Empowering the performance of advanced NMPC by multi-parametric programming – An application to a PEM fuel cell system Chrysovalantou Ziogou1,2, Efstratios N. Pistikopoulos3, Michael C. Georgiadis*,1,2, Spyros Voutetakis1, Simira Papadopoulou1,4
1
Chemical Process and Energy Resources Institute (CPERI), Centre for Research and
Technology Hellas (CERTH), PO Box 60361, 57001 Thessaloniki, Greece 2
Department of Engineering Informatics and Telecommunications, University of Western
Macedonia, Vermiou and Lygeris str., 50100 Kozani, Greece 3
Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial
College London, SW7 2AZ London, UK 4
Department of Automation, Alexander Technological Educational Institute of Thessaloniki, PO
Box 141, 54700 Thessaloniki, Greece
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ABSTRACT Fuel cell (FC) systems are part of a prominent key enabling technology for achieving efficient and carbon free electricity generation and as such their optimum operation is of great importance. This work presents the combination of two advanced model predictive control (MPC) methodologies to guarantee the optimal operation of a Polymer Electrolyte Membrane (PEM) fuel cell system. More specifically at the core of the proposed framework is a Nonlinear MPC (NMPC) formulation that solves online a nonlinear programming (NLP) problem using a simultaneous direct transcription optimization method. The performance of the NLP solver is enhanced by a warm-start initialization and a search space reduction (SSR) technique. A piecewise affine (PWA) approximation of the variable’s feasible space is used to define the boundaries of the search space computed offline using a multi-parametric Quadratic Programming (mpQP) method. The proposed unified framework is developed and deployed online to an industrial automation system. The response of the multivariable nonlinear controller is assessed through a set of experimental studies, illustrating that the control objectives are achieved and the fuel cell system operates at a stable environment regardless of the varying operating conditions. KEYWORDS: nonlinear model predictive control, fuel cell system, online implementation, dynamic optimization, multiparametric control
1. Introduction Fuel cell systems are a prominent and emerging technology that can be used for carbon free electricity production. Among the various fuel cells categories, distinguished by their electrolyte, the Polymer Electrolyte Membrane (PEM) fuel cell dominates in terms of shipments per year1, as
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its flexibility makes it appropriate for a range of markets, such as the transport sector, small stationary power applications and lately consumer electronics. The main features of a PEM fuel cell is its fast response, the low operating temperature, high efficiency and low corrosion2. Overall the operation of a fuel cell involves various phenomena related to electrochemical reactions, mass and heat transport. Therefore it is necessary to design and develop a control system able to handle a number of issues regarding power management, in the context of a safe operation which is achieved by proper fuel/air delivery and temperature control. Moreover, the need for control strategies is regarded as an important component of the Balance of Plant (BOP) for an integrated fuel cell system3. Over the past few years significant developments have been made in the area of fuel cell control. The output power of the fuel cell is controlled by power conditioning devices such as DC/DC converters or in the case of fuel cells with low output power, used for experimental purposes, by an Electronic load. The power output can be regulated by switching rules4, adaptive control5, an adaptive peak seeking controller6, nonlinear internal model control7 or model predictive control8,9. The problem of preventing air starvation is extensively studied and a number of control configurations have been suggested. The safe operation is guaranteed by proper air management, an objective which can be accomplished by a feedback scheme10, a reference governor11, a static feedforward structure12,13, adaptive control algorithms14,15 or model predictive control schemes16,17. Regarding thermal management various contributions have been reported that address the temperature objective using control schemes, such as PI controllers18,19, a Variable Structure Control with a rapid-convergent sliding mode controller20, a fuzzy based approach with a genetic algorithm21 and multi-parametric MPC controllers22.
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Besides the numerous works that perform simulation-based design and analysis of the behavior of the fuel cell system, there is a need for experimental verification through online deployment of the various proposed control schemes23. A number of experimental studies can be found in the literature that exemplify the use of various control configurations focusing on one objective at a time. The safety of the system which is maintained by avoiding air starvation has been experimentally achieved by an adaptive controller24, a sliding mode controller25,26, a super twisting algorithm27, a combined mpMPC with a PI control scheme28 and an NMPC approach based on a Volterra series model29. On the other hand, the temperature management can be achieved by a PI loop30 or an explicit MPC controller31 while the pressure is controlled using an adaptive state feedback scheme32. Most of these experimental control studies focus and analyze only one control objective while Ziogou et al.33 developed a multivariable NMPC scheme that concurrently handles the issue of power management in conjunction with the starvation avoidance objective which was experimentally verified. Another interesting work addresses the issue of maximum efficiency as a function of stoichiometry, temperature and humidity using an adaptive control scheme34. Overall the dynamics of the PEM fuel cell are highly nonlinear and are strongly affected by the operating conditions35,36. Recently the NMPC or receding horizon control has become the preferred strategy for a number of applications37 as it is able to accomplish simultaneously multiple objectives under intense operating conditions. Furthermore a noticeable improvement has been reported on online optimization algorithms38. Based on these facts an advanced control scheme that handles input constraints and fulfills economical and operating objectives is a suitable choice for an integrated fuel cell system.
