Reactive Power and Voltage Control of Power Systems Using Modified PSO

Transcription

1 J. Energy Power Sources Vol. 2, No. 5, 2015, pp Received: March 29, 2015, Published: May 30, 2015 Journal of Energy and Power Sources Reactive Power and Voltage Control of Power Systems Using Modified PSO Mohamed I. Mosaad Higher Technological Institute, Egypt on leave to Yanbu Industrial College YIC, Saudi Arabia Corresponding author: Mohamed I. Mosaad Abstract: This paper presents a reactive power and voltage control of power systems using modified particle swarm optimization. The modification in the particle swarm is changing the constrained to unconstrained optimization problem. The proposed method determines a control strategy with continuous and discrete control variables such as AVR operating values, tap positions of On-Load Tap Changer of transformers (OLTC), and the amount of reactive power compensation equipment. The method also considers voltage stability using a continuation power flow technique. The validity of the proposed method is tested over IEEE 14 bus system. A comparison between the propose method over conventional particle swarm optimization is introduced. Keywords: Constrained particle swarm optimization, reactive power and voltage control and continuation power flow. 1. Introduction Voltage and Reactive power Control (Volt/Var Control: VVC) determines an on-line control strategy for keeping voltages of target power systems at certain values at different operating conditions,such as load variation. VVC is usually realized based on power flow sensitivity analysis of the operation point considering execution time and the available data from the actual target power system. Recently, voltage stability problem has been dominating and the consideration of the stability has been required in VVC problem [1-2]. Since fast computation of voltage stability is required for VVC, Continuation Power Flow (CPF) [3], which is suitable for the calculation. CPF has been developed and verified with an actual power system [4]. VVC can be formulated as a mixed-integer nonlinear optimization problem with continuous state variables such as AVR operating values and discrete state variables such as OLTC tap positions and the amount of reactive power compensation equipment. The objective function can be varied according to the power system condition. For example, the function can be loss minimization of the target power system for the normal operating condition or minimizing the cost function for the generated power [5]. Conventionally, the methods for VVC problem have been developed using various methods such as fuzzy, expert system, mathematical programming, and sensitivity analysis [6-11]. However, a practical method for a VVC problem formulated as a mixed-integer nonlinear optimization problem has been eagerly awaited. Particle Swarm Optimization (PSO) is one of the Evolutionary Computation (EC) techniques [12]. PSO method is able to handle continuous state variables easily and search a solution in a solution space efficiently. However, it can be expanded to treat both continuous and discrete variables. Therefore, the method can be applicable to a VVC problem. Constrained Optimization (CO) problems are encountered in numerous applications. Structural optimization, engineering design, economics and allocation problems are just a few of the scientific fields in which CO problems are frequently met [13]. The CO problem can be represented as the following nonlinear programming problem:

2 Reactive Power and Voltage Control of Power Systems Using Modified PSO 183 min f(x) subject to the linear or nonlinear constraints g i (x) 0, i=1, m (1) The formulation of the constraints in Eq. (1), is not restrictive, since an inequality constraint of the form g i (x) 0, can also be represented as -g i (x) 0, and an equality constraint, g i (x) = 0,can be represented by two inequality constraints g i (x) 0 and -g i (x) 0. The most common approach for solving CO problems is the use of a penalty function. The constrained problem is transformed to an unconstrained one, by penalizing the constraints and building a single objective function, which in turn is minimized using an unconstrained optimization algorithm [14-16]. In this paper, Constrained PSO (CPSO), for a VVC problem is formulated as a mixed integer nonlinear optimization problem considering voltage stability. Voltage stability is considered using a CPF. The difference between this method and the PSO method is in the transformation of unconstrained PSO to CPSO in order to operate at constant values of bus voltages not in a range. Feasibility of the proposed method for VVC is demonstrated on IEEE 14 bus system with promising results. 