Udl, and the other method used the change in Rct vs. excitation current, Ict. These two indicators try to capture when the cell has reached the limit to transfer charge to or from the electrolyte. A Mass Transfer diffusion effect was identified and experimentally tested in VRLA batteries. The effect tends to have a time constant which is between the double layer time constant, τdl, and the first diffusion time constant, τdif. Incorporating this effect into the cell
140 model reduces the voltage prediction error by 70% when the cell is consistently operated near the charge region MT limit. 3.4.4 Diffusion Effect A discrepancy between the charge and discharge diffusion operation in NiMH cells led to the proposed direction sensitive model structure. This modeling form attempts to describe the differences in the way the charge accumulation in the metal matrix and electrolyte takes place. Experimental testing shows that the direction dependent modeling form can account for the diffusion voltage disparity.
141
Chapter 4, Power Capability Prediction 4.1 Short Term Modeling.................................................................................................142 4.1.1 General Short Term Cell Modeling.................................................................142 Inclusion of Diffusion Effects........................................................................143 4.1.2 Power Buffering Model Classifications ..........................................................143 Medium Frequency Power Buffering Model .................................................145 4.1.3 Effective Source Modeling..............................................................................145 Effective Source Resistance...........................................................................146 Effective Source Voltage ...............................................................................149 4.1.4 Effective Source Voltage Validation...............................................................150 Series Impedance Compensation ...................................................................151 Diffusion Compensation ................................................................................152 4.1.5 Evaluation of Four Model Structures ..............................................................154 4.2 Power Capability Calculations...................................................................................157 4.2.1 Voltage Limited Reactions ..............................................................................157 4.2.2 Current Limited Reactions ..............................................................................158 Mass Transport Limited Current....................................................................159 4.3 Power Capability Prediction Validation ....................................................................160 Models Used for Power Prediction Evaluations ............................................160 4.3.1 Current Limited Operation ..............................................................................160 4.3.2 Voltage Limited Operation..............................................................................163 4.3.3 Mass Transport Limited Operation .................................................................165 4.4 Summary ....................................................................................................................169
Section 1.2.2 reviewed the PNGV method for evaluating a battery's short term power capability. The series of off-line tests were used to evaluate the SOC dependent nature of the cell's effective source voltage and bi-directional effective series resistance. The cell model and analysis method is further modified for successful on-line power capability predictions with arbitrary current histories. The modifications to the equivalent electrical model of the cell, and the methods necessary to identify the components in the model with on-line data are discussed in this chapter.
142
4.1 Short Term Modeling In order to adequately predict short term power capability with the on-line data from dynamic driving cycles it is necessary to modify the off-line PNGV model of Figure 1.14.b. A short term equivalent electrical circuit model of an electrochemical cell is introduced which includes mass transport and diffusion effects to improve power prediction accuracy. The objective of the proposed model is to avoid the weaknesses of previous off-line (PNGV) model while providing sufficient modeling details for on-line power prediction techniques. 4.1.1 General Short Term Cell Modeling The proposed equivalent electrical model for short term power prediction is shown in Figure 4.1.
The model includes a diffusion impedance element and an MT diffusion
impedance element as described in Section 3.2.3. The model features the direction sensitive effective series resistance, Rs, the conventional open circuit voltage, Uoc, and predefined terminal voltage limits, Umax and Umin, as well as a current limit, Imax. The diode elements are considered to be ideal, and are used only to change the structure of the model for charge and discharge operation.
+ Uoc Figure 4.1
+ Udif Zdif
+ Umt Zmt -
Rs chrg
+ Us -
It
Umax
+ Rs dis
Ut -
Umin
Proposed on-line equivalent short term electrical model of the cell with effective source voltage, Us, and source resistance, Rs.
143 Inclusion of Diffusion Effects The history of the applied current to the cell effects the distribution of the acid concentration in the electrolyte, such that differences exist between the boundary and bulk SOC levels. The boundary SOC level highly effects the charge transfer resistance and the effective open circuit voltage. This effect is emphasized more if an MT limited reaction is reached. The diffusion and MT diffusion circuits need to be accounted for in order to obtain better power capability prediction results. In the introduced on-line effective source model, the effective source voltage, Us, is defined as the diffusion and MT modified open circuit voltage (Uoc - Udif -Umt).
To
emphasize the importance of including the diffusion effect, Figure 4.2 shows an example of how the discharge power capability is effected when in the presence of more discharge diffusion. The effective series resistance (discharge Rct) increases with discharge diffusion effects, and the diffusion modified open circuit voltage (Us) is falling, thus resulting in poorer discharge power performances. Pmax(t)
more discharge diffusion Us ≈ Uoc Us < Uoc 0
Figure 4.2
τdl
τdif
t
Maximum discharge power response with the presence of more discharge diffusion effects.
