Portal de Periódicos da CAPES

Decentralised optimal dispatch of multi‐area power systems based on non‐linear value‐function approximation

2019; Institution of Engineering and Technology; Volume: 13; Issue: 22 Linguagem: Inglês

10.1049/iet-gtd.2018.7108

1751-8695

Jianquan Zhu, Tao Zhu, Mingbo Liu, Wentian Lu,

Integrated Energy Systems Optimization

IET Generation, Transmission & DistributionVolume 13, Issue 22 p. 5083-5090 Research ArticleFree Access Decentralised optimal dispatch of multi-area power systems based on non-linear value-function approximation Jianquan Zhu, Corresponding Author Jianquan Zhu zhujianquan@scut.edu.cn School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this authorTao Zhu, Tao Zhu Guangdong Power Exchange Center, Guanzhou, 510000 People's Republic of ChinaSearch for more papers by this authorMingbo Liu, Mingbo Liu orcid.org/0000-0001-9097-9045 School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this authorWentian Lu, Wentian Lu School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this author Jianquan Zhu, Corresponding Author Jianquan Zhu zhujianquan@scut.edu.cn School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this authorTao Zhu, Tao Zhu Guangdong Power Exchange Center, Guanzhou, 510000 People's Republic of ChinaSearch for more papers by this authorMingbo Liu, Mingbo Liu orcid.org/0000-0001-9097-9045 School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this authorWentian Lu, Wentian Lu School of Electric Power Engineering, South China University of Technology, Guangzhou, 510640 People's Republic of ChinaSearch for more papers by this author First published: 24 October 2019 https://doi.org/10.1049/iet-gtd.2018.7108Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In recent years, the use of interconnected multi-area systems has increased significantly. To operate these systems efficiently, a proper coordination approach among different areas is required. With this context, a novel decentralised optimal dispatch algorithm is proposed in this study. The centralised economic dispatch model is decoupled into a series of sub-problems for different areas, allowing the independent decision of each area without a central operator. The detailed information of each area is not required for information exchange. To better estimate the impact of each area's decision on other areas, the approximate value functions of the tie-line powers are added into the local objective function of the sub-problem. This technique makes the decentralised solution of the proposed algorithm more explicit and rational compared with most existing distributed algorithms. Besides, the proposed technique does not require parameter tuning and has good convergence performance. Numerical simulations on several test systems and a real power system demonstrate that the proposed algorithm is favourable in terms of accuracy, adaptability, and computational efficiency. Nomenclature Indices and sets indices of areas i index of generating units k index of iterations r index of breakpoints for piecewise-linear functions set of generator and load buses set of load buses set of generator buses set of buses of area a connected with area a + 1 to N set of buses of area a connected with area 1 to a − 1 la set of internal lines of area a Variables generation cost of unit i immediate cost function of stage t tie-line power transmitted from area a to area b state variable representing the power transmitted by area 1…a − 1 to current area a through internal bus i tie-line power transmitted by areas before a to the following area b when solving sub-problem (a) vector whose components are state variable of stage t value function of sub-problem (a) slope corresponding to the breakpoint r value function of state conditional value function of decision variable of stage t flow over the interval optimal dual variable of the corresponding constraint when solving sub-problem (a) Parameters generation-cost coefficients of unit i active power of load bus i capacity of the tie line between area a and b power transmitted from current area a to the following area b through internal bus i maximum/minimum generation output of unit i capacity of line l capacity of the internal line of area a power shift factor for bus i on line l power shift factor for bus i on line la total number of breakpoints r breakpoint for the piecewise-linear function discount rate interval of adjacent breakpoints 1 Introduction Modern interconnected multi-area systems are usually composed of regional subsystems belonging to different administrative areas (e.g. China's northern–central–eastern interconnections) [1, 2]. In order to efficiently operate a multi-area system, multi-area economic dispatch (MA-ED) that