Optimization of reactive power compensation in distribution networks Andrija Volkanovski1, Marko Čepin, Borut Mavko 1 Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, Slovenia E-pošta: andrija.volkanovski@ijs.si, marko.cepin@ijs.si, borut.mavko@ijs.si Abstract. The optimal reactive-power compensation problem solving consists of finding the optimal combinations of the capacitors and their locations in a network in a way that technical requirements are satisfied and maximum economical effects are achieved. An approach to the optimal reactive-power compensation in distribution networks based on simulated annealing method and deterministic initialization is presented. The objective function considers the savings due to the energy losses and reduction of the peak load. The costs are divided into two categories: constant and variable depending on the capacitors size. The power summation method is used for calculation of the power flows and voltages. The method is applied on a test distribution network. By compensating the reactive power in distribution networks, several goals are achieved: voltages are increased to their nominal values, the network losses are decreased and power flows though interconnections are reduced. Key words: optimization, capacitor, distribution network, reactive power Optimizacija kompenzacije jalove moči v razdelilnih omrežjih Povzetek. Problem optimizacije kompenzacije jalove moči pomeni določiti najprimernejše število kondenzatorjev in njihove lokacije v omrežju, da je zadoščeno tehničnim zahtevam in minimizaciji stroškov. Razvit je bil pristop za optimizacijo kompenzacije jalove moči za razdelilna omrežja. Pristop temelji na optimizacijski metodi simuliranega izžiganja in na deterministični inicializaciji. V ciljni funkciji so upoštevani prihranki zaradi manjših izgub energije in zaradi zmanjšanja vršne obremenitve. Stroški so razdeljeni na konstantne in variabilne in so odvisni od velikosti kondenzatorjev. Izračun pretokov moči in napetosti temelji na metodi vsot moči. Metoda je uporabljena na testnem razdelilnem sistemu. S kompenzacijo jalove moči v razdelilnih omrežjih je doseženih več učinkov: napetosti so povečane na njihove nazivne vrednosti, zmanjšane so izgube v omrežju in zmanjšani so pretoki moči. Ključne besede: optimizacija, razdelilno omrežje, jalova moč kondenzator, 1 Introduction Capacitors are widely used in distribution networks to allow reactive power compensation, reduction of power and energy losses, improvement of service quality through voltage regulation, and delaying construction via system capacity increase. The goal is to choose the optimum capacitor size and allocation in order to maximize the benefits against the cost of capacitors installation. The early approaches are based on dynamic programming technique to consider the discrete nature of the capacitor size [1]. The analytical methods in conjunction with heuristics were applied [2], [3], [4]. The application of the simulated annealing (SA) and Received 5 September 2008 Accepted 8 January 2009 tabu search algorithm was developed [5], [6]. The genetic algorithm (GA) for searching the global optimal solution of capacitor placement problem was applied [7], [8], [9], [10], [11]. A heuristics SA method is developed and implemented for maximization of the optimized function, which is presented in the paper. 