Economic Dispatch with Dynamic Thermal Rating Milad Khakia, Petr Musilekb, Jana Heckenbergerovac, Raymond Pattersond a Department of Electrical and Computer Engineering, University of Alberta, Edmonton AB, Canada b Corresponding author: Department of Electrical and Computer Engineering, 2nd Floor ECERF (9107 - 116 Street), University of Alberta, Edmonton, Alberta, Canada T6G 2V4, Phone: (780) 492-5368, Fax: (780) 492-1811, E-mail:
[email protected] c Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic d School of Business, University of Alberta, Edmonton AB, Canada
Abstract
Restructuring of the power industry have resulted in more interest and demand for electrical energy. The attention is now focused on expanding the capacity of power transmission networks. The problem is that a number of environmental, social, and economic impediments prevent the construction of new generating and transmission facilities. Therefore, the challenge is to nd ways of optimizing the capacity of existing assets. This paper proposes an optimization method based on Mixed Integer Linear Programming techniques to incorporate weather-based dynamic thermal ratings into electrical production and transmission decisions. It considers standard costs of production along with other power generation constraints, such as delays and costs of start-up/shutdown of thermal plants. In addition, by using a spatially-resolved, high-resolution thermal model of the transmission system, the proposed method is able to dynamically identify thermal limits and temperature-dependent losses of the system. Load forecasting is currently used in dispatch centers to facilitate the scheduling of generators and to reduce the overall costs of the system. Multi-Snapshot Simulation enables the model to consider forecast values of the load and meteorological data; therefore, the resulting accuracy is improved and overall costs are reduced. The performance of the model is evaluated using a year of data from an electric utility. The simulation results indicate that overall costs of system operation can be signi cantly reduced using the additional dynamic capacity of the transmission lines. Preprint submitted to European Journal of Operational Research
Monday 3rd October, 2011
The use of this model allows cleaner and less expensive hydroelectric energy to be generated and transmitted through the existing transmission infrastructure. Keywords: OR in energy; mixed integer programming; distribution; economic dispatch; dynamic thermal rating 1. Introduction
From the generation perspective, various technologies are available. They have di erent capital and operation costs, performance characteristics, and environmental impact. An optimal generation scenario should involve technologies with minimal costs and least impact. Nevertheless, as generation technologies strongly depend on local conditions, e.g. availability of running water or reservoir for a hydro plant, realization of the ideal con guration is not always possible. As an example, less expensive energy can be generated at a distant hydro power plant and transmitted to the load center; however, the long distance transmission leads to higher voltage drop and power losses. To maximize the utilization of the existing transmission and generation assets, all such constraints must be considered. This paper extends the standard Mixed Integer Programming (MIP)-based optimization and transmission cost by taking into account the estimated ampacity of power transmission lines. By considering di erent generation technologies, this approach incorporates generation costs, ramp-up/ramp-down rates, and start-up/shut-down delays of various power plants. To determine weather-dependent transmission capacity and power losses, the proposed system uses a spatially-resolved, high-resolution thermal model of the transmission system. The performance of this advanced optimization system is demonstrated using a case study. The study involves a thermal plant close to a load center, and a remote hydro generation station. The simulation results con rm that a considerable decrease in generation costs can be achieved. This paper is organized in ve sections. Section 2 presents necessary background information including dynamic thermal rating, basic principles of power generation 2
dispatch, and cost models. The proposed optimization system is described in details in Section 3. The transmission network, which is used for the case study, is described in Section 4. Simulation results are presented and analyzed in Section 5. Finally, Section 6 brings main conclusions and outlines possible directions for future work. 2. Background
Electric power systems are comprised of sources of the energy (generators), energy delivery systems (both transmission and distribution), and energy sinks (loads centers). To optimize the operation of these systems under a stochastic electricity demand, a number of features related to the performance, operation, and state of their subsystems must be described. 2.1. Economic Dispatch