2. Problem Formulation of VVC VVC for a normal power system condition can be formulated as follows: n Minimize f Loss c Min i (2) i 1 where n: the number of branches; Loss i : power loss at branch i. subject to: (1) Voltage constraint Voltage magnitude at each node must lie within their permissible ranges to maintain power quality Vmin V V max (2) Power flow constraint. Power flow of each branch must lie within their permissible ranges. (3) Voltage stability The Determined VVC strategy should keep the voltage stability of the target power system. Total Power Loss (P loss ) of the power system is calculated for a certain VVC strategy using load flow calculations. Voltage and power flow constraints can be checked at the load flow calculation and penalty values should be added if the constraints are violated. P-V curve for the determined VVC strategy can be generated and checked whether the VVC is able to keep predetermined MW values or not. 3. Optimization Algorithms Heuristic methods may be used to solve combinatorial optimization problems. These methods are called intelligent, because the move from one solution to another is done using rules close to the human reasoning. The heuristic algorithms search for a solution inside a subspace of the total search space. Thus, they are able to give a good solution of a certain problem in a reasonable computation time, but they do not assure to reach the global optimum. The most important advantage of heuristic methods lies in the fact that they are not limited by restrictive assumptions about the search space like continuity, existance of derivative of objective function, etc. Several heuristic methods exist. Among them, we may quote Tabu Search method (TS) [17], Simulated Annealing (SA) [18], Genetic Algorithms (GAs) [19-20], and Particle Swarm Optimization (PSO) algorithms [21-22]. Each one has its own properties and drawbacks. The TS is basically a deterministic method, and experience shows that no random process might restrict the search in the set of solutions. The SA needs long computation time. Further, there are an important number of parameters that are difficult to determine, such as the cooling schedule. 4. Particle Swarm Optimization Method PSO is a stochastic global optimization method which is based on simulation of social behavior. As in genetic

3 184 Reactive Power and Voltage Control of Power Systems Using Modified PSO algorithm, PSO exploits a population of potential solutions to probe the search space. In contrast to the aforementioned methods in PSO no operators inspired by natural evolution are applied to extract a new generation of candidate solutions. Instead of mutation PSO relies on the exchange of information between individuals, called particles, of the population, called swarm. In effect, each particle adjusts its trajectory towards its own previous best position, and towards the best previous position attained by any member of its neighborhood [18]. In the global variant of PSO, the whole swarm is considered as the neighborhood. Thus, global sharing of information takes place and particles profit from the discoveries and previous experience of all other companions during the search for promising regions of the landscape. To visualize the operation of the method consider the case of the single objective minimization case; promising regions in this case possess lower function values compared to others, visited previously. Let x and y denote a particle coordinates (position) and its corresponding flight speed (velocity) VV x in the x direction and VV y in the y direction. Modification of the individual position is realized by velocity and position information. PSO algorithm for N-dimensional problem formulation can be described as follows. Let P be the particle position and VV is the velocity in a search space. Consider i as a particle in the total population (swarm). The i th particle position can be represented as P i = (P i1, P i2, P i3, P in ) in the N-dimensional space. The best previous position of the i th particle is recorded and represented as P besti = (P besti1, P besti2, P besti3 P bestij ). The index of the best particle among all the particles in the group is represented by g best. The velocity i th particle is represented as VV i = (VV i1, VV i2, VV i3...vv ij ). The modified velocity and position of each particle can be calculated using the current velocity and the distance from P best to g best as indicated in following formulas: VV ( t 1) ( t 1) ( i) ij w * VVij c1 * rand1 * ( Pbest P ij best i j ( i) c2 * rand 2 * ( gbest P ) i bestij i = 1, 2 I and j = 1, 2 N ) (3) ( t 1) ( t ) ( t 1) p ij ij v ij p (4) i = 1, 2 I and j = 1, 2 N where N, number of dimensions in a particle; I, number of particles; W, inertia weight factor; T, pointer of iterations; c 1, c 2, accelerating constant; rand 1, rand 2, uniform random value in the range of [0, 1]; ( v t ) ij ( p t ) ij, velocity of the j th dimension in the i th particle;, current position of the j th dimension in the i th particle at iteration t. Inertia weighting factor w has provided improved performance when using the linearly decreasing [18-19]. Its value is decrease linearly from about 1.2 to 0.1 during a run. Suitable selection of w provides a balance between global and local exploration and exploitation, and results in fewer iterations on average to find a sufficiently optimal solution. Its value is set according to the following equation: wmax wmin w wmax t (5) tmax In Ref. (5), the first term indicates the current velocity of the particle, second term represents the cognitive part of PSO where the particle changes its velocity based on its own thinking and memory. The third term represents the social part of PSO where the particle changes its velocity based on the social-psychological adaptation of knowledge [21]. 