2 Problem definition The optimized function is defined as a difference between the yearly savings due to the decreased losses of energy and the peak power in the network and the yearly expenses for installation and maintenance of the capacitors. Costs of capacitors are divided in two groups: costs independent from the size of the capacitors (protection, switching equipment) and the costs proportional to the size (procurement, installation, maintenance). These costs are included in the function through the rate of the actualization pa that considers the yearly payment rate for capacitors installation, maintenance, amortization and insurance. The optimized function is written as: ■{AWo-AW)+ cp jPmo - Pm) ml fa Z (c ■Qk ) (1) Where: ce - price of electric energy €/kWh. AWo - annual loss of energy in the uncompensated network kWh. AW - annual loss of energy in the compensated network kWh. cp - peak power price €/kW. Pmo - peak power in the uncompensated network kW. c k = 1 Pm - peak power in the compensated network kW. pa - rate of actualization of costs in %. ck - constant costs for installation of the capacitor in bus k. cv - price of the capacitor proportional to its size. Qkk - installed capacity of all capacitors in bus k. Nj - number of buses with capacitors. The power summation method [12] is used for calculation of voltages, power flows and losses of power and energy in the network. The load diagram divided into two segments, with two types of load, is used for calculation of the energy losses. The time axis is divided into intervals in which the constant load is assumed. The annual energy loss is expressed as: AW = £DPi ■ Ti (2) fi - evenly (normal) distributed random number in interval [0,1]. The initial population is calculated using the deterministic method. The deterministic initialization provides a good start to optimization algorithm, resulting in a faster and improved optimization [13], [14], [15]. The optimized function, after installation of capacitors and assuming change of flows only for the reactive power, can be written as: F = c N. m P2 + (Q _ )2 AWo _ £ £ T j=1 i=1 "tj P2 + (Q,V _ q )2 U2 Where: Ap - loss of the active power in the network in the interval i. T - length of the time interval i. Ns - number of time intervals. 3 The SA and deterministic initialization The heuristic methods are effective in finding the optimal solution for large combinatorial problems. The number of possible solutions in the analyzed problem / n\m is [n ) , where: n- maximum number of capacitors of the same type in the bus. N- number of the types of the capacitors. m- number of the buses in the network. The number of variables is equal to the number of buses m. The coding of the variables with real numbers is applied. Each variable X(n) is a real number in the interval 0 = £ Bk Qck Bk = 0,if iÏ Ak (7) k=1 Where: Ak - branches that constitute a path between bus k and the source bus. The maximum of the function, given with equation (6), is obtained through differencing it: 3Qci = 2-Ce-£ £ R Qj _ ^Bk-Qk B ■T, + I Qvv - £ Bk -Qck I- Bj +2-cpiR"^r^-p*cv (8) 1 = 1, 2, ... N The following system of linear equations is obtained with equalization of the first partial derivates of equations (8) to zero: N, £ Clk-Qck = D1, 1 = 1, 2, ... N k = 1 Where: (9) N, Bii ■B,i ■T' Clk " Ce ■££-U2 j=1 i =1 UiJ il Bik T j m Ri Bil Bik -- + cp ■£ U2 =1 Uiv (10) R, Ba Q-T- ij 'j _ srRrBu-Qy 1 p Z1. „ -Î---- ,,2 2 ' a v j=1 i=1 uj i=1 uiv 2 (11) By solving the equations (9) for previously chosen Nl bus locations, the size of capacitors is determined. The procedure for deterministic initialization has two steps: first buses, where capacitors will be installed, + k=1 --1 i=1 Optimization of reactive power compensation in distribution networks 59 are randomly selected and then, by solving equations (9), the optimal capacitors size is determined. From the available discrete capacitor sizes most adequate combination, the closest to the one calculated in step 2 of deterministic initialization, is selected and used as initial population. The generated reactive power is limited to the value that prevents 'return' (overflow) in the high-voltage network. This constraint can be written as: Q0 > 0 (12) Where: Qo - reactive power at the source bus. The