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While most of the previous literature contributions focus on a single control objective of the fuel cell system, this work considers a multi-variable control problem including several often conflict objectives. Therefore the contribution of this work is twofold. First a novel synergetic strategy between two well established control techniques (mpMPC and NMPC) is explored, aiming at the reduction of the computational time without compromising the accuracy of the obtained solution. Secondly the optimal operation of the fuel cell is experimentally verified using the proposed integrated framework at nominal and start-up conditions. The online multivariable controller which is deployed to an industrial automation system utilizes a full dynamic nonlinear model of the fuel cell and its behavior is validated under various operating conditions. The paper is organized as follows: Section 2 presents the experimental fuel cell unit, followed by a brief analysis of a dynamic nonlinear model and presentation of the control objectives. The subsequent section outlines the synergetic explicit Nonlinear MPC framework along with the details of the employed optimization methodology. Section 4 presents a detailed analysis through a set of experimental case studies. Finally Section 5 draws up concluding remarks of this work.
2. PEM Fuel Cell System – Experimental Setup and Dynamic Modeling In this work a small-scale fully automated fuel cell unit is used for the demonstration of the proposed controller. The integrated unit was designed and constructed at CPERI/CERTH and it has four main subsystems that handle the produced power, the gas supply, the temperature of the fuel cell and the humidity of the inlet gasses (Fig 1).
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Figure 1. Experimental Fuel Cell Unit The PEM fuel cell is by Electrochem® , has a Nafion 1135 (89μm thickness) membrane electrode assembly (MEA) with an active area of 25cm2, catalyst layer with platinum loading 1mg/cm2 20wt% Pt/C while the power is drawn from the fuel cell through a DC load (KIKUSUI®). The air and the hydrogen are supplied from pressurized cylinders using two mass flow controllers (MFCs). A set of hydrators along with heated lines are used to maintain the proper humidity of the inlet gases. Finally the temperature is maintained by an air cooling subsystem and an electrical heating resistance. Although each subsystem has its own control objective, interactions with other subsystems also exist. Therefore it is important to measure a variety of variables in order to monitor and control the system’s operation. These variables constitute the unit’s Input/Output (I/O) field, acquired by a Supervisory Control and Data Acquisition System (SCADA) system by GE Intelligent Platforms®. The available measurements as shown in Fig. 1 are: the flow from the MFCs ( m air ,in , m H 2,in ), the temperature of the hydrators ( Th,ca , Th ,an ), the line temperature, before and after the fuel cell ( Tca ,in , Tan ,in , Tca ,out , Tan ,out ) and the fuel cell temperature ( T fc ). Also the inlet and outlet pressure ( Pca ,in , Pan ,in , Pca ,out , Pan ,out ) are measured, along with the pressure difference between the anode and the cathode ( P ). Finally the DC load provides the measurement of fuel
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cell voltage ( V fc ). Furthermore a number of control actions are applied regarding the flow of the gases ( m air , m H 2 ) the heat-up and the cooling percentage ( xht , xcl ) and the current ( I fc ) applied by the DC converter which are derived based on the advanced model-based control framework presented in Section 3. Apart from these control actions the temperature of the hydrators and the inlet lines are maintained by independent PI controllers. One of the advantages of using an industrial automation system is that the data acquisition modules are also industrial grade. Such hardware components are properly manufactured in order to have a stable and consistent behavior through time. However, the uncertainty which is inserted by the sensors, the cabling noise and the nature of process could cause performance deterioration, especially when a linear controller is applied to a nonlinear process. In our case a nonlinear approach is used that incorporates a model that thoroughly describes the dynamic behavior of the fuel cell unit into consideration.
2.1 Nonlinear Dynamic Semi-empirical Model The behaviour of the aforementioned unit is described by an experimentally validated nonlinear dynamic model that couples a set of first principles equations with a set of semi-empirical equations. For reasons of completeness a brief description of the main equations is presented here while a detailed analysis can be found in our previous work39. The mass balances at the cathode’s gas flow channel are:
dmO 2,cach dt
dmN 2,cach dt
dmv ,cach dt
m O 2,cach ,in m O 2,cach ,out M O 2
I 4F
m N 2,cach,in m N 2,cach ,out
m v ,cach ,in m v ,cach,out m evap ,cach Afc M H 2O N v ,k
(1) (2) (3)
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where mO 2,cach , mN 2,cach , mv ,cach are the oxygen, nitrogen and water vapor masses at the cathode, m k ,cach ,in , m k , cach ,out , k [O2 , v ] , are the input and output mass flows, m evap ,cach is the rate of evaporation, N v ,ca is the vapor molar fluxes between GDL and the cathode, M O 2 , M H 2O are the oxygen and
water molar mass, I is the current, Afc is the membrane active area and F is the Faraday number. Similarly the mass balances for the anode channel are derived. The partial pressure of water vapor in the cathode GDL ( pv ,caGDL ) satisfies the respective mass balance equation:
I RT fc N v ,mem N v ,ca / GDL 2 FA fc
dpv ,caGDL dt
(4)
where δGDL is the thickness of the diffusion layers and N mem is the vapor molar flux at the membrane affected by the electro-osmotic drag (Nv,osm) and the back diffusion (Nv,diff) between the cathode and the anode:
N v ,mem N v ,osm N v ,diff
(5)
The dynamics of the temperature ( T fc ) results from the overall energy balance equation of the fuel cell:
m fc Cp fc
dT fc dt
an ca chem Qrad Qamb QR Qcl Pelec
(6)
where m fc denotes the mass of the fuel cell and Cp fc is the specific heat calculated for the system into consideration. The above equation takes into account the differences of the energy
) and the cathode ( ), the flow rates between the input and output streams at the anode ( an ca
rate of energy produced by the chemical reaction ( chem ), the rate of energy which is released to the environment through radiation ( Q rad ) and the rate of heat losses to the environment ( Q amb ).