4.1.2 Power Buffering Model Classifications A general definition of the power buffering time scale relative to the natural time
144 constants within the battery is given in this section. Three general categories for power buffering batteries are outlined. The PNGV power capability tests, as outlined in section 1.2.2, experimentally determined the effective series resistance of a cell based on 18 second discharging, and 2 second charging events. The duration's of the power pulses determined what portion of the Randles-Ershler impedance structure was excited and played a significant role in determining the terminal voltage. The arbitrary duration's of power pulses in realistic battery load cycles make it difficult to define a single set of effective series resistances. In this document the effective series resistances are defined according to the general spectral characteristics of the power buffering application. Table 4.1 shows how the effective series resistance, Rs, and the effective source voltage, Us, are defined for three power buffering applications based on knowledge of the characteristic power buffing frequency, ωbuf, and the double layer frequency, ωdl. Table 4.1
† ‡
Classifications of applications and definitions of Us and Rs. Application
ωbuf
τbuf
Rs
Us
Instantaneous Peak Power
> ωdl
< τdl
Rele
Uoc + Udif + Udl
Medium Freq. Power Buffering
~ ωdl
~ τdl
Rele + Rct
Uoc + Udif
Low Freq. Power Buffering
< ωdl
> τdl
Rele + Rct + Rdif †
Uoc + Udif ‡
- the first resistive element of the Zdif description. - the voltage across the low frequency terms representing Zdif. The application's power buffering frequency, ωbuf, is defined as the highest frequency
which contains a significant amount of spectral power. This characteristic power buffering frequency, ωbuf, or alternately the characteristic power buffering time scale, τbuf, should be
145 estimated from a typical battery power trace before the final selection of an appropriate short term cell model is made. The definitions of "medium" and "low" frequencies are relative to the battery's characteristic double layer frequency, ωdl = τdl-1 = (Rct⋅Cdl)-1 (see: Section 1.2.3, Figure 1.16). Medium Frequency Power Buffering Model For typical CS-HEV applications which utilize a fast response engine (FRE) (τbuf = 2 sec), and a low energy storage battery (τdl ≈ 2 sec) a Medium Frequency Power Buffering Model (MF-PBM) should be used. The FRE will generally satisfy the positive axle power demands up to 0.5 [rad/sec] (τ ~ 2 sec), and the battery is relegated to supplying all the high frequency content (as well as all brake regeneration events). See Figure 2.19 for an example CS-HEV axle power spectral content. If the CS-HEV application features a slow response engine (SRE) (τbuf = 30 sec) and a moderate storage battery (τdl ~ 5 sec), then a Low Frequency Power Buffering Model (LFPBM) may be more appropriate (τbuf > τdl). Recent trends in CS-HEV development are favoring FRE's with low energy storage batteries (Ford P2000, Toyota Prius, Honda Insight, Dodge ESX-II), hence the MF-PBM will be explored further in detail in this document. 4.1.3 Effective Source Modeling This section describes how the short term effective source voltage, Us, and the short term effective source resistance, Rs, can be constructed or estimated from on-line impedance data as modeled in Chapter 3. The end goal is to provide accurate estimations of Us and Rs for the power prediction algorithms presented in Section 4.2. The MF-PBM is used as an
146 example framework for this section. Effective Source Resistance For the MF-PBM, the direction dependent effective source resistance can be constructed with information from the battery's electrolyte and charge transfer resistances as outlined in Table 4.1. An alternate, less computationally intensive, method of estimating Rs is now presented; the lumped resistance estimation method.
These two Rs estimation
methods are outlined below and the relative performance of the techniques are given at the end of this section. Assuming that an on-line model parameter estimation technique is active and estimates of the electrolyte and charge transfer resistances are available (Section 3.1), the effective series resistance can be defined as shown in (4.1) and Figure 4.3.a. ^ =R ^ ^ R s dis ele dis + Rct dis
(It > 0)
[Ω]
(4.1.a)
^ ^ ^ R s chrg = Rele chrg + Rct chrg
(It < 0)
[Ω]
(4.1.b)
^ R ct chrg ^ R ele chrg ^ R ct dis
^ R ele dis a)
Figure 4.3
It
^ R s chrg
Ut
HPF
÷
LPF
^ R s
It
^ R s dis
sign
b)
^ U Rs
τHPF < τbuf
chrg dis
Effective source resistance modeling methods; a) electrolyte/double layer model, b) lumped source resistance model.
An alternative method to estimating the short term resistance in the absence of electrolyte and charge transfer resistance estimates is via a lumped resistance model. This lumped resistance is directly approximated from the voltage and current data via the use of an
147 operating point model. The use of high-pass filters effectively limits the spectral range of interest (the short term resistance estimation). The equations governing the estimation of the lumped short term resistance is given in (4.2) and shown in Figure 4.4.b. ^ = LPF ⋅HPF(Ut) R s dis HPF(It)
(It-HPF(It) > 0)
[Ω]
(4.2.a)
HPF(Ut) ^ R s chrg = LPF HPF(It)
(It-HPF(It) < 0)
[Ω]
(4.2.b)
where HPF cut off frequency ≅ ωbuf , and LPF cut off frequency is < ωsamp and ≤ ωbuf. Note that the correlation of the charge and discharge regions are done via the slightly lagged current signal (It-HPF(It)) as shown in Figure 4.3.b. This lag is needed to obtain good correlation performance in the presence of fast transients between regions. Later numerical evaluation will show that the lumped resistance model is not as accurate as the more detailed electrolyte/charge transfer model. The lumped resistance model may still be used with acceptable results in high power density batteries which have minimal double layer capacitance effects and very low series resistances in general. The sample low power density VRLA battery used in this study has a large Cdl effect, and is easily driven into MT limitations, hence the lumped resistance model does not adequately describe the dynamic nature of that chemistry.
