Distributed Computing Approach to Solve Unbalanced Three-Phase DOPFs

Transcription

1 Distributed Computing Approach to Solve Unbalanced Three-Phase DOPFs Abolfazl Mosaddegh Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada Claudio A. Cañizares Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada Kankar Bhattacharya Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada Abstract Distribution systems have been gradually improved with new technologies. They have been upgraded from the traditional system with low-level control to a smart-grid system with high-level control. In the present work, a mathematical model of an unbalanced three-phase distribution system, including ZIP loads and other components of distribution systems is used, and a Genetic Algorithm (GA) -based Distribution Optimal Power Flow (DOPF) model is applied to find the optimal integer solutions for discrete system control elements such as Load Tap Changers (LTCs) and Switched Capacitors (SCs) in a practical feeder. In order to reduce the computational burden and consequently the run-time, a communication Middleware System for smart grids is used to solve the GA-based DOPF problem on a decentralized computer system using a parallel computing approach. This system is responsible for running the model, managing all communication between the nodes, and transferring the results between various parts of the parallel system. Comparing with heuristic methods with faster sub-optimal solutions in a centralized computer system, the present work is expected to yield better optimal solution within acceptable practical runtimes. Index Terms Distributed computing, distribution optimal power flow, genetic algorithm, real-time application, smart grids. I. NOMENCLATURE Indices D L Delta-connected loads and capacitors f Per-phase controllable tap changers. h Time, h = 1,..., 24. i Set of operating limits. l Series elements. n Nodes. n D Delta-connected loads. n Y Wye-connected or single-phase loads. nd W Worker-nodes. p Phases, p = a, b, c. pp Phase-to-phase, pp = ab, bc, ca. r Receiving-end. This work was supported by Hydro One Networks, Energent Inc., Milton Hydro Distribution, the Ontario Power Authority (OPA), and the Ontario Centres of Excellence (OCE) under the Energy Hub Management System (EHMS) project ( EPEC 2015 London, ON, Canada /15/$31.00 c 2015 Crown s Sending-end. wr Conductors or cables. Y I Wye-connected constant current loads. Y L Wye-connected loads and capacitors, Y L = Y I, Y P, Y Z. Y P Wye-connected constant power loads. Y Z Wye-connected constant impedance loads. Parameters α Penalty factor. Q Size of each capacitor block in capacitor banks [Var]. T Per-unit voltage change for each LTC tap. T Maximum tap changer position. θ pt Specified load power factor angle [rad]. T Minimum tap changer position. A, B, C, D Three-phase ABCD parameter matrices; A and D are dimensionless, B in [Ω], and C in [S]. Cn Max Total number of capacitor blocks available in capacitor banks. CAP Fixed number of capacitor blocks in capacitor banks. CR Cross-over rate. G Maximum number of generations in GA. G m GA generations. I pt Load current at specified power and nominal voltage [A]. l rn Series elements whose receiving ends are connected to node n. l sn Series elements whose sending ends are connected to node n. M R Mutation rate. N d Distributed nodes in the Middleware System. P Load active power [W]. P S Population size. Q Load reactive power [Var]. T Fixed tap position. V pt Specified nominal voltage [V]. Z Load impedance at specified power and nominal voltage [Ω]. Variables

2 cap F I J K tap V Number of capacitor blocks switched on. Fitness function. Current phasor [A]. Objective function. Penalty function. Tap position. Voltage phasor [V]. II. INTRODUCTION Reliability and power quality at the distribution level are the focus of Distributed Automation (DA) which provides options for computation, communication, and control of distribution systems in real time [1]. An important aspect of DA is voltage and reactive power control in distribution systems, which has been traditionally performed by Load Tap Changers (LTCs), Switched Capacitors (SCs), fixed capacitors, and step-voltage regulators. All these traditional Volt/Var Control (VVC) devices operate locally with stand-alone controllers with the goals of improving voltage profiles and reduce power losses. Mixed-Integer Non-Linear Programming (MINLP) models have been used for determining the discrete settings of LTCs and SCs in an optimal integrated VVC approach [2] [4]. In general, VVC problems can be divided into two major