Dispatch of electric power system was traditionally scheduled in a way minimizing the total fuel cost. Economic dispatch (ED) has become an essential part of the operation and planning in the deregulated power industry. The main purpose of ED is to schedule the active generation units so that the demand and all operational constraints are satis ed, while the sum of all costs is minimized [1]. In other words, each individual unit's output is scheduled to minimize the generation expenses of the entire system [2]. Environmental concerns introduced new constraints to avoid unnecessary emissions. A review of algorithms for environmental-economic dispatch is provided in [3]. The calculation accuracy of generation costs has a considerable impact on eciency of ED algorithms. In order to solve the ED problems, various methods have been used, ranging from linear programming [4] through inverse incremental cost functions [5] to heuristic approaches, such as particle swarm optimization [1]. 2.2. Power Generation Costs
Operation of a power plant is associated with various types of costs [6]. Capital cost can be expressed as the cost of land, buildings, and equipment added to the expenses 3
for design, planning, and installation of the generator. Considered before building the generator, this cost is usually amortized over plant's lifespan. Operation and management cost is usually divided into two parts: variable and xed. Cost of fuel, along with handling, transportation, and storage charges are expressed as the variable cost, which depends on the amount of generation. On the other hand, the maintenance cost of a plant that does not depend on the plant's generation is called xed cost. It only exists if the generator is continuously in an active or stand-by state. Hence, xed costs are usually normalized to the amount of generation capacity and expressed as a monthly charge. For information on costs associated with di erent generation technologies, see [2], [7], [8], [9], and [10]. Start-up costs are the expenses attached to some types of generators, most notably thermal [11]. They are incurred during the transition of a unit from \o " to \on" state. The start-up costs are usually de ned using an exponential function of the generator's o time CS X off i
i
2
(t) = i + i 41
e
X off (t) i i
3 5;
(1)
the time constant, i, is a parameter of the i-th generator start-up function. Constants i and i are the parameters of the generator, dependent on its mechanical and electrical characteristics, and Xio is the duration of the generator o -state when it restarts. Generation increase and decrease rates are limited by two parameters called rampup, RUi , and ramp-down, RDi . The limiting factors, de ned only for thermal generator units (e.g. gas, coal, oil, etc.), are functions of generation capacity [9]. In general, there are several generating units in a power plant. Usually one of these units, called Spinning Reserve, is kept in standby mode and used occasionally. This unit acts as a safety factor and provides sucient power to cope with forced or planned outages [12]. Another limiting characteristic of thermal generators is up/down time delay. Before a thermal generating unit is committed or de-committed, it should retain its state for 4
a minimum time [11]. 2.3. Thermal Rating of Transmission Lines
The maximum allowed current to keep the temperature of a transmission conductor within an acceptable range is expressed in terms of ampacity or the thermal rating. The most common, traditional rating approach is called Static Thermal Rating (STR). In STR, the ampacity is xed over a time period (year, season), and independent of actual weather conditions. Due to operating practices, driven by safety and reliability requirements [13], the assumptions used to derive STR are very conservative. As a result, transmission systems that use STR are greatly underutilized. To overcome these problems, Dynamic Thermal Rating (DTR) methods employ direct measurements of conductor temperature or sag, or estimates of conductor's temperature based on meteorological measurements. Based on these inputs, the DTR engine determines the current-carrying ampacity of the transmission lines. DTR systems were viewed as an e ective tool to expand the throughput of transmission systems without the need to build new lines with high environmental and nancial costs [14]. Since its introduction in 1977 [15], DTR has been the subject of many research studies. Ciniglio et al. used DTR together with favorable weather conditions to increase the ampacity of Idaho regional transmission system in [16]. E ects of using DTR on di erent parts of transmission systems are examined in [17]. Applications of conventional DTR to the problems of electrothermal coordination, augmenting power transfer capability, and network congestion management are presented in [14] and [18]. Range of studies utilizing DTR methods varies from determination of current carrying line capacity [19], through assessment of line thermal aging [20], to comparison of STR and DTR performances [21]. The approach presented in this paper utilizes spatially distributed meteorological conditions along the transmission line to evaluate its thermal rating as the minimal ampacity corresponding to the ampacity of its bottleneck. Meteorological variables used for the calculations include horizontal wind speed and direction, ambient temperature, 5
