Practical application of evolutionary computing to reactive power planning L.L. Lai J.T. Ma Indexing terms: Power systems, Reuctive power planning. Improved genetic ulgoritlm. Evolutionury progruwiming Abstract: The paper presents a practical application of evolutionary programming (EP) to reactive power planning (RPP). The proposed approach has been used in a real power system in England. Simulation results, compared with those obtained by using an improved genetic algorithm (IGA) and a conventional gradient-based optimisation method, Broyden’s method, are presented to show that the present method is better for power system planning. Several cases simulating the real network situation of both normal operation and the operation with line outages have been studied. For all these cases, EP is much better than others. The comprehensive real-state simulation results show a great potential for applications of EP to real-life power system economical and secure operation, planning and reliability assessment. List of symbols NI = set of numbers of load level durations NE = set of branch numbers N, = set of numbers of possible VAR source instal- ment buses N, = set of numbers of buses adjacent to bus i, including bus i Npp = set of PQ-bus numbers Ng N, = set of numbers of tap-setting transformer Ns = set of numbers of total buses h = per-unit energy cost dl = duration of load level 1 gk = conductance of branch k V, = voltage magnitude at bus i eY = voltage angle difference between bus i and bus j e, = fixed VAR source instalment cost at bus i C,, = per-unit VAR source purchase cost at bus i Q,, = VAR source installed at bus i = set of generator bus numbers branches 0 IEE, 1998 IEE Proceedings online no. 19982368 Paper first received 22nd October 1997 and in revised form 31st March 1998 The authors are with the Energy Systems Group, City University, London EClV OHB, UK G,, B, = mutual conductance and susceptance between G,i, Bij = self-conductance and susceptance of bus i Q,, = reactive power generation at bus i Tk = tap-setting of transformer branch k NV,, = set of numbers of buses of voltage overlimits NQs,,.? = set of numbers of buses of reactive power bus i and bus j generation overlimits 1 Introduction It is well known that reactive power planning (RPP) is one of the most complex problems of power systems as it requires the simultaneous minimisation of two objec- tive functions [l]. The first objective deals with the min- imisation of operation cost by reducing real power loss and improving the voltage profile. The second objective minimises the allocation cost of additional reactive power sources. RPP is a nonlinear optimisation prob- lem for a large-scale system with a lot of uncertainties. Various mathematical optimisation algorithms have been developed for RPP, such as the use of linear [2], nonlinear [3] or mixed integer programming [4], and decomposition methods [5]. However, these techniques are known to converge to a local optimal solution rather than to the global one for RPP which have many local minima. Evolutionary algorithms (EAs) are computational intelligence methods for optimisation based on the mechanics of natural selection, such as mutation, recombination, reproduction and selection. Mutation randomly perturbs a candidate solution. Recombination randomly mixes their parts to form a novel solution. Reproduction replicates the most suc- cessful solutions found in a population. Selection purges poor solutions from a population. Starting from a random initial generation of candidate solutions, this process produces better candidates in later generations. Recently, evolutionary algorithms, such as evolutionary programming (EP) [6] and genetic algorithms (GA), have been used in reactive power problems [7-111. In [8], four techniques have been reported to make EP practicable in real-life systems. The present paper shows the benefit obtained in using EP for RPP in a practical power system. The practical network consists of 40 buses and 56 branches. The buses comprise 8 generator buses that are repre- sented by PV-buses and a slack bus and 18 load buses. There are 11 under-load-tap-changing transformers and 5 possible VAR source installation buses. There are totally 24 control variables. Though the system is not big enough to show that the EP can solve the RPP 753 IEE Proc.-Gener. Trurzsm. Distrih., Vol. 145, No. 6, November 1998
Transcript
Practical application of evolutionary computing to reactive power planning
L.L. Lai J.T. Ma
Indexing terms: Power systems, Reuctive power planning. Improved genetic ulgoritlm. Evolutionury progruwiming
Abstract: The paper presents a practical application of evolutionary programming (EP) to reactive power planning (RPP). The proposed approach has been used in a real power system in England. Simulation results, compared with those obtained by using an improved genetic algorithm (IGA) and a conventional gradient-based optimisation method, Broyden’s method, are presented to show that the present method is better for power system planning. Several cases simulating the real network situation of both normal operation and the operation with line outages have been studied. For all these cases, EP is much better than others. The comprehensive real-state simulation results show a great potential for applications of EP to real-life power system economical and secure operation, planning and reliability assessment.