categories: rule-based and network-model-based [5]. The rulebased methods use real-time measurements to control SCs and LTCs based on rules and operator experience, whereas the network-model-based methods use different parameters such as the topology of a distribution system, the system configuration, measurements, and statistical data. Typically, to obtain the optimal solution in the latter methods, the Distribution Load Flow (DLF) and the Distribution Optimal Power Flow (DOPF) models are used. To find the DLF solution, different methods have been proposed in the literature such as Newton-Raphson methods with computationally expensive algorithms [6], fastdecoupled methods [7], quasi-newton methods [8], and forward/backward sweep methods with faster convergence [9]. To reduce the computational costs, heuristic approaches have been proposed to solve VVC problems; neural network techniques [3], computational cost reduction [10], and decoupling of the VVC problem into sub-problems [4], [11] are among the wide range of heuristic techniques described in the literature. However, approaches with enumerative techniques are good enough to deal with smaller systems and limited number of LTCs and SCs. Recently, researchers have considered the DLF model as a constraint in the DOPF model [12], [12], [13]. In all these papers, the authors propose a quadratic penalty approach to reduce the computational burden and make the DOPF model suitable for real-time applications, relaxing the MINLP problem into a Non-Linear Programming (NLP) problem by rounding the integer variables to the closest discrete values; limits on the number of daily switching operations of SCs and LTCs are also considered in the DOPF solution. However, the drawback of rounding continuous variables into discrete variables is that the solution may be sub-optimal [14]. Evolutionary Algorithms (EAs) can also be used to solve MINLP problems without rounding of discrete variables. Genetic Algorithms (GAs), Particle Swarm Optimization (PSO), and evolution programming strategies are among the different EA-based methods. In [14], heuristic and GA-based solution methods for the DOPF problem are compared for the IEEE 13-node test feeder in a 24-hour timeframe. It is noted that although the GA-based method yielded a better solution, it required a significantly larger computational time. In order to meet the need to reduce the computational costs and yield a better solution, EA-based methods can be run on distributed computing environments. From the above discussions, the present work has the following objectives: Develop the DLF model of an unbalanced three-phase distribution system including ZIP loads and other components of distribution systems. The model should be a stand-alone executable program that can be used on different platforms (i.e., Windows, Linux, and Mac). Develop a DOPF model with a GA-based solution to find the optimal decisions regarding switching of control devices such as LTCs and SCs. Introduce a MapReduce model on a communication Middleware System for smart grids to reduce the computational time of GA model and make it suitable for real-time applications. The rest of the paper is organized as follows: Section III presents the unbalanced three-phase DOPF model, which includes the model of the distribution system and the DLF model; the GA-based approach to solve the DOPF problem is also discussed in this Section. Section IV describes the possible implementation of a GA-based solution on a distributed computing platform, referred to as a Middleware System, to make the model suitable for real-time applications. The main conclusions and contributions of the presented work are highlighted in Section V. III. UNBALANCED THREE-PHASE DOPF MODEL A. Distribution System Component and Load Flow Models Common distribution system components are series components such as conductors or cables, transformers, switches, and LTCs, and shunt components such as loads and capacitors. The series components are modeled using ABCD parameters, relating the sending-end and receiving-end voltages and currents, as follows [15]: [ ] [ ] [ ] Vs,p,l Al B = l Vr,p,l. p l (1) I s,p,l C l D l I r,p,l where A l, B l, C l, and D l are 3 3 matrices. Different kinds of conductors and cables are represented as π-equivalent circuits, one-phase and three-phase transformers are modeled in wye or delta connection, and switches are assumed as zero-impedance components. The ABCD parameters of LTCs depend on the setting of tap positions, as follows:

3 A f = 1 + T f T a,f T f T b,f 0 f T f T c,f (2a) B f = C f = 0 f (2b) D f = A 1 f f (2c) where T a,f, T b,f, and T c,f have the fixed integer values in the range T p,f to T p,f. For three-phase group-controlled tap changers, all the tap operations are the same. The shunt components are modeled for each phase in order to represent unbalanced three-phase loads. A polynomial ZIP model is used for loads, while capacitors are considered as constant impedance loads. The following is the model of wyeconnected constant power loads: V n,p I Y P,n,p = P YP,n,p + j Q YP,n,p n p (3) The following equations represents wye-connected constant impedance loads: Vn,p pt 2 Z YZ,n,p = P YZ,n,p j Q YZ,n,p n p (4a) V n,p = Z YZ,n,p I YZ,n,p n p (4b) with wye-connected constant current loads modeled as: I pt Y I,n,p = P YI,n,p+j Q YI,n,p ( n p (5a) Vn,p pt ) θ pt Y I,n,p = tan 1 Q YI,n,p P YI n p (5b),n,p I YI,n,p e j( Vn,p I Y,n,p) I = I pt Y I,n,p ejθpt Y I,n,p n p (5c) Wye-connected capacitor banks with SCs and fixed capacitors are assumed constant impedance loads in the DLF model, as follows: Q YZ,n,p = CAP YZ,n,p Q YZ,n,p n p (6) where the parameter CAP YZ,n,p takes non-negative integer values less than Cn Max, and is equal to 1 for fixed capacitors. Delta-connected loads and capacitors are modeled in a similar way. Kirchhoff s Current Law (KCL) is applied to define the line current balance at each node and phase, as follows: I r,p,l = I s,p,l + I YL,n,p + I DL,n,p n p (7) l rn l sn Y L D L Similarly, based on Kirchhoff s Voltage Law (KVL), the voltages across the elements connected at each node and phase are considered to be equal, i.e.: V r,p,lrn = V s,p,lsn = V n,p n p l (8) The DLF model described above is implemented in the MATLAB R environment which allows to generate an executable file that can be run on any platform. Consequently, there is no need to install MATLAB R on every computer node, since the DOPF model is envisaged to be executed across different nodes of the Middleware System, as explained in Section IV, it needs to be a stand-alone executable file independent of any particular computing environment. Using a particular environment to implement the DOPF, would require a license for each node, which would be quite expensive. B. GA-Based DOPF Implementation The DOPF model with a GA-based solution for a 24- hour timeframe is explained in this section. The output of DOPF is an optimal set of LTC tap positions and number of capacitor blocks switched on to supply the given loads. For this reason, T, which was a fixed parameter in the DLF model, is considered to be an integer variable, called tap; Likewise, CAP, which was a fixed parameter in the DLF model, is considered to be a non-negative integer variable, called cap. The fitness function F is a sum of the objective function to be used in the optimization algorithms and penalty functions. The latter is added to the objective function so that the value of F rises significantly when the operating limits such as voltage and feeder current limits are exceeded, and remains zero when the operating limits are satisfied; this way the value of the fitness function increases significantly when the operating limits are violated. The fitness function used in the GA approach is energy losses, which are minimized in order to obtain the optimal set of LTC tap positions and number of capacitor blocks switched on over a 24-hour timeframe, and hence is defined as follows: F = J + i α i K i (9) where J is the energy losses; α i is an appropriate penalty factor for the specified operating limit, which is typically a large value; and K i is a penalty function for that operating limit. If F has a large value, the corresponding individuals are discarded. Furthermore, as demand varies during the day, LTCs and SCs might switch frequently in order to keep the voltage profile within limits; however, maintenance costs of LTCs and SCs are considerable. Hence, limits for the number of switching operations of LTCs and SCs (5 times a day) and limits for step changing for LTCs and SCs for consecutive hours (± 3 steps) are also accounted for as penalty functions in (9). The real unbalanced distribution feeder of Fig. 1 with 41 nodes from [16], is used for this work; the system has three three-phase transformers equipped with 32-step LTCs and one single-phase transformer. By changing the GA parameters such as generation, population, elite selection in each generation, and stall generation, the energy loss over a 24-hour timeframe is obtained. Table I shows a comparison of the results for the decentralized system. The closeness of objective function values with different GA parameters shows the robustness of the model, but the solution time varies depending on