The sample NiCd and NiMH systems studied are higher
performance in nature and are able to entertain the use of the lumped effective sour resistance estimation. Figure 4.4. shows the voltage-current plane for the high pass filtered U-I signals of both NiCd and NiMH systems during a driving cycle test. The NiCd system at 60% SOC in Figure 4.4.a shows a distinct shift in the charge and discharge regions of the lumped
148 resistance estimation as well as some double layer capacitance effects (oval trajectory). The NiMH system at 50% SOC in Figure 4.4.b shows almost no difference in the charge and discharge resistances, and almost no capacitive effects, hence one can assume that the lumped Rs model may work well for this chemistry under nominal conditions (no MT limitations & nominal SOC range). Uhpf vs. Ihpf, NiCd, Q=0.60
Uhpf vs. Ihpf, NiMH, Q=50 0.15
0.15
Rs chrg 0.1
HPF Voltage [xUn]
HPF Voltage [xUn]
0.1
Rs dis 0.05
0
-0.05
0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15 -8
a)
-6
-4
-2
0
2
HPF Current [xIn]
Figure 4.4
4
6
8
b)
-8
-6
-4
-2
0
2
4
6
HPF Current [xIn]
U-I plane and lumped effective source resistance modeling in a Ni-Cd and NiMH system during a driving cycle test; a) NiCd, b) NiMH. (Q=0.6, T=298°K test=Test22s & Q=0.5, T=298°K test=PiMH50)
A comparison of the lumped source resistance and the electrolyte/charge transfer resistance modeling forms are shown in Figure 4.5 for a NiCd system at 60% SOC. The discharge region of operation shows moderate correlation between the two estimations, while the charge region of operation shows very good correlation between the two estimation methods. The sharp spikes in the electrolyte/charge transfer method is due to the fast transitions into and out of the charge and discharge regions during the driving cycle test. The charge transfer estimation routines were not as noise free as the lumped resistance model's
8
149 due to the presence of the low pass filter in 4.3.b. Often the estimate for Rct would sharply rise just before the current transitioned to the next region of operation. Thus the value for Rct in that region would not be updated until the current returned to that region of operation. 0.05
0.05
Rele chrg + Rct chrg 0.045
Rele dis + Rct dis
Eff. Source Res., Rs [xRn]
Eff. Source Res., Rs [xRn]
0.045 0.04 0.035 0.03 0.025 0.02 0.015
lumped (R s dis)
0.01 0.005 0
250
a)
300
350
400
450
Figure 4.5
0.035 0.03 0.025 0.02 0.015
b)
lumped (R s chrg)
0.01 0.005 0
500
Time [sec]
0.04
250
300
350
400
450
500
Time [sec]
Effective source resistance estimates vs. time and tracking in a NiCd system during a driving cycle test, a) discharge region, b) charge region. (Q=0.6, T=298°K test=Test22s)
Effective Source Voltage The effective short term source voltage, Us, of the cell can be estimated in many ways. Three methods are introduced and two are analyzed and compared. The spectral portion of the terminal voltage below the characteristic power buffering frequency, ωbuf, can be estimated from either the open circuit voltage source side of the Randles-Ershler equivalent electrical model, or from the terminal side. The first method begins with the open circuit voltage and removes the estimated diffusion and mass transport effects to estimate Us. This diffusion compensated method is illustrated in Figure 4.6.a and defined in (4.3).
150
^ U oc
^ U s
a) Diffusion Comp. Model
Figure 4.6
^ U Rs
^ ^ U ele Udl
^ ^ U dif Umt
Ut
LPF
^ U s
Ut
τLPF < τbuf
LPF
^ U s
τLPF < τbuf
b) Ele/Dl Comp. Model
c) Rs Comp. Model
Effective source voltage modeling methods; a) diffusion compensated model, b) electrolyte/double layer compensated model, c) lumped resistance compensated model.
^ =U ^ -U ^ ^ U s oc dif dis - Umt
^ (U dif dis >0)
[V]
(4.3.a)
^ =U ^ -U ^ ^ U s oc dif chrg - Umt
^ (U dif chrg 0)
[V]
(4.4.a)
^ = LPF(U + U ^ ^ U s t dl chrg + Uele chrg)
(It < 0)
[V]
(4.4.b)
where the break frequency of the LPF is ≥ ωbuf. 4.1.4 Effective Source Voltage Validation An essential part of predicting the short term power capability is the estimation of the effective source voltage, Us. Two methods for estimating the effective short term source voltage are experimentally compared in this section through specialized testing on three
battery chemistries.