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Also the heat supplied by the heating resistance ( Q R ) is included along with the heat which is removed by the air cooling system ( Q cl ). The last term Pelec is the amount of energy which is converted to electrical power. To determine the voltage ( V fc ) and subsequently the produced power of the fuel cell, the following equations are used:
V fc Enerst Vact Vohm Vconc
(7)
Vact 1 2T fc 3T fc ln( I ) 4T fc ln(cO 2 )
(7a)
Vohm (5 6T fc 7 I ) I
(7b)
Vconc 8 exp(9 I )
(7c)
where Enerst is the ideal Nernst voltage, Vact, Vohm, Vconc are the activation, ohmic and concentration losses and co2 is the oxygen concentration. In (7a)-(7c) k , k 1..9 represent
experimentally defined parametric coefficients39. This nonlinear dynamic model is used at the core of the proposed synergetic control scheme which is analyzed in Section 3. The described dynamic model is not system dependent as it can be adjusted to describe any other PEM fuel cell system by performing a number of sequentially executed actions. These actions are related to the determination of the physical characteristics of the PEM fuel cell, the unit specific parameters and finally the empirical parameters utilized by the electrical subsystem of the model. The value of the empirical parameters can be specified by a systematic parameter estimation procedure as long as experimental data or data from the manufacturer are provided. Finally the resulted estimated values are included to the model and a different set of data is necessary to explore the validity of the model against the fuel cell unit.
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2.2 Operation and Control Objectives
The durability and performance of a PEM fuel cell are influenced by the operating conditions. Therefore it is of vital importance to control the various subsystems responsible for maintaining a stable operating environment while ensuring an economically attractive operation. Overall the primary objective for the control system is to effectively address the issue of power generation in an optimum manner. In this context the optimality is defined by operating at a safe region, while minimizing fuel consumption and maintaining stable temperature conditions. It is important to prevent fuel and oxidant starvation as it affects the longevity of the fuel cell and can cause irreversible damage to the membrane40. Another significant factor that should be considered is the proper maintenance of the operating temperature, as it affects the long-term performance of the fuel cell. The operation at an elevated temperature accelerates the degradation phenomena and can influence the durability of the fuel cell. Furthermore the temperature directly affects the rate of chemical reactions and transport of vapor and reactants18. As the hydrators, used for the humidification of the gases, operate at a stable temperature point, it is important to maintain the fuel cell operating temperature, in order to keep the water content of the gases at a desired level.
Controller
Overall these four distinct goals are accomplished by the control configuration of Fig. 2.
Figure 2. Control configuration
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The desired power ( Psp ) is delivered by properly manipulating the current ( I ) which is applied to the FC by the converter (DC load) connected to the system. To avoid starvation caused by sub-stoichiometric reaction conditions the cathode and anode excess ratios ( 2 , H 2 ) are controlled and expressed as the ratio of the input flow of each gas to the consumed quantities due to reaction10: 2
H 2
m O 2, cach ,in M O2 (I / 4F ) m H 2, anch , in M H 2 (I / 2F )
(8a) (8b)
The safety of the operation is ensured by maintaining the excess ratios above one ( 2, SP 1, H 2, SP 1 ). Another prerequisite for the controller, related to the gas supply subsystem, is to minimize the surplus of the gasses. The goal is to reduce the amount of air and hydrogen which will not be consumed by the reaction, for the given operating conditions, based on the requested power drawn from the fuel cell. An open loop analysis was performed considering current changes for various stoichiometries for the anode and the cathode. Moreover, an additional design constraint is imposed to account a physical limitation of the hydrogen mass flow controller used in the unit (flow rate > 180cc/min). For each power-excess ratio curve the minimum feasible point of the excess ratio is determined. The set of all these points form a path approximated by a 4th degree polynomial: f H 2 ( Psp ) 0.045PSP4 0.661PSP3 3.72 PSP2 10.02 PSP 12.43
(9a)
f O 2 ( Psp ) 0.065 PSP4 0.896 PSP3 4.88 PSP2 12.29 PSP 14.6
(9b)
These polynomials are used to define the optimum oxygen and hydrogen excess ratio value which is applied online in a feedforward manner to the controller ( H 2, sp f H 2 ( Psp ) ,
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O 2, sp f O 2 ( Psp ) ). Once 2, sp , H 2, sp are determined by the feedforward mechanism, the setpoint are reached by manipulating the air and hydrogen flows ( m air , m H 2 ). The power generation and the starvation avoidance objective can be achieved by control actions that aim at an accurate set-point tracking of PSP , 2, SP , H 2, SP . On the other hand the temperature control involves two mutually exclusive subsystems, one for the heat-up and another for the cooling; therefore a number of requirements are considered besides the set-point tracking ability:
Maintain the temperature at the desired level having an allowable deviation of +/- 1C Avoid concurrent operation of the heat-up and cooling Exhibit stable operation Avoid large overshoot after step changes The heat-up and the cooling are controlled by manipulating the operating percentage of the heating resistance ( xht ) and of the cooling fans ( xcl ) of the system, respectively.