4 Substation Fig. 1. Practical distribution feeder [16]. Input Map Mapped Shuffle Shuffled Reduce Output Task Task Fig. 2. Schematic of the MapReduce model [17]. the GA parameters and solution approach used. In order to have better solutions, increasing the value of both generation and population parameters tends to be effective. However, comparison of Cases 2 and 3 shows that increasing the value of stall generation and number of elite candidate for the next generation can also help to find a better solution. Since the computational burden for the centralized approach is rather expensive (the fastest solution time is 14h 24m 19s for Case 1), a distributed computing system approach to solve this problem is proposed next. IV. IMPLEMENTATION OF GA-BASED DOPF APPROACH ON DISTRIBUTED COMPUTING PLATFORM Although the GA-based method with larger parameters yielded a better solution, it requires significant computational time. Hence, in order to implement the proposed DOPF for real-time applications, the GA approach could be implemented on a distributed computing environment, using a MapReduce model on a communication Middleware System [18]. The main steps of the MapReduce model are Map and Reduce. In Map, the master-node divides the input into smaller sub-problems and distributes them between the worker-nodes; worker-nodes then process the sub-problems and return the solutions to the master-node. In Reduce, the master-node collects the results from entire worker-nodes, combines them in an output format and releases the results. If the system has a tree structure, each worker-node acts as a new master-node for its sub-branches. Each workernode operates independently; hence, the entire Map step can run in parallel, and the distributed operation reduces the computational burden [19]. There maybe a Shuffle midstage between the Map and Reduce steps; in this phase, the output of the Map operation is sorted in parallel or exchanged between the nodes in order to prepare the data for a reduction phase. Figure 2 shows the different stages of the MapReduce model [17]. Figure 3 shows the flowchart for the implementation of DOPF model with a GA-based solution. After defining F, the parameters G, P S, CR, and MR are chosen. For each hour, the individuals, which are determined by the GA procedure, will be appropriately distributed from the master-node among the Nd worker-nodes of the Middleware System (ideally, there would be enough N d nodes to fully parallelize the algorithm). Each worker-node will run the DLF model, the master-node then will combine the results, and the results will be ranked by the GA procedure based on the value of F. Based on the ranks, a pool of individuals will be selected. Cross-over and mutation operators will then be applied to generate the off-spring. The best P S fits among parents and off-springs will be selected for the new generation. This process will be repeated until the solution converges, the criteria are met, or the number of GA iterations G m reaches the maximum generation number G. By distributing the individuals from the master-node to the worker-nodes, and neglecting small overhead times, the computational time could be divided by the number of workernodes, and hence the run-time will be significantly reduced. In distribution systems, real-time intervals can be considered to be 15 to 30 minutes; hence, if the Middleware System is used, it should be possible to yield the required solution of the DOPF model with a GA-based approach in this timeframe, as demonstrated in Table I. Thus, to test the DOPF model in the distributed computing approach, this is run on two computers with 12 worker-nodes (i.e., each computer has 6 worker-nodes) and with 24 worker-nodes ((i.e., each computer has 12 workernodes). Note that if the number of worker-nodes is greater than the number of population size, for Cases 1, 2, 4, and 5 with Nd = 24, 4 worker-nodes are idle. Also, in Case 7 with Nd = 12, 2 worker-nodes are idle; therefore, it is not necessary to run Case 7 with 24 worker-nodes. Observe as well that because of stall generation, some cases are terminated before reaching the maximum number of generations G; for instance, Case 3 is terminated in 25 generations, Case 4 is terminated in 73 generations, and Case 5 is terminated in 77 generations. Comparing the solution times, note in Table I that with 12 worker-nodes in Case 3 takes 3h 26m 12s, against the 31h 54m 47s required in the centralized approach, which means that it takes around 28 hours less to reach the same objective function value. The solution time can be reduced further to 2h 35m 32s in Case 3 with 24 worker-nodes. Case 1 with 24 worker-nodes gives the best solution time with proper DOPF solution, which