provides proper coordination among different areas is required. Ideally, a central operator with access to detailed data (e.g. the topologies and parameters of power networks, fuel costs of generators) of all areas could centrally operate the multi-area system. However, for the requirements of information privacy and decision independence of each area, this kind of operator is unlikely to be implemented [3]. Therefore, the multi-area systems are generally operated by the coordinated actions of decentralised operators in a decentralised MA-ED framework. The operator of each area makes decisions independently, and only a small amount of border information is exchanged between these areas. In this way, the multi-area system not only can be dispatched effectively but also can preserve each area's information privacy and decision independence [1–5]. Some optimisation methods have been used to solve economic dispatch (ED) problems. Traditional methods include dynamic programming [6, 7], Lagrangian relaxation (LR) [8], and the interior point method [9]. Intelligent methods include the particle swarm algorithm [10], genetic algorithm [11], neural network algorithm [12], and so on. However, most of these methods are implemented in a centralised manner, which are not suitable for the decentralised optimisation of MA-ED. Given that the decentralised algorithm aims to obtain the same solution as the centralised algorithm, while also preserving the information privacy and decision independence of each area, simulation results from a centralised procedure commonly are used to evaluate the accuracy of the decentralised algorithm [1–4]. To date, much literature has been dedicated to applying decentralised algorithms to power-system scheduling and dispatch problems. In [13], LR is used to relax the area-coupling constraints into the local objective function. By solving the sub-problem of each area alternately and updating the Lagrangian multipliers, the decentralised solution of the optimal power flow (OPF) is carried out. In [1], LR is developed into a dynamic form in which the Lagrangian multipliers are exchanged from 'static' point-wise constants to 'dynamic' linear expressions, and this algorithm shows better computational performance. To improve the convergence of LR, augmented LR (ALR) adds a quadratic penalty term to the local objective function and then updates the coefficients of the penalty function iteratively by using the difference of coupling variables until it satisfies the coupling constraints. ALR has been applied to solve multi-area unit commitment problems and OPF [14–16]. Unlike LR and ALR, optimality condition decomposition decomposes the Karush–Kuhn–Tucker condition of the original problem to achieve the decoupling of sub-problems, which is widely employed to solve OPF [17] and MA-ED [18]. In addition, in [2], a generalised Benders decomposition (GBD)-based decentralised algorithm is proposed. The GBD algorithm is modified by introducing a locally optimal cost for each area. When implemented in multi-area systems with a hierarchical control structure containing a coordinator, this algorithm exhibits favourable convergence. Meta-heuristic algorithms are also used for the optimisation of MA-ED [19, 20]. Compared with the traditional decentralised algorithms which need specific requirements in the convexity, continuity, and gradient information of the model, meta-heuristic algorithms can better deal with the non-convex, discontinuous and non-differentiable MA-ED problems. Unlike existing decentralised optimal algorithms in power systems, this paper studies the MA-ED problem from the perspective of a non-linear value-function approximation [21–24]. First, the MA-ED model is transformed into a multistage optimisation problem, and the tie-line powers are taken as state variables to describe the interaction of various stages (i.e. sub-problems of different areas). Thus, when the value functions of tie-line powers are obtained, we can learn how much the dispatch decision of each area affects the decisions of other areas. By adding these value functions to the objective function of each area according to the Bellman optimality principle, the centralised ED model can be decoupled into some decentralised sub-problems dispatched by various areas in a dispersed manner. The value functions of tie-line powers are unknown to us, and thus they are approximated iteratively during the process of a decentralised solution. When algorithmic convergence is achieved, both the approximate value functions and decentralised solutions of MA-ED are provided accordingly. Besides the advantages mentioned above in comparing with centralised algorithms, the proposed algorithm has some additional