limitation given by Eq. (12) is included in the optimization with the application of the penalization function [14]. The SA optimization routine implements the continuous simulated annealing global optimization algorithm [16]. The SA starts with a random selection of a trial point within the step length VM (a vector of length N) of the user-selected starting point. The function is evaluated at this trial point and its value is compared to its value at the initial point. In a maximization problem, all the uphill moves are accepted and the algorithm continues from new trial point. The downhill moves may be accepted; the decision is made accounting to the Metropolis criteria. Temperature T and the size of the downhill move are applied in a probabilistic manner [17]. Parameter T is crucial for a successful use of SA. It influences VM, the step length over which the algorithm searches for optima. The relationship between the initial temperature and the resulting step length is function-dependent [17]. The starting temperature that is consistent with the optimized function is determined with trials in order to identify the T value that produces a large enough VM. For the specified optimization problem, the optimal value of T=5 is determined [17]. 4 Results Figure 1 shows the test distribution network with three voltage levels (35kV, 10 kV and 0.4 kV) and parameters given in Table 1. The input data include: interconnections parameters (transformer impedance is given for the higher voltage side), nominal voltage and peak load of the buses, type of the load diagram of the bus load, and constant expenses for installation of the capacitors in the bus. The bus data in Table 1 correspond to the second (last) bus of the given interconnection. The load diagram is approximated with two segments. The duration of the load diagram segments, load factor for peak loads and voltage of the source bus are given in Table 2. The values of the parameters, defined in Equation 1, are assumed to: ce =0.05 €/kWh, cp =150 €/kW, cp =10 €/kVAr and pa =12 %. Results for SA are obtained for the value of seed=3910, initial temperature T=5 and reduction of temperature AT=0.1. Four types of capacitors with sizes of 50, 100, 160 and 250 kVAr are taken in the analysis. 6 11 14 19 22 26 Figure 1. Test Table 1 Input 13 16 network data for the test network 24 27 Line Un (kV) R (W) X (W) B (mS) Type P (kW) Q (kVAr) cfk (€) 0-1 35 3.19 3.67 31.3 2 0 0 20.000 1-2 10 0.77 10.00 0.0 2 0 0 15.000 2-3 10 1.12 0.75 6.1 2 0 0 15.000 2-4 10 1.12 0.75 6.1 2 0 0 15.000 2-5 10 1.12 0.75 6.1 1 1040 416 15.000 3-6 0.4 1.35 5.85 0.0 2 800 150 1.000 3-7 10 1.12 0.75 6.1 2 0 0 15.000 4-8 0.4 2.88 9.38 0.0 2 504 94.5 1.000 4-9 10 1.12 0.75 6.1 2 0 0 15.000 7-10 10 1.12 0.75 6.1 2 0 0 15.000 7-11 0.4 2.88 9.58 0.0 2 320 60 1.000 9-12 10 1.12 0.75 6.1 2 0 0 15.000 9-13 0.4 2.88 9.58 0.0 2 320 60 1.000 10-14 0.4 2.88 9.38 0.0 2 504 94.5 1.000 10-15 10 1.12 0.75 6.1 2 0 0 15.000 12-16 0.4 1.35 5.85 0.0 2 800 150 1.000 12-17 10 1.12 0.75 6.1 2 0 0 15.000 15-18 10 1.12 0.75 6.1 2 0 0 15.000 15-19 0.4 2.88 9.38 0.0 2 504 94.5 1.000 17-20 10 1.12 0.75 6.1 2 0 0 15.000 17-21 0.4 2.88 9.38 0.0 2 504 94.5 1.000 18-22 0.4 1.35 5.85 0.0 2 800 150 1.000 18-23 10 1.12 0.75 6.1 2 0 0 15.000 20-24 0.4 2.88 9.38 0.0 2 504 94.5 1.000 20-25 10 1.12 0.75 6.1 2 0 0 15.000 23-26 0.4 1.35 5.85 0.0 2 800 150 1.000 25-27 0.4 1.35 5.85 0.0 2 800 150 1.000 60 Volkanovski, Cepin, Mavko Table 2 Load diagram parameters Segment 1 Segment 2 Duration (h) 4344 4416 Factor for type 1 1 1 Factor for type 2 0.76075 + j0.05736 0.64453 + j0.10415 Uo (pu) 1.03 1.00 SA case A SA case B DW (kWh) 5755967 6458532 Pmax (kW) 9649 9956 Qo (kVAr) -2.113739 2.552657 F (€) 200355.6 121967.4 Table 3 shows that deterministic initialization results in a considerable increase