5 TABLE I COMPARISON OF DOPF SOLUTION RESULTS WITH DIFFERENT GA PARAMETERS. Case G PS Elite Stall Solution time Objective function selection generation Centralized Nd = 12 Nd = 24 (Energy losses [MWh]) h 24m 19s 1h 52m 28s 1h 24m 16s h 8m 39s 2h 39m 35s 1h 57m 5s h 54m 47s 3h 26m 12s 2h 35m 32s h 53m 28s 3h 21m 35s 2h 31m 16s h 7m 27s 3h 20m 37s 2h 38m 25s h 15m 59s 5h 34m 59s 4h 12m 13s h 6m 34s 5h 45m 32s GA Procedure Middleware System Distribute individuals from the master-node among worker-nodes nd = 1 W nd = 2 W Start Define F Choose G, PS, CR, and MR Determine the number of worker-nodes (Nd) G = 0 m Worker-Nodes Run DLF on worker-nodes and calculate the fitness function nd W = 3 nd W = Nd-2 Combine the results in the master-node Rank individuals regarding to F in the master-node Does GA converge? No nd W= Nd nd W = Nd-1 Apply cross-over & mutation operators to generate off-springs Choose best PS fits among parents & off-springs Yes G m = G m +1 G m No Print the 24-hour optimal LTC tap positions and number of capacitor blocks switched in capacitor banks Fig. 3. Flowchart of proposed implementation of DOPF model with a GAbased solution. shows the effectiveness of the proposed distributed computing approach to solve the unbalanced three-phase DOPF. End G V. CONCLUSIONS In this paper, a DOPF model with a GA-based approach was implemented to find the optimal integer solutions for controllable devices such as LTCs and SCs. A comparison of DOPF results with different GA parameters showed that to have a better solution with less computational time, decentralized systems would be more beneficial than centralized systems. A MapReduce model on a communication Middleware System for smart grids was introduced to reduce the computational Yes time of the GA model, and make the model suitable for realtime applications by distributing the individuals to the workernodes; this would divide the computational burden by the number of worker-nodes. ACKNOWLEDGMENT The authors would like to thank Dr. Hongbing Fan from Wilfrid Laurier University, Waterloo, Ontario, for help and support with the implementation of the model on the Middleware System. REFERENCES [1] IEEE/PES Distribution Automation Tutorial 2007/2008, Smart Distribution Wiki, 2007/2008, [online]. Available: Distribution Automation Tutorial 2007/2008. [2] K. R. C. Mamandur and R. D. Chenoweth, Optimal Control of Reactive Power Flow for Improvements in Voltage Profiles and for Real Power Loss Minimization, IEEE Trans. Power App. Syst., vol. PAS-100, no. 7, pp , July [3] Y. Y. Hsu and F. C. Lu, A Combined Artificial Neural Network-Fuzzy Dynamic Programming Approach to Reactive Power/Voltage Control in a Distribution Substation, IEEE Trans. Power Syst., vol. 13, no. 4, pp , Nov [4] R. Baldick and F. F. Wu, Efficient Integer Optimization Algorithms for Optimal Coordination of Capacitors and Regulators, IEEE Trans. Power Syst., vol. 5, no. 3, pp , Aug [5] I. Roytelman and V. Ganesan, Coordinated Local and Centralized Control in Distribution Management Systems, IEEE Trans. Power Del., vol. 15, no. 2, pp , Apr [6] M. R. Irving and M. J. H. Sterling, Efficient Newton-Raphson Algorithm for Load-Flow Calculation in Transmission and Distribution Networks, IEE Proc. C Generation, Transmission and Distribution, vol. 134, no. 5, pp , Sept [7] R. D. Zimmerman and H. D. Chiang, Fast Decoupled Power Flow for Unbalanced Radial Distribution Systems, IEEE Trans. Power Syst., vol. 10, no. 4, pp , Nov [8] S. Bruno, S. Lamonaca, G. Rotondo, U. Stecchi, and M. L. Scala, Unbalanced Three-Phase Optimal Power Flow for Smart Grids, IEEE Trans. Ind. Electron., vol. 58, no. 10, pp , Oct [9] W. H. Kersting, A Method to Teach the Design and Operation of a Distribution System, IEEE Trans. Power App. Syst., vol. PAS-103, no. 7, pp , July [10] I. Roytelman, B. K. Wee, and R. L. Lugtu, Volt/Var Control Algorithm for Modern Distribution Management System, IEEE Trans. Power Syst., vol. 10, no. 3, pp , Aug [11] M. M. A. Salama, N. Manojlovic, V. H. Quintana, and A. Y. Chikhani, Real-Time Optimal Reactive Power Control for Distribution Networks, Int. Journal Elect. Power Energy Syst., vol. 18, no. 3, pp , Mar [12] M. B. Liu, C. A. Cañizares, and W. Huang, Reactive Power and Voltage Control in Distribution Systems with Limited Switching Operations, IEEE Trans. Power Syst., vol. 24, no. 2, pp , May 2009.

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