151 The evaluation of the effective source resistance estimation is not
separately attempted. Rather the relative errors in the estimation of Rs will be witnessed in the following section with the evaluation of power capability prediction. The desired time scale of power performance prediction for the selected CS-HEV application is approximately 5-10 seconds, which is based on τbuf (Section 4.1.2). Specialized testing with occasional rest periods of 5-10 seconds (no excitation, It = 0) is used to compare the absolute and relative accuracy of the two effective source voltage estimation methods. Series Impedance Compensation The series impedance compensation method, or "Rs compensation," tries to eliminate the known dynamic events present in the terminal voltage, thus revealing the effective source voltage of the battery. This method was shown in Figure 4.6.b and defined in (4.4). The use of a low pass filter with a time constant less than the prediction time frame (~5 seconds) was used to help smooth the series impedance compensated estimate of Us. The Rs compensated methods tends to respond quickly to changes in the battery and will always converge upon the effective source voltage when the current excitation is removed (Ut approaches Us when It = 0).
The results of the specialized testing with
occasional rest periods, is given in Table 4.3. The test reveal that the method can predict the effective source voltage to within 0.26% absolute error (near limits of measurement accuracy), and 6% error relative to Uz, the overall impedance voltage.
152 Diffusion Compensation The equations describing the diffusion compensation method were given in (4.3). The diffusion compensated Us estimation method can be hampered by the sluggish update of the diffusion circuit parameter estimates and errors in the open circuit voltage estimate. An additional challenge for the diffusion compensation method is to properly track MT effects. If the MT effects are not properly identified by the parameter estimation methods, then large Us estimation errors can occur. The results of the specialized testing in Table 4.3 revealed that diffusion compensation method typically predicts Us to within 0.4% absolute error, and 8% error relative to Uz, the impedance drop voltage. This method performed slightly worse than the series impedance compensation method when predicting the effective source voltage Us. This method performed better though when used to predict the terminal voltage response of the cell.
153 Table 4.3
Prediction errors for diffusion and resistance compensated Us estimates.
RMS Voltage
NiMH
NiCd
VRLA
Prediction Errors
Terminal Voltage, Ut
Eff. Source Voltage, Us
Lumped Model
Full Model
Lumped Model
Full Model
absolute
0.0097 [pu]
0.0028 [pu]
0.0028 [pu]
0.0033 [pu]
rel. to Uz♣
24 %
7.0 %
7.0 %
8.2 %
absolute
0.0044 [pu]
0.0031 [pu]
0.0018 [pu]
0.0044 [pu]
rel. to Uz♦
8.5 %
6.0 %
3.5 %
8.5 %
absolute
0.0044 [pu]
0.0024 [pu]
0.0032 [pu]
0.0040 [pu]
rel. to Uz♥
9.3 %
5.1 %
6.8 %
8.5 %
♣ rms Ut = 0.958 [pu], rms Uz = 0.040 [pu], Q=0.50, T=298°K, test=UsPb50 ♦ rms Ut = 0.935 [pu], rms Uz = 0.052 [pu], Q=0.90, T=298°K, test=UsCd91 ♥ rms Ut = 0.917 [pu], rms Uz = 0.047 [pu], Q=0.85, T=298°K, test=UsMH85 Note: when MT effects are not modeled, the Rs model responds better than the diffusion model.
Figure 4.8 shows example voltage traces for the series impedance and diffusion compensation estimation methods while under both specialized verification testing and under MT limited operation (charge region). The most notable difference between the techniques is the presence of phase lag in the impedance compensation method due to the low pass filter. Both techniques performed well at estimating the short term effective source voltage, even in the presence of a MT limited reaction as shown in Figure 4.8.b. The diffusion compensation method included the MT diffusion model as described in Chapter 3 and was properly tuned to recognize the onset of the effect. The series impedance compensation model removed the dynamic electrolyte and charge transfer effects from the terminal voltage, thus the Us estimate still possesses the MT limitation information.
154 Effective Source Voltage Estimation
Effective Source Voltage Estimation 1.06
Ut
1
Rs Comp.
1.04
LPF(Rs Comp.)
1 0.98
0.99
Voltage [xVn]
Voltage [xVn]
1.02
evaluation point
^
Uoc
0.96 0.94
Dif. Comp.
0.92
LPF(Rs Comp.) 0.98
0.97
Dif. Comp.
0.96
0.9 0.88
MT limitation
0.95
rest periods
0.86 360
a)
380
400
420
440
460
480
Time [sec]
Figure 4.8
200
500
b)
220
240
260
280
300
320
340
360
Time [sec]
Effective source voltage estimation for a VRLA battery; a) specialized testing, b) typical MT reaction during a driving cycle test. (Q=.50, T=298°K, a) UsPb50, b) PvPb51MT) .
The effective source voltage estimate should have as little phase lag and magnitude error as possible. The effective source voltage will become one of the main factors in the following chapter's power capability calculations. An absolute accuracy of 1% is roughly a 10% error relative to the typical voltage limits applied to load leveling batteries. This 10% error will directly translate in poor power prediction capability. The following section tries to quantify the relative performances of different Us and Rs modeling forms.
4.1.5 Evaluation of Four Model Structures The two methods for estimating both the source resistance or source voltage are used to develop four equivalent short term cell model topologies. These four topologies are compared relative to the terminal voltage prediction errors and computational effort. The four compared topologies are outlined below and shown in Figure 4.7.