5. The Penalty Function Approach The search space in constrained problems consists of two kinds of points: Feasible and unfeasible. Feasible points satisfy all the constraints, while unfeasible points violate at least one of them. The Penalty Function technique solves the problem through a sequence of unconstrained optimization problems [23]. Penalty functions are distinguished into two main categories: stationary and non-stationary. Stationary penalty functions, use fixed penalty values through-out

4 Reactive Power and Voltage Control of Power Systems Using Modified PSO 185 the minimization, while in contrast, in non-stationary penalty functions, the penalty values are dynamically modified. In the literature, results obtained using non-stationary penalty functions are almost always superior to those obtained through stationary functions. A penalty function is, generally, defined as [14], F ( x) fc ( x) h( k) H ( x) (6) where f c (x), is the objective function to be minimized (losses), of the constrained optimization problem f(x); h(k) is a dynamically modified penalty value, where k is the algorithm s current iteration number; and H(x) is a penalty factor, defined as: h( x) m ( qi ( x)) ( qi ( x)) qi ( x) (7) i 1 where q i (x) = max{0, g i (x)}, i = 1,., m. The function q i (x) is a relative violated function of the constraints; θ(q i (x)) is a multi-stage assignment function γ(q i (x)) is the power of the penalty function; and g i (x) are the constraints described in Eq. (1). The functions h(k), θ(q i (x)) and γ(q i (x)), are problem dependent by try and error as will be indicated. In this paper, a non-stationary multi-stage assignment penalty function is used. 6. Overview of Continuation Power Flow Continuation Power Flow (CPF) utilizes power system loads as parameters and calculates a P-V curve by modification of the parameters using the continuation method. The continuation method is one of the methods in applied mathematics and it calculates transition of equilibrium points (for example, P-V curve) by modification of parameters. In order to avoid the ill condition around the nose point, arc length along the P-V curve is introduced as an additional state variable and the power flow equation is expanded. The continuation method is applied to the expanded power flow equation and the P-V curve can be generated rapidly without ill condition around the nose point. CPFLOW can generate a P-V curve automatically and can be applied to large-scale power systems easily [3-4]. The proposed method generates a P-V curve using the CPFLOW technique and calculates a MW margin for the determined control strategy. Then, the method checks whether the MW margin is enough or not compared with the predetermined value. Using the procedure, the method checks whether the target power system can keep voltage stability by the control or not. P L = λ * P o (8) where P o is the base case load power, λ is the loading parameter and P L is the power direction. 7. Formulation of VVC Using CPSO 7.1 State Variables The following control equipment is considered in the VVC problem. (1) AVR operating values (continuous value); (2) OLTC taps position (discrete value); (3) The amount of reactive power compensation equipment SC, (discrete value); (4) The voltage constrains used in PSO are modified in this CPSO as the bus voltages are kept at pre specified fixed value (not in range). According to Eq. (8), the desired voltage values are one variable m = 1. The above state variables are treated in load flow calculation as follows: AVR operating values are treated as voltage specification values. OLTC tap positions are treated as tap ratio to each tap position. The amount of reactive power compensation equipment is treated as corresponding susceptance values. Initial AVR operating values are generated randomly between upper and lower bounds. The value is also modified in the search procedure between the bounds. OLTC tap position is initially generated randomly between the minimum and maximum tap positions. The value is modified in the search procedure among existing tap positions. The amount of reactive power compensation equipment is also generated from 0 to the number of existing equipment at the substation initially. The value is also modified in the search procedure between 0 and the number of existing equipment.