contributions as follows: (i) The MA-ED model is transformed, for the first time to our knowledge, into a multistage optimisation problem, and then it is decomposed into decentralised sub-problems based on the non-linear value-function approximation. (ii) The approximate value function of tie-line power is able to estimate the impact of one area's decision on the other areas, making the decentralised solution of each area explicit and rational. Moreover, the approximate value function can be updated according to the dual-variable information of sub-problems, which is effective for fast convergence of the proposed algorithm. (iii) The proposed technique does not require parameter tuning. Its accuracy, convergence, and adaptability have been demonstrated in several test systems and an actual power system. The remainder of this paper is organised as follows. In Section 2, the decentralised MA-ED model is formulated. In Section 3, the methodology for the decentralised solution is presented, and in Section 4, the simulation results on several test systems and on an actual power system are discussed. Section 5 concludes the paper. 2 MA-ED model 2.1 Centralised ED model The ED model is designed to minimise the generation cost shown in (1), which is subject to the system operational constraints shown in (2)–(4) (1) (2) (3) (4)Equation (1) describes the objective function, where the generation cost of unit i can be expressed as . Equation (2) represents the power-balance constraints. Equation (3) describes the output limit of each unit. Equation (4) represents the network transmission constraints based on the DC power-flow model, which is used widely in MA-ED problems. The DC power-flow model describes the relationships among the nodal voltage phase angle, branch power, and other variables of the network. From these relationships, the influence of power generation and load on the power of each branch can be further deduced, and then the power shift factors shown in (4) can be obtained accordingly [1–4, 25]. 2.2 Area-decoupling principle For the decentralised solution of MA-ED, the centralised model in (1)–(4) should be transformed into decentralised models. It is essential to achieve decoupling of sub-problems in various areas. The multistage optimisation theory is employed to solve this problem. Multistage optimisation is a decision-making process with a chain decision structure. Multistage optimisation has two basic elements: the division of stages and the definition of state variables that represent the interaction of those different stages. Regarding MA-ED, after the stages and the state variables are determined, the optimisation of the former areas will affect the optimisation of the latter areas through tie-line powers. Accordingly, the MA-ED is regarded to be the multistage optimisation problem. In Fig. 1, a system with three areas, denoted A, B, and C, is used to illustrate the area-decoupling principle. First, the MA-ED problem is divided into three sub-problems corresponding to each of these areas. Then, the optimisation sequence of the sub-problems is given arbitrarily (e.g. A–B–C). This makes the MA-ED become a chain decision-making process. By regarding the sub-problem of each area as a stage, MA-ED is converted into three stages. Motivated by the fact that each area affects the other areas through the tie lines, the tie-line powers are taken as state variables. In this way, a complete multistage optimisation problem is constructed to decouple the MA-ED. Fig. 1Open in figure viewerPowerPoint Analogy between MA-ED and multistage optimisation In the field of multistage optimisation, the Bellman optimality principle is widely used to decouple the sub-problems of different stages. This principle is expressed by Bellman's equation as follows: (5)In (5), the value function is the key to the decoupling. In general, the input variables of the value function are the state variables of the current stage. The target value of the value function is the total cost (or revenue) of subsequent stages. Accordingly, the impact of each stage's decision on the other stages can be characterised, and then the multistage problem can be decoupled into a series of sub-problems as shown in (5). Equation (5) describes the impact of the current state on the following stages by introducing the value function. When dealing with the decentralised optimisation of MA-ED, the value functions are introduced to characterise the impact of each area's decision on the other areas through the tie lines. Thus, these sub-problems can be decoupled. Details are given in the following section. 