3. Advanced Model-Based Control Framework Based on the above considerations this work utilizes a combined control framework of two MPC-based strategies in order to take advantage of their synergistic benefits and overcome their limitations. The first methodology is an online Nonlinear Model Predictive control (NMPC) strategy, based on the detailed nonlinear model, which is very appealing due to its ability to handle dynamic nonlinearities of the process under consideration41. NMPC is the main controller of the system. The second methodology is an explicit or multi-parametric Model Predictive Control (mpMPC) strategy. It can provide the optimal solution in real-time, as the solution is
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computed offline and can be located online by simple look-up functions42,43. Its purpose on the proposed synergetic framework is to aid the NMPC controller to reduce its computational requirements. Both methodologies are able to handle state and input constraints and can achieve a set of predetermined objectives.
3.1
Nonlinear Model Predictive Control
Nonlinear Model predictive control (NMPC) is part of a family of optimization-based control methods44,45
that involves the online solution of a constrained finite-time optimal control
problem over a prediction horizon (Tp) partitioned into Np intervals. The optimization yields an optimal control sequence (uk .. uk+Nc) over a control horizon (Tc) partitioned into Nc intervals and the first control action (uk) is applied to the system. The following formulation of the NMPC problem44,46 is considered: Np
Nc 1
j 1
l 0
min J ( yˆ k j ysp ,k j )T QR ( yˆ k j ysp ,k j ) ukTl R1uk l s.t.:
x f d x, u ,
y g x, u
(10) (11a)
ek ( y pred ymeas ) k
(11b)
yˆ k j y pred ,k j ek
(11c)
where u,y,x are the manipulated, the controlled and the state variables, y pred , ymeas , ysp are the predicted, the measured variables and the desired set-points and QR, R1 are the output tracking and the input move weighing matrices. The minimization of functional J (eq. 10) is also subject to constraints of u, x and y. In the current work a direct optimization simultaneous method is selected in conjunction with an augmented Lagrangian solver47 and the problem formulation involves a direct transcription method based on orthogonal collocation on finite elements (OCFE) 48,49 for the discretization of
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the model. The input, state and output variable profiles are approximated with a family of Lagrange polynomials on finite elements ( i 1..NE ). Within each finite element these dynamic profiles are further discretized around collocation points ( N cop ), determined as the shifted roots of Legendre polynomials: Ncop
x
x(t )
i, j
j 0
Ω j (t )
j (t ), i 1..NE , j 0..N cop , t [ti , ti 1 ]
Ncop
(t ti ,k )
k 0, k j
(ti , j ti ,k )
(12)
(13)
where t is the scalar independent dimension defined in the fixed domain [0, t f ] , xi , j is the value of the state vector at collocation point j of the ith finite element and is the basis function. Respectively the algebraic variables ( z i , j ) and input variables ( u i ) are approximated. The length of each element is hi ti ti 1 . After this discretization the optimization problem is expressed as a large-scale but sparse NLP problem50,51,52:
x
min i, j i, j ,z
NE Ncop
,u i
w x i 1 j 1
Ncop
s.t. :
k 0
k
i, j
i, j
, z i , j , u i , i 1..NE , j 1..N cop
( i , j ) xi ,k =hi f d u i , xi , j , z i , j
0=f a u i , xi , j , z i , j x1,0 x0 , x(t f ) xi ,0
Ncop
x j 0
i 1, j
Ncop
x j 0
(14) (15a) (15b)
NE , j
j (1)
j (1), i 2..NE
xl xi , j xu , zl z i , j zu , ul u i uu
(15c) (15d) (15e)
where is the objective function, f d and f a are the differential and the algebraic equations. To enforce zero-order continuity of the state variables at the element boundaries, connecting equations are used (15d). The NLP problem (14) should be solved within the period defined by
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the sampling time of the system in order to avoid delays and deterioration of the control performance53.