These do not
155 represent an exhaustive set of possible structures, rather they were chosen as a fair mix of possible topologies in order to explore the relative merits of the Rs and Us estimation techniques. •
Linear Impedance Model: this baseline topology assumes a basic Randles-Ershler structure, and does not benefit from any of the enhancements outlined in Chapter 3. Rs is estimated from the electrolyte/charge transfer elements, and Us is estimated via a diffusion compensated method.
•
Non-Linear Impedance Model: this direction dependent model has full descriptions of the effects noted in Chapter 3 including the MT effect. Rs is estimated from the electrolyte/charge transfer elements, and Us is estimated via a diffusion compensated method.
•
Lumped Us-Rs Model: this computationally simple model reduces the battery to two components given only the terminal voltage and current. Rs is estimated via the lumped model, and Us is estimated via the lumped Rs compensation method.
•
Non-Linear Impedance with Lumped Us Model: this hybrid model structure mixes estimation methods, and does not include MT effects. Rs is estimated via electrolyte/charge transfer elements, and Us is estimated via the lumped Rs compensation method. MT Dif. Us
Dif. Rs
1)Linear Impedance Model
LPF(Ut-URs) Us
Rs
2)Non-Linear Imped. Model
LPF(Ut-URs) Rs
3)Lumped Us-Rs Model Figure 4.7
Us
Us
Rs
4) Non-Linear w/ Lumped Us
Four effective source estimation topolgies used for comparison.
The terminal voltage prediction errors and the computational effort required by the
156 four effective source model structures are experimentally compared for three different chemistries in Table 4.2.
The ability of each topology to predict the dynamic voltage
behavior of the cell as it was exposed to a typical driving cycle determines the relative performance. The computational effort was evaluated by the summing the number of floating point operations (FLOPS) required to estimate all the necessary parameters each sampling period.
The sampling frequency for the driving cycle tests was 10 Hz which fell in
recommended range of 5 to 20 [Hz] for the selected batteries with double layer time constants on the order of 0.5 to 4 seconds. The separation of the Us and Rs effects was based on a power buffering frequency of ωbuf = 0.2 [rad/sec] (τbuf = 5 [sec]). Table 4.2. FLOPS and voltage prediction errors for four effective source structures. 1) Linear
2) Non-Linear
3) Lumped Rs
4) Non-Linear
Prediction Errors
Impedance
with MT effect
& Us Model
with Lumped Us
FLOPS per samp.
101
125
29
82
0.010 [pu]
0.0055 [pu]
0.018 [pu]
0.0051 [pu]
20 %
11 %
36 %
10.2 %
0.0076 [pu]
0.0029 [pu]
0.0084 [pu]
0.0069 [pu]
12.1 %
4.6 %
13.3 %
9.2 %
0.0030 [pu]
0.0024 [pu]
0.0086 [pu]
0.0058 [pu]
4.8 %
3.9 %
13.9 %
9.4 %
NiMH
NiCd
VRLA
RMS Voltage
absolute rel. to Uz♣ absolute rel. to Uz♦ absolute rel. to Uz♥
♣ RMS(Ut) = 1.003[pu], RMS(Uz)= 0.050[pu], Q=0.50, T=298°K, test=PiPb50 ♦ RMS(Ut) = 0.982[pu], RMS(Uz)= 0.063[pu], Q=0.50, T=298°K, test=PiCd50 ♥ RMS(Ut) = 0.976[pu], RMS(Uz)= 0.062[pu], Q=0.50, T=298°K, test=PiMH50
Table 4.2 reveals that the second model structure which incorporated the non-linear and direction dependent impedance effects was the best performer, but required the most FLOPS per sample. The second best performer was the fourth model structure which used
the non-linear estimates of Rele and Rct, and lumped Us estimate.
157 Both of these model
structures are used in the evaluation of the power prediction routines, see Section 4.3. The selection of effective source model structure should not be based solely on sample tests shown in Table 4.2. The whole range of excitation and operation should be considered before a final model structure is selected to insure adequate prediction performances.
4.2 Power Capability Calculations Predicting the short term power capability of the battery as described in the section 4.1 requires knowledge of the battery's effective series resistance, Rs, and equivalent source voltage, Us. The known voltage limits, (Umin, Umax), or the current limits, (Imax or Imt) bound the maximum allowed power flow of the battery. Calculating the maximum power transfer in both current and voltage limited reactions are explored in this section. 4.2.1 Voltage Limited Reactions The definition of the voltage limited short term power capabilities are given in (4.5). The Power Acceptance Index (PAI) and the Power Delivery Index (PDI) acronyms are used to differentiate between discharge and charge power capabilities of the cell. The power indices are normalized to the base quantity, Pn, as seen in the following plots. The voltage limited range of operation assumes that the series resistance presents a significant voltage drop when power is applied to the cell, such that the extreme voltage specifications are met before the cell reaches either the maximum current level, or the MT current limit. The equations governing the voltage limited power flow are given in (4.5).