List of symbols
NI = set of numbers of load level durations NE = set of branch numbers N, = set of numbers of possible VAR source instal-
ment buses N, = set of numbers of buses adjacent to bus i,
including bus i N p p = set of PQ-bus numbers
Ng N , = set of numbers of tap-setting transformer
Ns = set of numbers of total buses h = per-unit energy cost dl = duration of load level 1 gk = conductance of branch k V, = voltage magnitude at bus i eY = voltage angle difference between bus i and bus j e, = fixed VAR source instalment cost at bus i C,, = per-unit VAR source purchase cost at bus i Q,, = VAR source installed at bus i
= set of generator bus numbers
branches
0 IEE, 1998 IEE Proceedings online no. 19982368 Paper first received 22nd October 1997 and in revised form 31st March 1998 The authors are with the Energy Systems Group, City University, London EClV OHB, UK
G,, B, = mutual conductance and susceptance between
G,i, Bij = self-conductance and susceptance of bus i
Q,, = reactive power generation at bus i Tk = tap-setting of transformer branch k N V , , = set of numbers of buses of voltage overlimits NQs,,.? = set of numbers of buses of reactive power
bus i and bus j
generation overlimits
1 Introduction
It is well known that reactive power planning (RPP) is one of the most complex problems of power systems as it requires the simultaneous minimisation of two objec- tive functions [l]. The first objective deals with the min- imisation of operation cost by reducing real power loss and improving the voltage profile. The second objective minimises the allocation cost of additional reactive power sources. RPP is a nonlinear optimisation prob- lem for a large-scale system with a lot of uncertainties. Various mathematical optimisation algorithms have been developed for RPP, such as the use of linear [2], nonlinear [3] or mixed integer programming [4], and decomposition methods [5]. However, these techniques are known to converge to a local optimal solution rather than to the global one for RPP which have many local minima. Evolutionary algorithms (EAs) are computational intelligence methods for optimisation based on the mechanics of natural selection, such as mutation, recombination, reproduction and selection. Mutation randomly perturbs a candidate solution. Recombination randomly mixes their parts to form a novel solution. Reproduction replicates the most suc- cessful solutions found in a population. Selection purges poor solutions from a population. Starting from a random initial generation of candidate solutions, this process produces better candidates in later generations. Recently, evolutionary algorithms, such as evolutionary programming (EP) [6] and genetic algorithms (GA), have been used in reactive power problems [7-111. In [8], four techniques have been reported to make EP practicable in real-life systems.
The present paper shows the benefit obtained in using EP for RPP in a practical power system. The practical network consists of 40 buses and 56 branches. The buses comprise 8 generator buses that are repre- sented by PV-buses and a slack bus and 18 load buses. There are 1 1 under-load-tap-changing transformers and 5 possible VAR source installation buses. There are totally 24 control variables. Though the system is not big enough to show that the EP can solve the RPP
problem for a real large power system, it at least shows that the EP is capable of dealing with the problems of a real power system under real operation states. It is believed that, for a large scale system, the main differ- ence is the increase in computation time and memory requirements. However, remarkable advances in com- puter technology will soon eliminate or minimise this problem. This paper provides much better insights than those described in [8-1 I], and is much more valuable to industry.