2.3 Decentralised MA-ED model According to the above-mentioned area-decoupling principle, the value functions of the tie-line powers are added into the local objective function of each area; then the centralised model shown in (1)–(4) can be transformed into the following decentralised sub-problems: (6) (7) (8) (9) (10) (11)where is a column vector with all entries equal to one with the appropriate dimensions. Equation (6) is the objective function of area a. It considers the generation cost of area a and the impact of this cost on other areas. Equation (7) represents power–balance constraints. Equation (8) describes the output limit of each unit. Equation (9) describes the transmission constraints of the tie lines. Equation (10) captures the transition of the system state that reflects the evolution of the system. It is actually a boundary condition of sub-problem (a) according to the multistage optimisation theory. Equation (11) represents the transmission constraints of the internal lines of area a. 3 Decentralised optimisation of MA-ED 3.1 Approximation of the value function As shown in (6)–(11), the value function of the sub-model reflects the impact of the current area's decision on the other areas, which is essential to coordinate the decentralised optimisation of the subareas. Since the true value function is unknown, it is replaced with an approximate value function. Then this approximate value function can be updated iteratively to find the decentralised solution of the MA-ED as the true value function is approached. When the approximate value function is used to replace the true value function , (6) can be rewritten as follows: (12)In (12), is an approximate value function of the state variable . As described, the state variable represents the tie-line power between areas. In the case of multi-area interconnection, the state variable is a multidimensional vector. It is difficult to update the value function of this multidimensional vector. Also, this vector cannot clearly describe the impact of one area's decision on each of the other areas. To address this problem, is replaced with , which is the approximate value function of the one-dimensional state variable [23]. Then (12) can be described as follows: (13)where is the one-dimensional state variable, which is an element in . Since this paper's original problem of decentralised optimisation is a convex optimisation problem, it can be proven that in (13) is a convex function [22, 24]. For this convex value function, it is simple and effective to use the piecewise-linear function shown in Fig. 2 to approximate it. Then (13) can be replaced with the following: (14) (15) (16)After the previous approximation, (6) described in Section 2.3 is replaced by (14)–(16). Hence, the decentralised MA-ED model with the objective function in (14) and the constraints in (7)–(11), (15), and (16) is formulated. Whether or not the decentralised model can be equivalent to the original centralised model depends on the approximation accuracy of the value function. In the next subsection, the updating method of approximate value function is described to yield a decentralised MA-ED solution. Fig. 2Open in figure viewerPowerPoint Piecewise-linear convex function 3.2 Updating the value-function approximation In this study, the dual-solution information is used to update the approximate value function. Specifically, when solving the approximate decentralised MA-ED model composed of the objective function in (14) and constraints in (7)–(11), (15), and (16), the optimal dual variables of the constraints in (9) are employed to update the slopes of the approximate value function. Note that gives a sub-gradient of the objective function shown in (14) at . This result matches the physical meaning of the slopes of the previous area's value function. Hence, is used as an estimate of the sub-gradient of at point and smooth the slopes of by [22]. Details are summarised as follows. Let be a one-dimensional piecewise-linear value function characterised by a series of breakpoints and the corresponding slopes . Let be an estimate of the sub-gradient of at the point R and be a smooth step-size. Then, the slopes are updated as follows: (17) 3.3 Procedure of the proposed algorithm On the basis of the previous description, the implementation of the decentralised MA-ED via non-linear value-function approximation is as follows. Step 1. Initialisation : Set the sequence of sub-problem optimisation arbitrarily and choose the initial values for the piecewise-linear value functions, which include the total number of breakpoints and the corresponding slopes. Set the convergence tolerance and the iteration index k = 1. Step 2. Area sub-problem solution: For area a = 1 to N : (2-1): Receive the state-variable information from the last area a − 1 and then formulate its sub-problem (a) with the objective