158 Us⋅Umin - Umin2 Rs dis
(P > 0)
[W]
(4.5.a)
Us⋅Umax - Umax2 Rs chrg
(P < 0)
[W]
(4.5.c)
Pdis max = ⋅
Pchrg max =⋅
PDI =
Pdis max Pn
[pu]
(4.6.a)
PAI =
Pdis max Pn
[pu]
(4.6.b)
Figure 4.9 shows the block diagram for the calculation of the PAI and PDI metrics. The diagram shows the system limits (Umax, Umin, Imax, Imt) and the estimated high rate series ^ ^ resistances (R s chrg, Rs dis). Imt chrg
Umax -
^ U s
^ R s chrg -1
-Imax
PAI
+
max
^ R s chrg
^ U s
+
PDI -
Umin
^ -1 R s dis
Imt dis
a) voltage limited reactions Figure 4.9
Imax
+ +
^ R s dis
-
PAI PDI
min
b) current limited reactions
Normalized power index estimation block diagrams using estimated source voltage and resistance information.
4.2.2 Current Limited Reactions A second set of relations are needed to describe the power flow of the cell when the series impedance voltage drop is small, or when MT current limits are low. Under these circumstances the limitations on power flow is govern by a current limited reaction and given
159 in (4.7). Pdis max =⋅Imax dis·(Us - Imax dis·Rs dis) †
(P > 0)
[W]
(4.7.a)
Pchrg max = ⋅Imax chrg·(Us - Imax chrg·Rs chrg) ‡
(P < 0)
[W]
(4.7.b)
† ‡
where Imax dis is the least positive of Imax and Imt dis. where Imax chrg is the least negative of -Imax and Imt chrg.
The selection of the "maximum" current is dependent upon the relative magnitude of MT limited current, Imt, and the system current limit, Imax. Both charge and discharge current limits must be evaluated separately. A typical current limit selection procedure is graphically shown in Figure 4.10. Imt dis Imax
time
-Imax Imt chrg
Figure 4.10 Selection of directional current limits based on maximum system current and mass transport current limitations. Mass Transport Limited Current One should note that the MT limited operation actually manifests itself as a dramatic increase in the charge transfer resistance, Rct. This sudden increase in the series resistance of the battery then results in a dramatic voltage drop or rise, hence a MT limited event will actually result in a voltage limited power flow reaction. The current limited relations of (4.7) are still used in order to predict the maximum power flow just before the onset of the MT
160 limited effect.
4.3 Power Capability Prediction Validation The calculated power capability predictions are compared with the actual short term power performances from a set of specialized tests. The specialized tests occasionally drove the battery to artificially high power levels in both the charge and discharge directions. The power flow was then limited if either a voltage or current limit was reached. Three operating conditions are compared; voltage limited, current limited, and MT limited. Models Used for Power Prediction Evaluations Model structure (2) and (4) from Section 4.1.5 (see Figure 4.7) were used to predict the short term power capability of the battery. Both structures used the direction dependent and non-linear electrolyte and charge transfer resistance estimates to calculate the effective series resistance of the battery.
Model structure (2) used a diffusion and MT structure to
estimate the effective source voltage, Us, from the open circuit voltage, Uoc (diffusion compensated model). Model structure (4) used the series resistance voltage drop information to estimate Us from the terminal voltage Ut (Rs compensation model). These two modeling methods are compared in the following tables. 4.3.1 Current Limited Operation A specialized driving cycle test was applied to three battery chemistries operating with a set of wide voltage limits [Umin=0.8, Umax=1.15], and an artificially low system current limit, Imax=2 [pu]. The tests drove the batteries into several current limited reactions during the course of a typical driving cycle. An example power flow test for a VRLA system is
161 shown in Figure 4.11. The short term power prediction envelopes are seen in the third trace of Figure 4.11. The solid envelope is the prediction of the non-linear impedance model with diffusion compensated Us estimate. The dashed envelope is the power prediction of the lumped Us model with non-linear impedance estimate for Rs.
Both methods were easily able to track
the current limited power capability of the battery. The relations governing the maximum power flow, (4.7), show that maximum power is relatively insensitive to the series resistance of the battery. Rather the maximum current limit is the dominant factor in estimating the maximum power flow. The mean power prediction errors for the current limited power flow experiments are shown in Table 4.4. Both short term power prediction modeling methods featured very low mean errors.