2 Problem formulation
For the convenience of the reader, the problem formu- lation will be summarised as follows [S-IO]. The objec- tive function in RPP problem comprises two terms. The first term represents the total cost of energy loss as follows:
wc = h 4 f i o s s , l (1) LEN/
where Ploss,l is the network real power loss during the period of load level 1. The Ploss,l can be expressed in the following equation in the duration d/:
f l o s s = gk (y2 + - 2 K 4 C 0 S k ) i j ) ( 2 ) k € N E
L = ( 2 . 3 )
The second term represents the cost of VAR source installations which has two components, namely, fixed installation cost and purchase cost:
IC = (e; + CciQcz) (3) ZENc
If there is no VAR source installations, the installation cost is zero. The objective function can be expressed as follows:
min fc = IC + Wc
s.t. o = Q~ - V , V, ( G ~ ~ sin8;j - B~~ cosoij) j€NZ
i E NPQ
&>Tin 5 Qci 5 Q Z a z
QZin 5 Q,i <_ QZax i E Nc
i E N,
TTin 5 T k 5 ,Tax ynin < - V, 5 Kmax
IC E NT i E N B
(4) where reactive power flow equations are used as equal- ity constraints; VAR source installation restrictions, reactive power generation restrictions, transformer tap- setting restrictions and bus voltage restrictions are used as inequality constraints. pinci can be less than zero and if Q,, is selected as a negative value, say in the light load period, variable reactance should be installed at bus i. Although the active power flow is not explicitly included in the equation, it is implicitly included in the full AC power flow computation. The transformer tap- setting T, generator bus voltages Vg and VAR source installations Q, are control variables so they are self- restricted. The load bus voltages Vloud and reactive power generations Qg are state variables, which are restricted by adding them as the quadratic penalty terms to the objective function to form a penalty func- tion. Eqn. 4 is therefore changed to the following gen- eralised objective function:
754
s t . 0 = Q, - V, V, (GzJ sink),, - B,, COS^',,) ,EN,
i E NPQ (5)
where A,, and A,, are the penalty factors, which can be increased in the optimisation procedure; VF' and Qgpi' are defined in the following equations:
ymzn if V, < Vlnzn Vzmax if V, >
In general, the purchase cost for various size of VAR is a step-like function, therefore a nondifferentiable func- tion. The transformer tap-settings are of discrete values too. It can be seen that the generalised objective func- tion Fc is a nonlinear and noncontinuous function. Furthermore, it contains a lot of uncertainties because of uncertain loads and other factors.
3 Evolutionary programming (EP) EP is a computational intelligence method in which an optimisation algorithm is the main engine for the proc- ess of three steps, namely, natural selection, mutation and competition. According to the problem, each step could be modified and configured to achieve the opti- mum result. Each possible solution to the problem is called an individual, pi.
3.1 Initialisation The fitness score ,f; of each p i is obtained by a fitness function. The initial control variable population is selected by randomly selecting p i = [ V p J , Q,", TI, i = 1, 2, ..., m, where m is the population size, from the sets of uniform distribution ranging over [ V'"'', Pus], [Qp;", e,"'""] and [Tilin, T'70-y]. The fitness score f , of each p i is obtained by running P-Q decoupled power flow.
3.2 Statistics The maximum fitness, minimum fitness, sum of fitness and average fitness of this generation is calculated.
3.3 Mutation Each pi is mutated in order to generate a new popula- tion.
The EP algorithm has four main stages:
3.4 Competition Each individual p i in the combined population has to compete with some other individuals to get its chance to be transcribed to the next generation.
The number of generations is determined by test or by monitoring the convergence of the system e.g. the difference between maximum and minimum fitness.
To make EP practicable, in addition to the two tech- niques: adaptive mutation scale and relative fitness values [9], two extra techniques, adaptive population size and competition size, have also been developed by the authors [SI.
The application of evolutionary algorithms to reactive power planning for the simplified South-West UK power network was carried out. The network parame- ters are given in Appendix 7. The following seven cases have been studied: Cuse I: A uniform load profile for 12 months at 100°/o of the load level. Cuse 2: A uniform load profile for 12 months at 130% of the load level. Case 3: A stepped load profile with 6 months at 100% and 6 months at 130% of the load level. Case 4: A uniform load profile for 12 months at 100% of the load level, with the circuit outages of (11-12) and (1 1-38). Case 5: A uniform load profile for 12 months at 100% of the load level, with the circuit outage of (12-15). Case 6: A uniform load profile for 12 months at 130% of the load level, with the circuit outages of (11-12) and (1 1-38). Case 7: A uniform load profile for 12 months at 130% of the load level, with the circuit outage of (12-15). The loads in Cases 1, 4 and 5 are:
Pkoa,j = 41.1534, Qload = 7.073 The loads in Cases 2, 6 and 7 are:
One year’s energy loss cost is employed to assess the possibility of installing the VAR sources. The variable limits and duration of load levels are given in Table 10.