in the yearly savings. The value of the reactive power returned in the network Q0 is negative for case A, implying a backflow of the reactive energy from the network to the power system. The small value of Q0 can be neglected and the imposed limitation is assumed to be satisfied. The necessity for deterministic generation of the initial population is confirmed. Allocation and sizes of the capacitors for the SA case A are given in Table 4. Table 4 Optimal allocation and sizes of capacitors SA case A (kVAr) 50 100 160 250 Bus Number of capacitors Qc (kVAr) 2 6 7 5 7 3690 3 7 2 0 0 510 8 0 1 0 0 150 11 2 0 0 0 50 13 0 0 1 0 150 14 0 0 1 0 150 16 5 0 0 0 250 19 2 0 0 0 200 21 0 0 1 0 150 22 5 1 0 0 160 24 2 0 0 0 160 26 0 0 1 0 200 27 1 2 0 0 150 Total 5990 The voltages in uncompensated Uu and compensated network Uc, and the increase in voltages AU in a compensated network are given in Table 5. Table 5 Voltages in an uncompensated and compensated network The analysis is done with (case A) and without (case B) deterministic initialization. The obtained results in Table 3 include the yearly loss of energy, peak load in the network and reactive-power flow from the highvoltage network to the distribution network. Table 3 Obtained results Bus Uu (pu) Uu(kV) Uc(pu) Uc(kV) AU(V) 0 1.03 36.05 1.03 36.05 0 1 0.98979 34.643 0.99853 34.948 305 2 0.94972 9.497 0.97924 9.792 295 3 0.88169 8.817 0.92267 9.227 410 4 0.88857 8.886 0.92891 9.289 403 5 0.93391 9.339 0.96392 9.639 300 6 0.85731 0.343 0.89956 0.36 17 7 0.82602 8.26 0.87615 8.762 502 8 0.85987 0.344 0.90167 0.361 17 9 0.83527 8.353 0.88437 8.844 491 10 0.7757 7.757 0.83489 8.349 592 11 0.80669 0.323 0.85806 0.343 20 12 0.78727 7.873 0.84504 8.45 577 13 0.81619 0.326 0.86647 0.347 21 14 0.74186 0.297 0.80397 0.322 25 15 0.73444 7.344 0.79776 7.978 634 16 0.75928 0.304 0.81938 0.328 24 17 0.75314 7.531 0.81875 8.188 657 18 0.70274 7.027 0.76931 7.693 666 19 0.69817 0.279 0.76508 0.306 27 20 0.72824 7.282 0.79637 7.964 682 21 0.71802 0.287 0.78709 0.315 28 22 0.67035 0.268 0.74049 0.296 28 23 0.6867 6.867 0.75495 7.55 683 24 0.69158 0.277 0.76362 0.305 28 25 0.7129 7.129 0.7826 7.826 697 26 0.65331 0.261 0.72544 0.29 29 27 0.68113 0.272 0.7544 0.302 30 Table 5 shows that the bus voltages in the compensated network, compared to the uncompensated one, are improved to nominal values. The voltages (pu) increase as the bus nominal voltage decreases. The power flows through interconnections, for the uncompensated network, are shown in Table 6. The losses in the uncompensated network, as shown in Table 6, are AP=2195.4 kW and AQ=3114.4 kVAr. Table 7 shows the power flows in the compensated network, with annual losses AP=1746.4 kW and AQ=2496.3 kVAr. Decrease of the losses in compensated network, compared to the uncompensated are APd=448.9 kW and AQd=618.1 kVAr. Table 3 shows that yearly savings, resulting from the decrease in energy losses and peak load, are in the range of 200000 €. Table 6 Power flows (kVA) in uncompensated network Line PlineS [kW] QlineS [kVAr] PlineE [kW] QlineE [kVAr] APline [kW] AQline [kVAr] 0-1 10395.4 4845.8 10072.5 4474.3 322.9 371.5 1-2 10072.5 4493.5 9994.5 3479.9 78.1 1013.6 2-3 4695.7 1626.2 4389.1 1420.9 306.6 205.3 2-4 4242.7 1428.1 3993.8 1261.5 248.8 166.6 2-5 1056.1 426.5 1040.0 415.7 16.1 10.8 3-6 812.2 202.7 800.0 150.0 12.2 52.7 3-7 3576.9 1218.7 3371.2 1081.0 205.7 137.8 4-8 514.2 127.9 504.0 94.5 10.2 33.4 4-9 3479.6 1134.2 3289.6 1007.0 190.0 127.2 7-10 3046.5 1006.0 2877.5 892.8 169.0 113.1 7-11 324.7 75.6 320.0 60.0 4.7 15.6 9-12 2965.0 932.4 2809.9 828.5 155.1 103.9 9-13 324.6 75.2 320.0 60.0 4.6 15.2 10-14 517.8 139.3 504.0 94.5 13.8 44.8 10-15 2359.8 754.1 2245.5 677.6 114.2 76.5 12-16 815.5 217.2 800.0 150.0 15.5 67.2 12-17 1994.4 611.9 1915.8 559.2 78.6 52.7 15-18 1726.0 533.1 