162 Power Capability Prediction, Current Limited
Voltage [xUn] Voltage [xVn]
2
Power Power[xPn] [xPn]
1.1
Current [xIn] Current [xVn]
1.2
1 0.9 0.8 0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
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Time [sec] Figure 4.11 Power capability prediction experiment with current limited power flow, for a VRLA system (Q=0.50, T=298°K, test=PiPb50). Table 4.4
Mean power prediction errors under current limited operation
mean Power
Power Delivery, PDI
Power Acceptance, PAI
prediction errors
Lumped Model
Full Model
Lumped Model
Full Model
VRLA♣
0.8 %
0.3 %
2.6 %
0.9 %
NiCd ♦
0.7 %
0.8 %
0.4 %
0.6 %
NiMH ♥
0.3 %
0.5 %
0.4 %
0.5 %
♣ mean PDI = 1.83[pu], mean PAI= -2.11[pu], Q=0.50, T=298°K, test=PiPb50 ♦ mean PDI = 4.22[pu], mean PAI= -5.20[pu], Q=0.50, T=298°K, test=PiCd50 ♥ mean PDI = 4.22[pu], mean PAI= -5.12[pu], Q=0.50, T=298°K, test=PiMH50
163 4.3.2 Voltage Limited Operation A specialized driving cycle test was applied to three battery chemistries operating with a set of artificially narrow voltage limits [Umin=0.9, Umax=1.02], and nominal current limit, Imax = 6[pu]. The tests drove the batteries into several voltage limited reactions during the course of a typical driving cycle. A voltage limited power flow test for a NiCd system is shown in Figure 4.12. The short term power prediction envelopes are seen in the third trace of Figure 4.12. The two models produced slightly different power prediction envelopes as the estimates for the effective source voltage varied throughout the cycle. Voltage limited power flow is more difficult to predict than current limited as seen in the relations of (4.5). The power flow is highly dependent upon both Us and Rs. Any errors in these terms results in poor predictions. Additionally, during the application of the maximum power flow pulse (~5 sec), the values for Us and Rs are changing due to the effects of extreme power flow, thus resulting in a dynamic prediction error addressed in the following sub-section. Table 4.5 shows the 10 second power prediction errors for the voltage limited reactions. The 10 second prediction errors in Table 4.5 were calculated by the comparison of the predicted power capability at the onset of the power pulse, with the actual maximum battery power at the end of the power pulse.
164 Prediction of Power Capability Voltage [xVn] Current [xVn]
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5 0 -5
5 Power [xPn]
Power [xPn]
Current [xIn]
Voltage [xUn]
1.05
0 -5
Time [sec] Figure 4.12 Power capability prediction experiment with voltage limited power flow, for a NiCd system (Q=0.50, T=298°K, test=PvCd50).
165 Table 4.5
Mean power prediction errors under voltage limited operation
mean Power
NiMH ♥
NiCd ♦
VRLA♣
prediction errors
Power Delivery, PDI
Power Acceptance, PAI
Lumped Model
Full Model
Lumped Model
Full Model
10 sec
8.1 %
10 %
38 %
28 %
Instant.
5.6 %
11 %
16 %
8.1 %
10 sec
22 %
8.8 %
18 %
31 %
Instant.
20 %
3.1 %
10 %
1.5 %
10 sec
24 %
24 %
18 %
10 %
Instant.
15 %
1.5 %
9.1 %
2.2 %
♣ mean PDI = 2.13[pu], mean PAI= -0.968 [pu], Q=0.50, T=298°K, test=PvPb51MT ♦ mean PDI = 2.31[pu], mean PAI= -2.18 [pu], Q=0.50, T=298°K, test=PvCd50MT ♥ mean PDI = 2.52[pu], mean PAI= -2.55 [pu], Q=0.50, T=298°K, test=PvMH51MT
Temporal Changes in Us and Rs The 10 second power prediction errors for the voltage limited reactions is poor. This error originates from three sources; temporal changes in Us, temporal changes in Rs, and the artificially narrow voltage limits. These errors sources are expanded below and further addressed in Chapter 6 as suggestions for improvements. During the 10 second power pulse tests, the value of the effective source voltage changes by approximately 1% (absolute scale) for both the VRLA, NiCd and NiMH experiments. This variation is due to the extreme power level during the pulse effecting the diffusion circuit. An additional weakness in the voltage limited reaction testing was the use of abnormally narrow voltage limits to insure voltage limited reactions. The net effect is a large change in the voltage headroom during the power pulses as shown in Figure 4.13. The
166 voltage headroom changes experienced in the VRLA experiments was approximately 6%, and in the NiCd and NiMH experiments was approximately 14%. The 10 second power capability errors in Table 4.5 did not account for these variations.
Current [xIn]
Voltage [xUn]
Prediction of Power Capability 1
voltage headroom
0.95 0.9
Us
Umin
5 0 -5
Power [xPn]
4 3 2
lumped model PDI
1 355
full model PDI 360
365
370
375
Time [sec] Figure 4.13 Close-up detail from the voltage limited test of Figure 4.12 showing variation in the effective source voltage and voltage headroom. Figure 4.14 shows how the effective source resistance changed during the same 10 second interval for both the lumped Rs model and full model (Rele + Rct). These resistance estimates typically changed by 5-10% over the power pulses for the three chemistries. These errors are directly transferred into the power prediction errors. The net series contribution of the voltage and the resistance changes represent 11-25 % error, hence the instantaneous power prediction errors of Table 4.5 are significantly reduced.
167 0.022
0.021
Rele dis + Rct dis
Resistance [xRn]
Resistance [xURn]
0.02
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0.017
Rs dis
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0.015
355
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Time [sec] Figure 4.14 Temporal change in effective source resistance estimates during the specialized 10 second pulse tests of Figure 4.12. The circles in Figure 4.13 show the values where the PDI predictions were sampled before the 10 second test, and where the actual 10 second power capability of the battery was sampled. A second error evaluation is included in Table 4.5 which evaluates the relative power prediction error at the end of the 10 second test. These prediction errors do account for the dynamic changes in the voltage head room and Rs values. 4.3.3 Mass Transport Limited Operation Mass Transport limited operation mimics the voltage limited reaction of the previous section even though the calculation is based upon the predicted Mass Transport current limit, Imt. A VLRA battery with high SOC (Q=0.7) was subjected to the power prediction testing and several charge region MT limited reactions were experienced as shown in Figure 4.13.