function as shown in (14) and with the constraints given in (7)–(11), (15), and (16). (2-2): Solve the sub-problem (a) and determine the optimal values of state variables and dual variables of the constraints in (9). (2-3): Send to the next area a + 1 and to the last area a − 1, respectively. Step 3. Update : Each area a updates its value-function approximation as described in (17) by using the dual-variable information . Step 4. Convergence check: If (18) is satisfied, stop the algorithm. Otherwise, increment k and then return to Step 2: (18) 4 Case studies To demonstrate the performance of the proposed algorithm, numerical simulations on multiple systems are carried out, including a simple four-generator system, several IEEE test systems, and an actual four-area, 2298-bus system. The proposed algorithm is coded using GAMS [26] installed on a Dell Precision T1700 workstation with 16 GB of memory and a 3.40-GHz Intel Xeon processor E3-1270 v3, and all problems are solved using IBM ILOG CPLEX Optimisation Studio. In these cases, the total number of breakpoints of the piecewise-linear value function is set to 100, and the corresponding initial slopes are set from −15 to −5 incremented by 0.1. Meanwhile, this study uses the step-size update rule from [23], and the convergence tolerance is set to 0.001. 4.1 Accuracy validation for the proposed algorithm To verify the accuracy of the proposed algorithm, it is compared with the centralised algorithm in a simple four-generator system. The example considered is shown in Fig. 3. Areas A and B are interconnected through tie line 3–6. The system includes four generators, three loads, and seven lines. Tables 1 and 2 give the data for the generators and the loads, respectively. The reactances of all lines are assumed to 0.13 p.u. on a base of 100 MVA. The capacities of internal lines are set to 100 MW, whereas the capacity of the tie line is 50 MW. Fig. 3Open in figure viewerPowerPoint Simple four-generator system Table 1. Generator data Generators G1 G2 G3 G4 Pmin, MW 10 10 10 10 Pmax, MW 110 150 150 110 a, $/h 100 100 200 200 b, $/MW h 15 25 30 50 c, $/MW2 h 0.002 0.002 0.004 0.004 Table 2. Load data Loads L1 L2 L3 power, MW 60 50 120 Table 3 compares the results from the centralised procedure with those from the decentralised procedure. The results (e.g. total cost and unit output) are identical, which indicates that the proposed algorithm has good accuracy. Table 3. Comparison of centralised and decentralised algorithms Centralised Decentralised total cost, $ 5844.00 5844.00 iterations — 3 unit output, MW 110/50/60/10 110/50/60/10 total power, MW 230 230 Fig. 4 shows the variation of total cost obtained by the centralised algorithm and the proposed decentralised algorithm when the tie-line capacity is increased between areas A and B. Upon increasing the tie-line capacity from 10 to 100 MW, the congestion is relieved and more power can be transmitted from area A with lower generation cost to area B. Therefore, the total system cost decreases monotonically. When the tie-line capacity exceeds 100 MW, all units of area B reach their lower output limits and additional power cannot be received by area B. Then, the total system cost will no longer decrease upon increasing the tie-line capacity. Meanwhile, it is evident from Fig. 4 that the results obtained by both algorithms are identical, which validates the accuracy of the proposed algorithm. Fig. 4Open in figure viewerPowerPoint Total cost versus tie-line capacity 4.2 Adaptability analysis of the proposed algorithm First, to analyse the sensitivity of the proposed algorithm to initial conditions of the piecewise-linear approximate value function , the four-generator system mentioned earlier is employed. The initial conditions of are determined by the total number of breakpoints r and the corresponding slopes . Therefore, the following assumptions can be made to test the proposed algorithm: The total number of breakpoints r is set to 50, 100, and 200, and the range of the corresponding slopes is divided into three cases according to their comparison with the generation-cost coefficient bi Case 1: starts from −100 and is incremented by 0.1. Case 2: starts from −40 and is incremented by 0.1. Case 3: starts from −15 and is incremented by 0.1. Table 4 presents the results of using the proposed algorithm to solve these three cases with r = 50, 100, and 200. The relative errors of the proposed algorithm are given to verify accuracy, which include the relative objective errors ('obj.error') and the relative output errors ('out.error'). These errors can be calculated as follows: (19) (20)where fdecen and fcen are the optimal objective values of the proposed algorithm and the centralised algorithm, and