1658.2 487.8 67.8 45.4 15-19 519.5 145.1 504.0 94.5 15.5 50.6 17-20 1397.1 417.5 1355.1 389.4 42.0 28.1 17-21 518.7 142.3 504.0 94.5 14.7 47.8 18-22 819.9 236.2 800.0 150.0 19.9 86.2 18-23 838.3 252.1 821.0 240.5 17.4 11.6 20-24 519.8 146.1 504.0 94.5 15.8 51.6 20-25 835.3 243.9 819.3 233.2 16.0 10.7 23-26 821.0 240.8 800.0 150.0 21.0 90.8 25-27 819.3 233.5 800.0 150.0 19.3 83.5 Table 7 Power flows (kVA) in compensated network Line PlineS [kW] QlineS [kVAr] PlineE [kW] QlineE [kVAr] APline [kW] AQline [kVAr] 0-1 9946.5 2267.7 9691.0 1973.8 255.5 293.9 1-2 9691.0 1993.0 9629.3 1191.5 61.7 801.4 2-3 4497.8 718.5 4255.5 556.2 242.3 162.3 2-4 4076.4 528.2 3879.1 396.0 197.3 132.1 2-5 1055.1 425.8 1040.0 415.7 15.1 10.1 3-6 811.1 197.9 800.0 150.0 11.1 47.9 3-7 3444.4 608.9 3283.5 501.1 161.0 107.8 4-8 513.3 124.8 504.0 94.5 9.3 30.3 4-9 3365.7 521.8 3215.2 421.0 150.6 100.8 7-10 2959.3 427.9 2828.9 340.6 130.4 87.4 7-11 324.1 73.8 320.0 60.0 4.1 13.8 9-12 2891.1 348.0 2769.7 266.7 121.4 81.3 9-13 324.1 73.5 320.0 60.0 4.1 13.5 10-14 515.7 132.7 504.0 94.5 11.7 38.2 10-15 2313.2 688.5 2219.6 625.9 93.6 62.7 12-16 813.3 207.7 800.0 150.0 13.3 57.7 12-17 1956.3 59.6 1896.3 19.4 60.1 40.2 15-18 1702.6 489.8 1647.4 452.8 55.2 37.0 15-19 516.9 136.6 504.0 94.5 12.9 42.1 17-20 1380.0 385.7 1345.7 362.7 34.3 23.0 17-21 516.2 134.3 504.0 94.5 12.2 39.8 18-22 816.3 220.7 800.0 150.0 16.3 70.7 18-23 831.1 232.8 817.0 223.3 14.1 9.4 20-24 517.0 136.8 504.0 94.5 13.0 42.3 20-25 828.8 226.5 815.7 217.8 13.0 8.7 23-26 817.0 223.6 800.0 150.0 17.0 73.6 25-27 815.7 218.1 800.0 150.0 15.7 68.1 5 Conclusions 6 References A new approach for optimal compensation of the reactive power in the distribution networks is presented. The optimized function is defined as a difference between the yearly savings resulting from the decreased losses and peak power, and the yearly costs for installation and maintenance of the capacitors. The combination and allocation of the capacitors resulting in maximum yearly savings are integrated into the optimization. The simulated annealing algorithm is applied for function maximization. A procedure for deterministic initialization of the algorithm is developed. The method is tested on an example distribution networks. The obtained results demonstrate an improvement due to application of the deterministic initialization. The results confirm the need for application and optimization of the reactive power compensation in the distribution networks. The bus voltages are improved, losses are decreased and the available transfer capacities of the interconnections are increased. The decrease in energy losses and peak load in the distribution network results in substantial yearly savings. [1] Duran H., Optimum number, location and size of shunt capacitors in radial distribution feeders: a dynamic programming approach, IEEE Trans Power Apparat. Syst., 1968, 87, Pages 1769-1774. [2] Cook R.F., Optimizing the applications of shunt capacitors for reactive volt-ampere control and loss reduction, AIEE Trans., 1961, 80, Pages 1961-1969. [3] Chang N.E., Locating shunt capacitors on primary feeders for voltage control and loss reduction, IEEE Trans. Power Apparat. Syst., 1969, 88, Pages 15741577. [4] Bae Y.G., Analytical method of capacitor application on distribution primary feeders, IEEE Trans. Power Apparat. Syst., 1978, 97, Pages 1332-1337. [5] Chiang H.D., Wang J.C., Cockings O., Shin H.D., Optimal capacitor placements in distribution systems, parts I and II, IEEE Trans. Power Deliver., 1990, 5, Pages 634-649. [6] Huang Y.C., Yang H.T., Huang C.L., Solving the capacitor placement problem in a radial distribution system using tabu search approach, IEEE Trans. Power Syst., 1996, 11, Pages 1868-1873. [7] Boone G., Chiang H.D., Optimal capacitor placement in distribution