168 Prediction of Power Capability 1.2
Ut
Voltage [xVn]
Voltage [xUn]
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Time [sec]
Time [sec] Figure 4.15 Power capability prediction experiment with mass transport modeling, for a VRLA system exposed to a modified driving cycle (Q=0.70, T=298°K, test = Test17h). The MT diffusion voltage, Umt as outlined in Section 3.2.3, was included in the estimation of Us and has been included in the first trace of Figure 4.15. Umt is seen to have a profound impact on the estimate of Us. The charge region power is limited by the charge region mass transport current limit and it's associated effect on Rct and the terminal voltage. The ability of the power prediction routines to dynamically predict the maximum charge region power flow is documented in Table 4.6.
169 Table 4.6
PAI prediction errors relative to charge region MT modeling.
mean PAI errors
Lumped Model
Full Model
Full Model w/o MT
VRLA
36 %
25 %
48 %
* mean PAI test level =-0.968 [pu] VRLA system, Q=0.70, T=298°K, Test17h
The short term power prediction model with diffusion compensated Us estimate featured the least error. Both the Rs compensated model and the diffusion compensated model without MT effects delivered poorer results. The Rs compensated model does feature the ability to partially track MT effects due to the low pass filter of the terminal voltage, but is clearly not as good as the former diffusion compensated model with MT effects included.
4.4 Summary Several key contributions with respect to on-line power capability prediction were outlined in this chapter. Three equivalent source model classifications were introduced which are based on the relative magnitude of the battery's characteristic double layer frequency to the application's power buffering frequency.
The effective short term resistance and the effective source
voltage of the battery were defined for each of the above three classifications and special attention was given to the Medium Frequency Power Buffering Model. Several methods for estimating the effective source resistance and voltage were introduced. The relative merits of each approach were evaluated relative to the computational load and voltage prediction error for three battery chemistries. In general, the estimation of the effective source resistance, Rs, was best accomplished with the direction dependent and non-linear descriptions of the electrolyte and charge transfer resistances. The estimation of
170 the effective source voltage was adequately accomplished via two means, both having strengths and weaknesses relative to their ability to estimate Us under abnormal conditions (rest periods, or MT limited operation). The relations governing the maximum power flow of the battery given the known voltage and current limits of the system were introduced. Current limited operation of the battery led to accurate power predictions on the order of the measurement error. Voltage limited operation of the battery led to power prediction errors on the order of 10-20% error partially due to the sensitivity of the series resistance estimation and the dynamic nature of the effective source voltage, Us.
Mass Transport limited operation of the battery led to
power prediction errors on the order of 25% error, largely due to the dramatic change in the battery characteristics during MT operation.
171
Chapter 5, Weighted Effective Efficiency 5.1 Efficiency Calculations ..............................................................................................171 5.1.1 Source or Sink Centric Energy Efficiencies ....................................................172 5.1.2 Instantaneous Power Efficiency ......................................................................173 Effects of Diffusion........................................................................................175 5.1.2 Weighted Efficiency ........................................................................................176 Power Histogram and Energy Accumulation.................................................177 Forget Factors for the Power Histogram........................................................179 Updating the Histogram and Weighted Efficiency ........................................179 5.2 Weighted Efficiency Evaluation ................................................................................179 5.2.1 Controlling Battery Efficiency ........................................................................181 5.3 Summary ....................................................................................................................182
The non-linear and direction sensitive impedance modeling techniques outlined in Chapter 3 lead directly to a battery's power processing efficiency. This chapter presents methods of evaluating the instantaneous efficiency characteristics and an energy weighted effective efficiency of the battery. The objective of this chapter is to provide a meaningful battery operating efficiency metric for the load-leveling application controller.
5.1 Efficiency Calculations The equivalent electrical model of Figure 3.16 is used to calculate the short term efficiency as a function of battery current. The effective series resistances in the model are the loss elements, and the open circuit voltage is the power source (or sink during charging). One must note that the model also possesses energy storage elements in the diffusion, mass transport, and double layer circuits. The following sub-sections introduce how the energy storage and loss elements help determine the effective operating efficiency of a battery
172 system. 5.1.1 Source or Sink Centric Energy Efficiencies The energy efficiency, ηeng, of a power buffering battery system as shown in Figure 5.1 can be defined in two ways.
Energy source and energy sink centric relations are
presented in (5.1). The two relations differ in numerical values when a net change in stored energy of the battery, ∆Estorage, is realized over the evaluation period. Psource
Psink Pstorage
Figure 5.1
Power flow diagram for a power buffering battery application. Esink+∆Estorage Esource
ηeng =
(source centric)
[-]
(5.1.a)
ηeng = E
(sink centric)
[-]
(5.1.b)
Esink source-∆Estorage
Both energy efficiency relations have peculiar attributes; •
(5.1.a) tends to return large negative results if the battery contributes significantly to the sink, i.e. Esource