and short/long-wave radiation. Ampacity of each line segment is then computed using procedure outlined in IEEE Standard 738 [22]. The spatially resolved thermal model of the entire transmission system [23] allows a more accurate modeling of transmission losses using temperature-dependent line resistance. 2.4. Voltage drop limitations
One of the major concerns in long-distance transmission is the voltage control. Voltage level is a ected by the shunt capacitance and series inductance of a transmission line. Voltage uctuations are primarily caused by the ow of reactive power. The series reactance's temperature variations are insigni cant and can be neglected [24]. Based on the reactive and real parts of the power ow on a speci c transmission line, voltage increase or decrease is expected from the receiving terminal. Voltage level reduction of more than 5% or in some extreme cases 10% is not desirable [25]. Generally, by increasing the length of a power transmission line the amount of ow which is sucient to saturate the line, will decrease [26]. 3. Optimization Method
In this section, we present two important extensions to the optimization method based on the minimum cost ow network model [27]. The rst extension stems from the use of spatially-resolved thermal model of the transmission network. This allows for accurate estimation of network ampacity limited by a bottleneck with variable location. It can also be used to calculate temperature-dependent resistance of transmission lines, and thus a better representation of power losses in the model. The second extension consists in multi-snapshot simulation. Using available forecasts of load and ampacity, it allows the generation shares of each power plant to be calculated in advance. 3.1. Model Nomenclature
The proposed model to solve the ED problem is based on Mixed Integer Programming (MIP). The model contains an Optimization Function [28] and a set of constraints. 6
Optimization constraints implemented in MIP model are introduced and described in following paragraphs. The Optimization Function is de ned to minimize the total cost of power generation, including hydro and thermal power plants. At the same time, it must ensure that sucient power for the demand node(s) in the network is supplied continuously. Each generator has a limited range of generation capability, speci ed using a Power Generation Boundary. For hydro power plants, it is simply de ned as the minimal and maximal possible generation. The thermal generation pro le is more complicated, with a semi-discrete characteristic. This means that the cost of generation below the minimal limit is as high as for the minimum itself. For generation above the minimum, the cost rises continuously and linearly. Thermal plants require a start signal ahead of time. Multi-snapshot simulation used in this study enables the model to dispatch thermal generator's early start-up/shutdown signals in advance. The Power Generation Boundary constraint is designed in a way to make sure that the model will work accordingly. In a power generation and transmission system, the total generated power must be equal to the demand plus the transmission losses in the network. Generation/Demand Equilibrium constraint ensures that this equilibrium is maintained during simulation. The amount of losses in a transmission line is a function of the current owing through the line, its resistance, and length of the transmission line. Line's resistance is, in turn, a function of the local weather conditions along the line and the current passing through the conductor. Loss Calculation constraints use a piecewise implementation of the non-linear resistance dependency to calculate the transmission losses. In order to simulate the minimum cost ow network for a power transmission circuit, the ow of power into a node and out of it must be in balance. However, each node of the transmission network might include a source/sink of power ow such as a generator/load center. While considering all ows associated with a node, the transmission losses must be taken into account as well. The Power Bus Flow Balance constraint forces the model to calculate the power ow values strictly based on these rules. The Network 7
constraint limits the model to route the power through existing physical transmission lines and to avoid virtual lines that might be available in the power transmission matrix. The Line Ampacity constraint is included to impose the maximum throughput of a transmission line. It can be determined using static (STR) or dynamic (DTR) thermal rating methods. Due to the mechanical characteristics of thermal plants, the rate of increase and decrease of generation amount is more restricted than in the case of hydroelectric generators. The Ramp Up/Down constraints keep the rate within acceptable limits. The Voltage Drop constraint is used to impose a limit on maximal voltage drop, speci cally for long transmission lines. While the MIP model itself is linear, it must represent some non-linear features. These include transmission losses, line temperature, and voltage drop. Convex nonlinear functions are converted to convex piecewise linear functions (PWL) that can be used in a linear MIP model. The Piecewise Linear Conversion Set constraints are de ned to obtain the linearized values from the input non-linear function. Constant and variable parameters used in the optimization model de nition are summarized in Tables 1 and 2. Physical Structure