3.2 Multi-parametric Model Predictive Control Explicit or multi-parametric MPC (mpMPC) is a multi-parametric programming technique that can compute off-line the control law for a given optimization problem using a piecewise affine (PWA) approximation defined over a polyhedral partition of the feasible variable space. The optimization problem is solved off-line and the control actions are obtained as functions of the parameters of the process and the regions in the state/output space where these parameters are valid, as a complete map of them42,43. The mpMPC approach involves the use of a discrete-time constrained linear time-invariant system: xt 1 Axt But Dwt yt Cxt
(16)
where u,y,x,w are the manipulated, the controlled, the state, and the disturbance variables and
A, B, C , D are fixed matrices. Considering the linear model (16), the MPC problem (10) can be recast as a multi-parametric Quadratic Problem (mpQP) which can be solved with standard multi-parametric programming techniques54 and involves a systematic exploration of the parameter space ( ). The resulting map consists of a set of convex non-overlapping polyhedra, critical regions (CR), which are bounded by a unique set of active constraints while the corresponding control law is piecewise linear in the form: ACR ,i bCR ,i u CCR ,i dCR ,i i 1, 2, N CR
(17)
where NCR is the number of critical regions, ACR , bCR , CCR , dCR are constants defining each region
CR, i and the derived optimal control action within.
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3.3 Synergetic Framework (exNMPC) The scope of the proposed integrated framework is to combine the benefits that each control methodology has, namely the accuracy and full coverage of the system’s operation for the NMPC approach and the fast execution time of the mpMPC approach. Furthermore, as the basis of the NMPC is the solution of a NLP problem it is important to reduce the computational effort55 between successive iterations, which constitutes the primary objective of the proposed synergy. The structure of the proposed synergetic framework is illustrated in Fig 3. Variable Bounds based: - Feasibility or - PWA approximation
ySP
exNMPC
Optimizer (NLP Solver)
Nonlinear Process Model Constraints
Obj. Function
u
Unit Model
exNMPC
ymeas e y pred x
Figure 3. exNMPC Framework The main barrier for the applicability of NMPC is its computational requirements imposed by the online solution of the optimization problem at every iteration. Several very promising works have appeared in the literature that approach this issue in a systematic way. An indicative only list include the use of a warm-start homotopy path method56, a PWA approximation for warmstarting57, a partially reduced Sequential QP (SQP) method58, new developments in interior-point methods59 and advanced preprocessing49. Similarly the proposed framework focuses on the same issue, the reduction of the computational effort for the solution of the NLP problem between successive iterations. This is achieved by the use of a warm-start initialization procedure of the NLP solver and a Search Space Reduction (SSR) technique of the feasible space. In order to take advantage of the information gained from the previous iteration a warm-start initialization is performed at each interval which can significantly reduce the number of iterations towards the
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optimum point60. In the case of a tracking problem, where step changes exist, the previous computed solution, although feasible and optimal, may be away from the new set point. Therefore after a step change the search for a new optimum solution is expanded throughout the variable’s feasible space. The main idea of the proposed framework is to define the region in a variable’s feasible space that includes the optimum solution for a given objective function, by applying an SSR technique. The decomposition of the search space into favorable and non-favorable areas has been previously studied for the improvement of global optimization problems using domain optimization algorithms61, agent-based systems62 and evolutionary methods63. The proposed synergy reduces the search space to a smaller subset around a suggested solution provided by a PWA approximation of the system’s feasible space. Thus, the NLP solver has a reduced variable space to explore in order to locate feasible points with acceptable solution quality. Based on the fact that the special treatment of the bounds can lead to substantial computational savings64 the proposed synergy aims at the adjustment of the search space through the modification of the upper and lower bounds at every iteration. In this context an mpMPC controller is used prior to the solution of the NLP problem in order to provide a suggested solution ( ump ) which is transformed into upper and lower bounds ( buact ,low , buact ,up ) augmented by a deviation term ( ebu ):
ebu
bu f ,up bu f ,low by f ,up by f ,low
e y ,max
(18)
where bu f ,up , bu f ,low are the feasible upper and lower bounds of variable u , and by f ,up , by f ,low are the respective bounds for variable y . The term ey ,max is the maximum model mismatch between the linearized and the nonlinear model and it is determined by an offline simulation study that
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involves the whole operating range of y . The bounds are modified at every iteration and as a consequence the search space of u is reduced to a smaller subset based on:
ump ebu buact ,low bu f ,low ump ebu buact ,up bu f ,up
, (ump ebu ) bu f ,low , (ump ebu ) bu f ,low , (ump ebu ) bu f ,up
(19)
, (ump ebu ) bu f ,up
where buact ,low , buact ,up are the active bounds for u . Therefore the optimizer has a set of updated bounds for the respective manipulated variable u . Apart from the bounds modification the rest of the NLP problem formulation remains the same. The proposed strategy at sampling interval k is summarized in Algorithm 1. Algorithm 1 SSR based on PWA and NLP problem Input: Warm-start solution ( xk , uk , yk , Hessian H ), measured
variables ( ykmeas ), parameters ( pk ), set-points ( ysp , k ) Output: Vector of manipulated variables uk 1
1: Calculate error ek and yˆ k 2: Locate CRi for parameter vector k and obtain ump 3: Calculate buact ,low , buact ,up 4: Modify bounds ul buact ,low , uu buact ,up 5: Solve NLP problem (14) 6: Obtain u1k 1 from uk 1 [u1k 1 ,..., ukNE1 ]
Based on the above analysis the explicit solution can direct the warm-start procedure for the solution of the NLP problem and thus improve its performance. The proposed method is used subsequently to control a PEMFC system.