Table 1: Initial power flow results
Generations and power losses
Qg
Case 1 41.4661 1.9938 0.3127 -5.0792
Case 2 54.0514 10.6546 0.5520 1.4599
Case 4 41.5897 4.4044 0.4363 -2.6685
Case 5 41.7271 4.8165 0.5734 -2.2565
Case 6 54.3195 15.3653 0.8200 6.1708
Case 7 54.5568 16.1692 1.0574 6.9743
Voltages outside limits
Case6 bus 1 2 3 4 5 7 8
Vi 0.91 0.91 0.92 0.91 0.92 0.94 0.93
bus 9 10 11 29 30 31 34
V; 0.91 0.91 0.90 0.86 0.85 0.87 0.88
Reactive power generations outside limits
Bus 24 39 40
Case 1 1.209
Case 2 1.540 3.432 2.531
Case 4 1.216 1.336
Case 5 1.422 1.396
Case 6 2.559 3.278 2.560
Case 7 3.163 4.974 3.028
The initial generations and power losses and the var- iable violations are obtained as in Table 1. In Cases 4- 7, with the same initial conditions as in Cases 1 and 2, more reactive power generations go out of their limits.
In Case 6, 14 buses are running under lower voltage limits, which can be seen as the serious security prob- lems in the network. The initial conditions of Case 3 are the same as those for Cases 1 and 2.
Table 2: Optimal results: generator bus voltage
Bus
Case 1 EP
IGA Broyden
IGA Broyden
IGA Broyden
IGA Broyden
IGA Broyden
IGA Broyden
Case2 EP
Case3 EP
Case4 EP
Case5 EP
Case6 EP
Case7 EP
IGA Broyden
18 20 22
1.04
1.04 1.01
1.08
1.08 1.09
1.09
1.05 1.08
1.05 1.05
1.05 1.05
1.05 1.06
1.08 1.09
1.07
1.09
1.09
1.06
1.07 1.06
1.07 1.07 1.03 1.03
1.04 1.05 1.03 1.05 0.94 1.03
1.02 1.03
1.03 1.06
1.06 1.04 1.06 1.04 1.07 1.06
1.01 1.04
1.05 1.06 1.07 1.08
1.04 1.05
1.07 1.09 1.07 1.06
1.03 1.03 1.04 1.09
1.06 1.07
0.99 0.96
24 39
1.07 1.06
1.06 1.06 1.02 1.02
1.06 1.04 1.06 1.04 1.01 1.00
1.06 1.04
1.06 1.04 1.00 0.98 1.06 1.06 1.06 1.06
1.02 1.01
1.05 1.06 1.05 1.05 1.00 1.04
1.05 1.04 1.06 1.04 1.02 1.01
1.04 1.05
1.04 1.05
0.98 1.02
40
1.05
1.05 1 .oo 1.01
1.02 1.02
1.02
1.04 0.98 1.06
1.05
1 .oo 1.05 1.05
1.03
1.04 1.04
1.03
1.06
1.05 1.01
Table 3: Optimal results: VAR source installations
Bus
6 13 26 29 34
Case 1 EP 0 0 0 0 0 IGA 0 0 0 0 0 Broyden 0.031 0.090 0.273 -0.361 -0.131
Case2 EP 0 0.469 0.360 0 0 IGA 0.004 0.537 0.291 0.001 0.045
Five buses, buses 6, 13, 26, 29 and 34 are selected as the possible VAR source installation buses. The generic range from -75 to +I50 MVAR is used as the VAR source installation limit. The costs of energy loss and installation can be found in Table 10. An improved
155
Table 4: Optimal results: generations and power losses
Case 1
Case 2
Case 3 100% base load
Case 3 130% base load
Case 4
Case 5
Case 6
Case 7
EP
IGA
Broyden
EP IGA
Broyden
EP
IGA
Broyden EP
IGA
Broyden
EP
IGA
Broyden
EP
IGA
B royd en
EP IGA
Broyden
EP
IGA
Broyden
ps 41.4059
41.4074
41.4502
53.9653
53.9667 54.0190
41.4020 41.4312
41.4681
53.8906
53.8797
54.0183
41.4302 41.4294
41.5275
41.6440
41.6471
41.6604
54.0575 54.0582
54.1485
54.2644
54.2669
54.4380
Q,
-0.6530 -0.6430
1.2184
6.2594 6.1231 7.1217
-1.2180
-2.9131
-0.5019
6.4781
4.7339
7.8697
1.3176 1.2481
3.0974
1.6891
1.8728
3.3103
7.0583 7.0732
12.9958 7.2582
7.3707
12.0052