systems by genetic algorithm, Int. J. Electr. Power Energy Syst., 2001, 15, Pages 155-162. [8] Sundhararajan S., Pahwa A., Optimal selection of capacitors for radial distribution systems using genetic algorithm, IEEE Trans. Power Syst., 1994, 9, Pages 1499-1507. [9] Miu K.N., Chiang H.S., Darling G., Capacitor placement, replacement and control in large distribution systems by a GA based two-stage algorithms, IEEE Trans. Power Syst., 1997, 12, Pages 1160-1166. [10] Levitin G., Kalyuzhny A., Shenkman A., Chertkov M., Optimal capacitor allocation in distribution systems using a genetic algorithm and a fast energy loss computation technique, IEEE Trans. Power Deliver., 2000, 15, Pages 621-628. [11] Kim K.H., Rhee S.B., Kim S.N., You S.K., Application of ESGA hybrid approach for voltage profile improvement by capacitor placement, IEEE Trans. Power Deliver., 2003, 18. [12] Rajičič D., Ačkovski R., Taleski R., Voltage Correction Power Flow, IEEE Transaction on Power Systems, 1994, 9(2), Pages 1056-1062. [13] Rajičič D., Ačkovski R., Optimalna kompenzacija radijalnih distributivnih mreža dekompozicijom funkcije cilja, XIX Savjetovanje elektroenergetičara Jugoslavije, Bled, 8-13 May, 1989, Paper ref. 39.14. [14] Todorovski M., Rajicic D., An initialization procedure in solving optimal power flow by genetic algorithm, IEEE Transactions on Power Systems, 2006, 21(2), Pages 480 - 487. [15] Volkanovski A., Mavko B., Boševski T., Čauševski A., Čepin M., Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system, Reliability Engineering & System Safety, 2008, 93(6), Pages 779-789. [16] Corana A., Marchesi M., Martini C., Ridella S., Minimizing Multimodal Functions of Continuous Variables with the 'Simulated Annealing' Algorithm, ACM Transactions on Mathematical Software, 1987, 13(3), Pages: 262 - 280. [17] Volkanovski A., Optimization of reactive power injections in distribution networks using heuristic methods and deterministic initializations, Zaklučna naloga - Modelska analiza, Fakulteta za matematiko in fiziko Univerze v Ljubljani, 2007, Pages 25. Borut Mavko received his Dipl.Ing. and M.Sc. degrees in electrical engineering from the Faculty of Electrical Engineering, University of Ljubljana in 1967 and 1971 respectively. He received his M.Sc. and Ph.D. in nuclear engineering from Georgia Institute of Technology, Atlanta, USA, and University of Maribor, Slovenia, in 1972 and 1979 correspondingly. Since 1967 he is employed by Jožef Stefan Institute, currently on position of Head of the Reactor Engineering Department. He is professor at the Faculty for Mathematics and Physics, University of Ljubljana, on graduate program of nuclear engineering. His research interests include nuclear safety, transient and accident analysis, probabilistic safety analysis and thermal hydraulics. Marko Čepin received his Dipl.Ing. degree in electrical engineering from the Faculty of Electrical Engineering, University of Ljubljana in 1992. He received his M.Sc. and Ph.D. in nuclear engineering from the Faculty for Mathematics and Physics, University of Ljubljana in 1995 and 1999 respectively. Since 1992 he is employed at the Reactor Engineering Department, Jožef Stefan Institute. His research interests include nuclear safety, Probabilistic Safety Assessment, technical specifications optimization and integration and development of safety analysis techniques for improvement of systems safety and reliability. Andrija Volkanovski received his Dipl.Ing. and M.Sc. degrees in electrical engineering from the Faculty of Electrical Engineering, University "St. Kiril and Metodij"- Skopje in 1999 and 2005 respectively. Since 2006 he is employed at the Reactor Engineering Department, Jožef Stefan Institute. His research interests include nuclear safety, power system reliability and complex systems analysis.