3.4 PEMFC Problem Formulation
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According to the control objectives ( ySP [ PSP , T fc , SP , O 2, SP , H 2, SP ] ) defined in Section 2.2, there are five manipulated variables ( u [ I , m air ,in , m H 2,in , xht , xcl ] ) and four controlled variables ( y [ PSP , T fc , O 2 , H 2 ] ). Two of the manipulated variables ( xht , xcl ) are selected to have varying bounds during the operation of the system. These variables are mutually exclusive and they mainly affect one of the controlled variables ( T fc ). Based on the SSR technique analyzed in Section 3.3 a PWA function is required to approximate the temperature behavior of the fuel cell system using the aforementioned mpMPC approach. Prior to the solution of the mpQP problem a linear discrete-time state space ( ssTfc ) model is derived by a model identification procedure using input-output data obtained from simulations of the nonlinear dynamic model. The resulting ssTfc model has two input variables ( xht , xcl ), one output ( T fc ), one disturbance ( Tamb ) and two states ( x1 , x2 ). The mpQP problem involves six parameters [ x1 x2 Tamb T fc T fc , sp ] while the resulting feasible space, defined by , is partitioned into NCR=23 critical regions. Based on these critical regions ( NCR ) the bounds of xht , xcl are adjusted accordingly while the bounds of the other three variables ( I , m air , m H 2 ) are fixed at their feasible bounds. The nonlinear dynamic model of the PEMFC, briefly presented in Section 2.1, is comprised of nine differential equations and one algebraic, discretized at 10 finite elements ( NE ) having 4 collocation points ( N cop ) each, resulting to 441 variables and 381 equations. The Jacobian matrix, which is computed analytically, has 3375 non-zero elements (density: 2.009%). Finally the prediction ( Tp ) and control horizon ( Tc ) is 5sec and 500ms respectively.
4. Simulation and Online Experiments
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In the current section the performance of the proposed exNMPC scheme is illustrated initially through a simulation case study and subsequently through two experimental cases. Their scope is to explore the overall behavior of the fuel cell system, controlled by the proposed exNMPC scheme, at nominal temperature conditions and during startup. During the experiments (simulation and online) the feedforward scheme for the adjustment of the hydrogen and oxygen excess ratio is also employed. The interconnection of the controller to the industrial SCADA system is achieved through a custom made OPC-based interface that was designed to establish the online communication between the NLP solver and the automation system. The sampling time is set at 500ms for each measurement.
4.1 Case study 1 - Simulation Results The scope of the first case study is to present the effect of the proposed search space reduction (SSR) approach to the performance of the system. Also the importance of the warm-start initialization of the NLP problem is illustrated and finally a comparison between the various initialization procedures is shown. The case study involves a scenario with few step changes at the power demand and at the fuel cell temperature. The response of an NMPC controller (method W1) is compared to the proposed exNMPC controller (method W2). Both use the same nonlinear dynamic model and the same optimization method and the simulation horizon is 10min.
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Power (W)
4
P (W2:exNMPC) P Set-point
3 2
Tfc (degC)
P (W1:NMPC)
0
100
200
300 Time (s)
400
500
600
300 Time (s)
400
500
600
66 64 Tfc (W1:NMPC) Tfc (W2:exNMPC) Tfc Set-point
62 60 0
100
200
Figure 4. a) Power demand profile and produced power by exNMPC and NMPC. b) Fuel cell
temperature Fig. 4 illustrates that the power demand is delivered upon request by using both initialization methods (warm-start with (W2) and without the SSR (W1)) in terms of accuracy and the temperature is maintained at the desired set-point with a negligible error (±0.4°C). Fig. 5 illustrates the corresponding oxygen and hydrogen excess ratio profiles that are adjusted
Lamda (O2,H2)
according to the power demand based on the minimum air and hydrogen considerations. 3 2.5 2 1.5 1
0
100
200
300 Time (s)
400
500
600
Figure 5. Oxygen and hydrogen excess ratio profiles
Figs. 4 and 5 show that both controllers (NMPC and exNMPC) have similar behavior in terms of accuracy which is verified by the mean squared error (MSE) as presented in Table 1. Table 1. Mean Squared Error of each control objective (methods W1 and W2)
NMPC (W1)
exNMPC (W2)
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Power
14mW
16mW
0.3°C
0.2°C
λO2
2.7 10 3
1.9 10 3
λΗ2
8.8 10 4
7.6 10 4
Temperature
The subsequent analysis shows the importance of the proposed SSR technique and its effect to the solution of the NLP problem regarding the computational requirements. The aforementioned simulation scenario is tested with four different methods regarding the initialization of the optimization problem. First a cold start optimization is performed (C1) at each iteration. In the second configuration the cold start initialization is complemented by the SSR approach (C2). The next two tests are performed with a warm start initialization method, with (W2) and without (W1) the SSR technique. Figs 4 and 5 present the results from methods W1 and W2. The performance of each method regarding the optimization time, the iterations and the calls to the objective function over the 10min is reported in Table 2. Table 2. Comparative analysis of the various initialization methods (C1, C2, W1, W2)
C1:Cold
C2:Cold+ SSR
W1:Warm
W2:Warm+SSR
Max opt time (ms)
3164
1026
493
382
Avg time (ms)
356
304
86
66
Total Iters
25187
21209
19329
10329
Max Iters.