5 186 Reactive Power and Voltage Control of Power Systems Using Modified PSO 7.2 VVC Algorithm Using CPSO The proposed VVC algorithm using PSO is expressed as follows: Step 1: Initial Searching points (agents) and velocities are generated using the above-mentioned state variables randomly; Step 2: Modification of the objective function f(x) to penalty function F(x), as indicated in Eq. (7) that includes P loss and the transformation of the voltage range Vmin V Vmax, to a specified value to the searching point for each agent is calculated using load flow. If the constraints are violated, penalty is added to the loss (evaluation value of agent); Step 3: P best is set to each initial searching point. The initial best evaluated value (loss with penalty) among p bests is set to g best ; Step 4: Velocities are calculated using (4); Step 5: New searching points are calculated; Step 6: P loss to the new searching point and the evaluation value is calculated; Step 7: If the evaluation value of each agent is better than the previous p best, the value is set to p best. If the best p best is better than g best, the value is set to g best. All of g bests are stored as candidates for the final control strategy; Step 8: If the iteration number reaches to the maximum iteration number, then exit. Otherwise, go to Step 4; Step 9: The P-V curve is generated for the best g best among the stored g bests (candidates). If the MW margin is larger than the predetermined value, the control is determined as the final solution. Otherwise, select next g best and repeat the procedure. 8. Simulation Results The proposed VVC method is applied to a modified IEEE 14 bus system shown in Fig.1. The operating conditions of the system are indicated in Table 1 as [16]. Bus 1, is the slack bus, buses 2, 3, 6 and 8 are PV buses while the other buses are load buses. The followings are control variables. (1) Continuous AVR operating values of node 2, 3, 6 and 8 are 1.09 [pu], fixed value without limits due to the advantages of CPSO. (2) Discrete tap position of transformers between node 4 and 7, 4 and 9, and 5-6. They are assumed to have 20 tap positions. (3) Discrete number of installed SC in nodes 9 and 14. Each node is assumed to have three 0.06 [pu] SC [16]. The PSO s parameters used: c 1 = c 2 = 2; w was gradually decreased from 1.2 towards 0.1. Some variants of PSO, impose a maximum value on the velocity, VV max, to prevent the swarm from explosion. In Fig. 1 Modified IEEE 14 bus system. Table 1 System parameters of IEEE 14 bus system. Bus No. Volt. P.u Node specification P pu Q pu SC

6 Reactive Power and Voltage Control of Power Systems Using Modified PSO 187 this search VV max was always fixed, to the value of VV max = 4. The size of the swarm was set equal to 100, 100 runs were performed, and the PSO algorithm ran for 100 iterations, in each case. A violation tolerance was used for the constraints. Thus, a constraint g i (x) was assumed to be violated, only if g i (x) > 10-5.The calculation time is about 3 seconds using PC (Pentium 2.67 GHz). The penalty function parameters are: If q i (x) < 1, then γ(q i (x)) = 1, otherwise γ(q i (x)) = 2. h( k) k k Moreover, if q i (x) < 0.001, then θ(q i (x)) = 10, else, if q i (x) < 0.01, then θ(q i (x)) = 20, else, if q i (x) < 1, then θ(q i (x)) = 100, otherwise θ(q i (x)) = 300 Table 2 Optimal control for IEEE 14 bus system. Method Proposed method PSO Variables CPSO AVR AVR AVR AVR Tap Tap Tap SC SC Loss AVR 2: AVR operating values [pu] at node 2; Tap 4-7: Tap ratio between node 4 and 7; SC 9: Susceptance [pu] at node 9; Loss: Power loss [pu]. V (p.u) Bus 12 Bus Loading Parameter (p.u) Fig. 2 P-V curve for the optimal control (Nodes 12, 13). A comparison between the results obtained by PSO method [16] and the proposed CPSO is indicated in Table 2. The proposed method generates an optimal control for the operating condition. P loss of the original system is [pu] and CPSO is less than both normal and PSO method. The major advantage of the proposed CPSO is constant voltage operation at fixed valued, not in a range. The proposed method generates a P-V curve for the optimal control strategy using the continuation power flow technique. It is verified that the strategy can keep voltage stability as depicted in Fig. 2. It shows an example of P-V curve for nodes 12 and 13 with the optimal control strategy proposed. 