x decen and x cen are vectors of the unit outputs of the proposed algorithm and the centralised algorithm, respectively. As shown in Table 4, the proposed algorithm can converge in three iterations in all three cases, and the relative errors are small enough to declare that the proposed algorithm can yield solutions as good as those in the centralised algorithm. The proposed algorithm makes good use of the convexity of the problem in the process of approximating the true value function. Owing to the convexity of the problem, the corresponding value function is convex, too. Therefore, the piecewise-linear function is used to approximate the true value function. On the one hand, the convexity of the piecewise-linear function is maintained when it is updated, so that it can always approach the optimal value in the right direction. On the other hand, the slopes of the piecewise-linear function are updated by using the relevant dual variables of the sub-problems. Accordingly, the piecewise-linear function can quickly approach the true value function at the optimal point. The characteristics of these two processing techniques enable the proposed algorithm to quickly obtain satisfactory results with different initial conditions. Hence, it can be reasonably claimed that the proposed algorithm is not sensitive to the initial values of the value function. Table 4. Simulation results of different initial conditions Case 1 Case 2 Case 3 centralised total cost, $ 5844.00 5844.00 5844.00 decentralised with r = 50 total cost, $ 5844.00 5844.00 5844.00 obj.error, % 0.00 0.00 0.00 out.error, % 0.00 0.00 0.00 Iterations 2 2 3 time, s 0.52 0.52 0.78 decentralised with r = 100 total cost, $ 5844.00 5844.00 5844.00 obj.error, % 0.00 0.00 0.00 out.error, % 0.00 0.00 0.00 iterations 2 2 3 time, s 0.52 0.52 0.78 decentralised with r = 200 total cost, $ 5844.00 5844.00 5844.00 obj.error, % 0.00 0.00 0.00 out.error, % 0.00 0.00 0.00 iterations 2 2 3 time, s 0.52 0.52 0.78 Second, to study the adaptability of the proposed algorithm to different system interconnections, four IEEE four-area test systems are demonstrated. These systems, designated 4A-(a), 4A-(b), 4A-(c), and 4A-(d), are constructed by duplicating the IEEE RTS-96 system described in [27]. Their topologies are shown in Figs. 5a–d, respectively. It is evident that from 4A-(a) to 4A-(d), the interconnection becomes increasingly complex. Table 5 lists the configurations of these test systems. The system parameters, unit data, and load data are also from [27]. To enforce the power interchange between areas, the generation cost coefficient bi of area B is set to be 1.5 times that of the other areas. Fig. 5Open in figure viewerPowerPoint Topology of the four-area systems (a) 4A-(a) system, (b) 4A-(b) system, (c) 4A-(c) system, (d) 4A-(d) system Table 5. Configuration of four IEEE four-area test systems Units Buses Lines Tie lines 4A-(a) 128 97 158 5 4A-(b) 128 97 159 6 4A-(c) 128 97 160 7 4A-(d) 128 97 161 8 Table 6 illustrates the decentralised solution of these IEEE test systems. Observe that the proposed algorithm can provide satisfactory results with very small relative errors in all cases. Moreover, when solving systems from 4A-(a) to 4A-(d), the iteration number required by the proposed algorithm increases only slightly, which demonstrates the good adaptability of the proposed algorithm to systems with complex interconnections. Table 6. Simulation results of four IEEE four-area test systems 4A-(a) 4A-(b) 4A-(c) 4A-(d) centralised cost, $ 103,361.93 103,361.93 103,361.93 103,361.93 decentralised cost, $ 103,361.93 103,361.93 103,361.93 103,361.93 obj.error, % 0.00 0.00 0.00 0.00 out.error, % 0.02 0.03 0.02 0.03 iterations 22 23 25 27 time, s 10.30 10.80 11.80 12.53 Third, to analyse the adaptability of the proposed algorithm to the optimisation sequence of sub-problems, the previously mentioned 4A-(d) system is employed. As shown in Table 7, the proposed algorithm is tested under four scenarios with different optimisation sequences of sub-problems. ABCD in Table 7 means area A is solved first, followed by B, C, and D one by one. Table 7 shows that under different optimisation sequences of sub-problems, the obj.error of the proposed algorithm is always 0.00% (less than one ten-thousandth). Thus, it can be reasonably claimed that the proposed algorithm achieves the optimal values of the total cost. Under these four scenarios, out.error ranges from 0.02 to 0.04%. This means the maximum error of the output among all units is no more than four ten-thousandths. Since some units have similar power generation cost parameters, a small deviation among their output still can keep the total cost constant for the given calculation precision. Table 7.