845
539
541
373
Avg Iters.
206
203
17
8
Max Fun. Calls
2100
1589
1857
884
Avg Func. Calls
234
189
46
28
From Table 2 it is observed that the maximum and the average optimization time are decreased when the search space is adjusted both in cold start and warm start optimization. But when the
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optimization is performed using the cold start method, the maximum optimization time is beyond the system sampling time specifications. Therefore it is not practical to use a cold start initialization, method C1 or C2. On the other hand, the warm start initialization (W1, W2) of the optimizer shows superior performance compared to the cold start. Furthermore a significant improvement is observed at the maximum and average optimization time in method W2 compared to method W1. When method W2 is applied the search space is adjusted around the suggested solution which is derived by the PWA approximation of the system and the total number of iterations is decreased by 47%. Figs 6 and 7 illustrate the optimization time and the
Opt Time (s)
required iterations of methods W1 and W2. 0.6 exNMPC (W2) NMPC (W1)
0.4 0.2 0
0
100
200
300 Time (s)
400
500
600
Figure 6. Optimization time per interval and exNMPC (W2) NMPC (W1)
Iterations
400 300 200 100 0
0
100
200
300
400
500
600
Figure 7. Iterations per interval for methods M3 (NMPC) and M4 (exNMPC)
In steady state both methods (W1, W2) have similar behavior with optimization time ~50ms and ~100ms respectively. But when a step change occurs or a disturbance affects the temperature of the system, W2 method requires less iterations. On the other hand, W1 requires more iterations and performs much more function calls in order to determine the optimum value.
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From the above analysis it is obvious that the proposed exNMPC framework (W2 method) outperforms the others, for the given problem. The results indicate that the proposed controller (exNMPC) can be applied online to the fuel cell unit. Thus the next two case studies explore the online behavior of the system using the selected exNMPC controller.
4.2 Case study 2 – Online behavior of the exNMPC controller The purpose of the second case study is to explore the response of the proposed exNMPC controller with respect to the four control objectives that are defined at Section 2.3. In this case study a random power profile is demanded by the fuel cell system while the temperature is maintained at a certain level (55C). The duration of the experiment is 5 minutes. Fig. 8a illustrates the demanded and the produced power by the fuel cell while Fig. 8b illustrates the temperature of the fuel cell.
Power (W)
5 4 3
1
Temperature (degC)
Power Psp
2 0
1
2
3 Time (min)
4
5
2
3 Time (min)
4
5
56 Tfc Tfc,sp
55.5 55 54.5 54
0
1
Figure 8. Power generation and temperature profile
A negligible error between the demanded and the delivered power is observed with a mean square error (MSE) of 12mW. Also the temperature deviates from the desired set-point between 0.4C to +0.1C, which is within the initial objectives (±1C). The oxygen and hydrogen excess
24
ratio profiles are calculated based on the desired power profile and Fig. 9 illustrates that the
O2 Excess Ratio (-)
system follows those trajectories.. 4 lamdaO2 lamdaO2,sp
3.5 3 2.5
H2 Excess Ratio (-)
0
4
1
2
3 Time (min)
4
5
2
3 Time (min)
4
5
lamdaH2 lamdaH2,sp
3 2 1 0
1
Figure 9. Oxygen and hydrogen profiles
From Figs 8 and 9 it is obvious that the control objectives are achieved and the system operates
10 8
Air flow (cc/min)
6 4 2
600
H2 flow (cc/min)
Current (A)
in an optimum manner. The corresponding control actions are shown in Figs 10 and 11.
300
0
1
2
3 Time (min)
4
5
0
1
2
3 Time (min)
4
5
0
1
2
3 Time (min)
4
5
400 200
250 200 150
Figure 10. Control actions: current, air and hydrogen flows
25
As the power demand varies the controller calculates the optimal current (Fig 10a) which is set to the DC load that subsequently applies it to the fuel cell. At the same time the air and hydrogen flows (Fig 10b,10c) are adjusted based on the excess ratio profiles.