9. Conclusions This paper presents a reactive power and voltage control using constrained particle swarm optimization for considering voltage stability. The proposed method formulates VVC problem as a mixed integer nonlinear optimization problem and determines control strategy with continuous and discrete control variables such as AVR operating values, OLTC tap positions, and the amount of reactive power compensation equipment with constant voltage controlled due to the modification of constrained PSO. The method also considers voltage stabsility using a continuation power flow technique. The feasibility of the proposed method for VVC is demonstrated on simple power systems with promising results. References [1] T. Van Cutsem, An approach to corrective control of voltage instability using simulation and sensitivity, IEEE Trans. on Power Systems 10 (2) (1995) [2] B.D. Thukaram, et al., Optimal reactive power dispatch algorithm for voltage stability improvement, Journal of Electrical Power &Energy Systems 18 (7) (1996) [3] H.D. Chiang, et al., CPFLOW: A practical tool for tracing power system steady-state stationary behavior due to load and generation variations, IEEE Trans. on Power Systems 10 (2) (1995)

7 188 Reactive Power and Voltage Control of Power Systems Using Modified PSO [4] H. Yoshida, Y. Fukuyama, et al., Practical continuation power flow for large-scale power system analysis, in: Proc. of IEE of Japan Annual Convention Record, No. 1313, (in Japanese) [5] M.I. Mosaad, M.M. ElMetwally, A.A. ElEmary, F.M. ElBendary, On line optimal power flow using evolutionary programming techniques, Thammasat International Journal of Science and Technology 15 (1) (2010) [6] K. Tomsovic, A fuzzy linear programming approach to the reactive power/voltage control problem, IEEE Trans. on Power Systems 7 (1) (1992) [7] B. Cova, et al., Contingency constrained optimal reactive power flow procedures for voltage control in planning and operation, IEEE Trans. on Power Systems (10) (2) (1995) [8] J.L.M. Ramos, et al., A hybrid tool to assist the operator in reactive power/voltage control and operation, IEEE Trans. on Power Systems (10) (2) (1995) [9] Q.H. Wu, et al., Power system optimal reactive power dispatch using evolutionary programming, IEEE Trans. on Power Systems (10) (3) (1995) [10] H. Vu, et al., An improved voltage control on large-scale power system, IEEE Trans. on Power Systems 11 (3) (1996) [11] T.L. Le, et al., Network equivalents and expert system application for voltage and VAR control in large-scale power systems, IEEE Trans. On Power Systems 12 (4) (1997) [12] J. Kennedy, R. Eberhart, Particle swarm optimization, in: Proc. of IEEE International Conference on Neural Networks, Perth, Australia, 1995, Vol. IV, pp [13] S. Sakthivel, D. Mary, Particle swarm optimization algorithm for voltage stability enhancement by optimal reactive power reserve management with multiple TCSCs, International Journal of Computer Applications 11 (3) (2010) [14] J.M. Yang, Y.P. Chen, J.T. Horng, C.Y. Kao, Applying family competition to evolution strategies for constrained optimization, Lecture Notes in Computer Science 1213 (1997) [15] C.A. Floudas, P.M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, Berlin Heidelberg New York, [16] H. Yoshida, K. Kawata, Y. Nakanishi, Particle swarm optimization for reactive power and voltage control considering voltage stability, in: IEEE International Conference on Intelligent System Applications to Power Systems (ISAP 99), pp [17] F. Glover, Tabu Search, University of Colorado, Boulder, CAAI Report 88-93, [18] S. Kirkpatrick, C.D. Gellat, M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983) [19] M.I. Mossad, M. Azab, A. Abu-Siada, Transformer parameters estimation from nameplate data using evolutionary programming techniques, IEEE Transactions On Power Delivery 29 (5) (2014) [20] D.E. Goldberg, Genetic Algorithms in Search Optimization and Machine Learning, Addison-Wesley Publishing Company, Inc., [21] Z.L. Gaing, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Transactions on Power Systems 18 (3) (2003) [22] E. Muthukumaran, T.K. Raja, K.P. Kumar, S.M. Kannan, Analysis of capacitor allocation in radial distribution feeder using PSO with voltage constraint, in: 1st International Conference on Electrical Energy Systems (ICEES), 3-5 Jan. 2011, pp [23] J.A. Joines, C.R. Houck, On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA s, in: Proc. IEEE Int. Conf. Evol. Comp., 1994, pp