Table 1: Variable parameters used in the MIP model
Parameter
Description i; j; k Variables de ned to indicate bus numbers P Gi (u) Power generation at bus i [MW] P G(u) Total generated power [MW] CSi (u) Start-up cost associated with generating unit i [$] CVi Variable cost coecient of generator i [$/MW] C (u) Total cost of the power generation [$] Z (t) Total generation cost of the system [$] P Gmax ( u) Maximum allowable generation at bus i [MW] i P Gmin ( u ) Minimum allowable generation at bus i [MW] i C P resi (u) Binary variable de ning the availability state of generator i P Gi;supp (u) Supplementary power generation variable at bus i [MW] Ii;j (u) Current owing through the line between buses i and j [A] Vi (u) Voltage level at node i [V] Li;j (u) Power loss of the line between buses i and j [MW] L(t) Power loss of the entire network [MW] P F Li;j (u) Power ow through the line between buses i and j [MW] PS Variable used for linear approximation in piecewise function's lookup table PZ Binary variable used for linear approximation in piecewise function's lookup table
8
Table 2: Constant parameters used in the MIP model
Parameter t Ai;j (u) C P rvi (u)
Description Simulation time step length [min] Ampacity of line between buses i and j [A] Binary state of generator i CFi Fixed cost of generator located at bus i [$/min] Di;j Length of line between buses i and j [km] DSi;j;k Length of the k-th line segment of transmission line between buses i and j F Number of snapshots for which the start/stop command can be initiated in advance IT abi;j (u; k) An array of constant numbers that form the input part, line current, of the piecewise lookup table. RT abi;j (u; k) An array of constant numbers that form the output part, line resistance, of the piecewise lookup table. M A large number used in the MIP Model N Number of buses in the network NSi;j Number of transmission line segments between buses i and j P Di (u) Power demand at bus i [MW] P Fi (u) Power factor of the load at demand node i P Gmin Minimum allowable generation at bus i [MW] i (u) P Hi;j Binary constant indicating electrical connection between buses i and j Ri;j (u) Resistance of the line between buses i and j [ =km] rk;i;j (u) Unit resistance of the line calculated at segment k of line between buses i and j [ =km] RUi , RDi Ramp-up/down limit of the generator located at bus i [MW/h] t The current time of the simulation's rst snapshot T The maximum simulation period Ti;j (u) Temperature of the line between buses i and j [K ] u The current snapshot in each snapshot step U Number of snapshots in each simulation step
3.2. System Block Diagram
Figure 1 shows a block diagram of the optimization system used in this study. It has four major parts: Inputs, the main program, the AMPL optimizer, and the Outputs. Inputs are comprised of meteorological data, demand pro le, and the power system con guration. Weather information along the transmission line corridors is obtained from NARR [29], while the demand pro le is calculated based on the typical weekly load pro le [30]. The estimated weekly consumption data of St. John's. Power system con guration is xed during the simulation and initializes some system variables in the main C++ program such as Network Con guration, Generator and Demand Speci cations, and the DTR Library. AMPL part is interfaced with the main program, which provides input data in an appropriate format. AMPL Optimizer generates the outputs, which are afterwards handled by the main program and saved in the database.
9
Figure 1: Block diagram of the optimization system used in this study
3.3. Model Description
The goal of the proposed model is to minimize the generation costs. It is described by the following optimization function Z (t) = min
N tX +U X u=t+1 i=1
[CFi t + CVi P Gi(u) + CSi(u)] ;
(2)
The control variables of the MIP model are the power ow of the transmission lines, PFL, and the generation amounts of each generator, PG. In order to process predicted values for demand and weather conditions, each simulation step consists of several snapshots U . The rst snapshot contains the inputs for the current simulation step, while the remaining snapshots represent the state of transmission network in the following hours. The number of snapshots to be processed in a simulation step is de ned by parameter U . For each simulation step, constant F de nes the number of snapshots for which the state of power plants can not be changed. For simplicity, rst simulation step, t = 1, is assumed in following descriptions. The minimum and maximum generation limit for each unit, at snapshot step u, are described using Power Generation Boundary P Gi (u) C P resi (u) P Gmax i (u)
10