Operation (%)
100 Heat-up Cooling
80 60 40 20 0
1
2
3 Time (min)
4
5
Figure 11. Percentage of operation of the heat-up resistance
For the given temperature the cooling fans do not operate and the temperature is maintained at 55C by properly adjusting the percentage of operation of the heat-up resistance. As the demanded power increases so does the produced heat. Therefore the resistance supplies less heat, as seen by the percentage of operation that decreases (Fig. 11). This can be clearly seen during the 1st and the 2nd minute of the experiment where the heat-up percentage is close to 2% as the required heat is produced by the fuel cell. Fig. 12 shows the solution time for the aforementioned
Optimization time (s)
experiment. 0.2 0.15 0.1 0.05 0
0
1
2
3 Time (min)
4
5
Figure 12. Optimization time for each sampling interval
The assessment of the performance of the controller is summarized in Table 3. Table 3. Performance of the exNMPC controller
Performance metrics
26
Max. opt. time
149ms
Average opt. time
63ms
Max. Iterations
341
Max. func. calls
712
The maximum optimization and the average time indicate that the sampling constraint is satisfied. From the above analysis it is concluded that the fuel cell system is able to provide the demanded power using the minimum fuel and oxidant consumption. Furthermore, the exNMPC controller exhibits a stable behavior while achieving the predetermined objectives within the required sampling time.
4.3 Case study 3 - System start up In the third case study the use of the proposed exNMPC scheme for the start-up of the system is presented. The objective is to heat-up the system in a controlled way while serving a stable power demand. Initially the temperature is the same with the environmental one (~27°C) while the goal is to reach 60°C. The demanded power is 3.2W and the resulting oxygen and hydrogen excess ratio profiles are set to 3.2 and 1.55 respectively. Fig. 13 illustrates the power generation profile. 3.4
3
3 P (W)
Power (W)
3.2
2.8 2.6
Power Psp
2.4 0
5
2 1 0
0
20
40 Time (min)
10 15 Time (min)
60
20
25
Figure 13. Produced power during temperature increase from 27°C to 60°C
27
As in the previous case study, the controller is able to deliver the demanded power even at low temperature conditions and during the temperature rise. The MSE for the power is 15.3mW while the values for λO2 and λH2 is 4.69 10 4 and 2.52 10 4 , respectively. Fig. 14 illustrates the temperature trajectory towards the set-point.
Tfc Tfc,sp
55 50
Te(degC)
Temperature (degC)
60
45 40 35
61 60 59 58 57 12
13
30 5
10
14 15 Time (min)
15 Time (min)
16
20
17
25
Figure 14. Temperature rise using exNMPC controller
From Fig. 14 it is clear that the desired temperature is accurately reached. The rise time of the temperature is 13.4min with an average steady-state error of 0.2°C. The percentage of operation for the heat-up resistance is shown is Fig. 15.
Operation(%)
100
50 Heat-up Cooling 0 0
5
10 15 Time (min)
20
25
Figure 15. Operating % of the heat-up resistance during temperature increase from 27°C to 60°C
Finally from the computational point of view the maximum and the average optimization time is 181ms and 48ms respectively as shown in Fig 16. Optimization (s)
0.2 0.15 0.1 0.05 0
5
10
15 Time (min)
20
25
28
Figure 16. Optimization time during temperature increase from 27°C to 60°C
It is observed that the optimization time increases when the temperature approaches the set-point and decreases after the temperature settles to the desired value. Based on the results of the above analysis it clear that the exNMPC scheme can be used for the start-up of the system as it is able to compensate the difference between the measured and the desired temperature without increasing the computational requirements.
5. Conclusions
This work presents an alternative way of using two advanced MPC methodologies for a multivariable fuel cell control problem. The results illustrate that the response of the NMPC controller can be enhanced when it is combined with an SSR technique. To improve the convergence speed without sacrificing the quality of the solution, a new algorithm is proposed to dynamically adjust the search space of selected variables, based on a pre-computed augmented low-complexity PWA approximation of the feasible space. The behavior of the proposed synergetic formulation is illustrated through a multivariable nonlinear control problem involving the optimal operation of the PEM fuel cell system. A comparative simulation study reveals the potential of the proposed method which is deployed online to the fuel cell unit. The salient characteristics of the exNMPC scheme are revealed through a set of experimental case studies during system start-up and at nominal conditions. It is shown that the computational requirements of the exNMPC are within the desired sampling time constraints without compromising the full achievements of the control objectives. Overall the proposed framework guarantees that the fuel cell system can deliver the demanded power upon request while operating at a safe region using the optimum amount of air and hydrogen and simultaneously maintaining a stable temperature environment.
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ACKNOWLEDGMENT
This work is implemented through the Operational Program "Education and Lifelong Learning" and is co-financed by the European Union (European Social Fund) and Greek national funds, program “Archimedes III” (OPT-VIPS). This information is available free of charge via the Internet at http://pubs.acs.org/. AUTHOR INFORMATION Corresponding Author
*e-mail:
[email protected], FAX 302310498380, Tel 302310498316 Notes
The authors declare no competing financial interest.
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