8i 2 [1; N ]; u 2 [1; U ];
(3)
P Gi (u) P Gmin i (u) P Gi;supp (u) u 2 [1; F ]; C P resi (u) = C P rvi (u)
8i 2 [1; N ]; u 2 [1; F ]:
(4) (5)
Equation (3) guarantees that each generator is working within expected operating range. Without enforcing constraints (4) and (5), each generator is virtually capable of producing power from extremely low values close to zero to maximum generation capacity. Parameter C P resi(u) determines whether a generator is in standby mode, 0, or in generation mode, 1. The parameter P G2;supp, usually used for thermal power plants is an auxiliary variable de ned to store the extra amount of generation that is greater than the minimum generation value. In addition, it is possible to add the spinning reserve power to the maximum limit, P Gmax i (u). Constant F de nes start-up time of thermal power plant. Due to the start-up/shutdown constraint, the generator's state can not be changed. In order to implement such characteristic, C P rvi parameter is introduced. This constant parameter is initialized based on the values of C P resi. Using equation (5) the MIP model ensures that previous status of the generator becomes xed in snapshots [1; F ] of the current simulation. The ramp-up and ramp-down limits of a generator, are enforced using the following two inequalities P Gi (u) P Gi (u RDi P Gi (u
1) RUi; 8i 2 [1; N ]; 8u 2 [1; U ]; 1)
P Gi (u);
8i 2 [1; N ]; 8u 2 [1; U ]:
(6) (7)
Note that in order to implement this constraint, the initial generation amount of each plant needs to be speci ed. These constraints are only applied to thermal power plants. To keep the model uniform, variables RUi and RDi are also de ned for hydroelectric and other types of generators without ramp-up/down limitation. However, rather large 11
values are assigned to the variables to practically remove the limitations, e.g. maximum capacity of a generator multiplied by two. In each simulation step, the generation/demand equilibrium of all snapshots is ensured by the following constraint N X i=1
P Gi (u) =
N X N (u) + X Li;j (u): 8u 2 [1; U ]; PF
N PD X i i=1
i=1 j =1
i
(8)
Transmission line losses calculation is implemented by constraint which follows 2 (u) Li;j (u) = Ri;j (Ii;j (u)) Ii;j
8(i; j ) 2 ([1; N ]; [1; N ]); 8u 2 [1; U ]:
(9)
It is clear, that this constraint includes two non-linear convex functions. In order to keep the model linear, application of a linearization method, such as piecewise linear function approximation, is necessary. The linearization process is described in full detail in subsection 3.4. For each simulation snapshot u at node i, Power Bus Flow Balance constraint ensures that the sum of the generated and incoming power, without considering losses, is equal to the sum of power consumed at this node plus the power owing out of it. In order to avoid double-counting of losses, the line loss term should be inserted only on one direction of the power ow. In our model, the losses are added to the right side of the equation P Gj (u) +
N X i=1
P F Li;j (u) =
N P Dj (u) X + (P F Lj;k (u) + Lj;k (u)) P Fj k=1
(10)
8i; j; k 2 [1; N ]; 8u 2 [1; U ]:
Next constraint, describing Network Physical Structure (11), forces the model to route the power ow through existing paths in the network and avoid using virtual paths in the transmission matrix. Physical structure of the network is de ned by a 12
binary constant called P H . P Hi;j = 1 when there is an electrical connection between buses i and j . Otherwise P Hi;j = 0. P F Li;j (t) M P Hi;j ;
8(i; j ) 2 ([1; N ]; [1; N ]):
(11)
Line Ampacity constraint, ensures that the current ow of the line does not exceed
its ampacity. The current carrying capacity of a conductor can be determined using either STR or DTR methods Ii;j (u) Ai;j (u);
8(i; j ) 2 ([1; N ]; [1; N ]); 8u 2 [1; U ]:
(12)
Finally, active voltage drops of the transmission lines have been considered in this study. The Voltage Drop constraint is de ned as follows Vi
p 3Ii;j Ri;j (u) Vi 0:95 8i; j 2 ([1; N ]; [1; N ]) 8u 2 [1; U ]:
(13)
Only active voltage drops of the transmission lines have been considered in this model. 3.4. Piecewise Linear Variable Constraints
As mentioned in section 3.1, piecewise linear functions provide a way to approximate non-linear constraints using linear programming methods. Chapter 17 of [28] is dedicated to de nition and implementation of piecewise functions for MIP problems. The variables and functions which are converted to piecewise linear include: voltage drops, transmission losses, line resistance, and line temperature. The procedure of piecewise linear transformation is illustrated using line resistance. Resistance of a transmission line is not constant; it is a function of conductor temperature, which is a ected by weather conditions [22] and the load current. In order to calculate both resistance and current load of the line, a piecewise linear conversion between them is required at every snapshot. For resistance, this dependency is described
13
in form of Piecewise Linear Conversion Set constraints Ii;j (u) =
X
Ti;j (u) =
X
(IT abi;j (u; k) P Z (u; k) + (IT abi;j (u; k + 1)
k
k
(RT abi;j (u; k) P Z (u; k) + (RT abi;j (u; k + 1)
P S (u; k)