Ploss
0.2743
0.2749
0.2996
0.4839
0.4841
0.51 49
0.2806 0.2898
0.3021 0.491 1
0.5003
0.5361
0.3764 0.3762
0.4149
0.5000
0.5032
0.5210
0.6475
0.6486
0.7470
0.8549
0.8573
0.9787
Qloss
-7.7385
-7.7234
-6.0658
-2.1 1305
-2.1936
-0.3322
-7.5457
-7.6737
-5.7626 1.9715
-2.1485
0.3109
-5.7730 -5.7676
-4.0604
-5.3838
-5.2005
-4.5365
0.3927
0.4469
4.1623 1.3044
1.4044
5.351 4
Table 5: Iteration and computation time (minutes) (486/ 50MHz)
Case1 Case2 Case3 Case4 Case5 Case6 Case7
EP 9.5 10.3 24.8 12.7 11.4 14.1 13.6
IGA 16.2 19.9 41.3 21.9 20.0 25.2 24.3
Broyden 2.3 2.5 4.9 2.8 2.7 3.0 2.9
Table 6: Transformer tap-setting
genetic algorithm (GA) [lo] and a conventional optimi- sation method, Broyden’s method [12], are used as the comparison methods. The optimal generator bus voltages, VAR source installations and generations and power losses are given in Tables 2-5. The transformer tap-settings are given in Table 6. In all the cases, the voltage and reactive power violations have been elimi- nated.
Table 7: Comparison between EP, GA and Broyden’s
WcSave (f/year) F, (€1 14 419 208 2 018 304 Case 1 EP
IGA
Broyden
IGA
Broyden
IGA
Broyden
Case2 EP
Case 3 EP
Case4 EP
IGA
Broyden
IGA
Broyden
IGA
Broyden
IGA
Case5 EP
Case6 EP
Case7 EP
1 986 768
688 536
3 579 336
3 568 824
1 949 976
2 444 040
1 960 488
696 420
3 148 344
3 158 856
1124784
3 857 904
3 689 712
2 754 144
9 066 600
9 008 784
3 836 880
10 643 400
10 517 256
Broyden 4 136 472
14 601 400
17 125 600
26 770 400
26 638 300
29 921 800
22 524 700
24 728 300
25 480 700
19 814 800
19 825 500
23 963 100
26 277 300
28 477 400
28 617 100
37 222 100
37 320 500
42 507 000 48 627 600
48 781 700
56 321 400
Table 7 gives the energy save and cost comparisons between EP and Broyden’s, which is obtained from the
756 IEE Proc. -Geiier. Trrriisn?. Distrili., Vol. 14J, No. 6, November 1998
following equation: waue = hdl (e:; - 82;)
Fc = IC + Wc In all ‘real-life’ simulation cases, both EP and GA
have obtained much better results than Broyden’s method. It can be concluded that both EP and GA have reached the global or near global minimum points, while Broyden’s method is stuck in some local minimum points. The results of EP are a little better than those of GA. However, both EP and GA use much more computation time than Broyden’s. Between GA and EP, the former uses almost twice the computa- tion time as the latter does. Therefore, EP is better than other methods.
5 Conclusions
An EP approach has been developed for solving the RPP problem for practical power systems. The applica- tion studies on a real power system in England show that EP always leads to a global or near global opti- mum point of the multiobjective RPP problem under different situations. EA is inherently a parallel process. Advances in distributed processing architectures could result in major reduced execution times, and it could mean a large reduction in the amount of computation to obtain a global optimal solution instead of a local one. EP has a better chance to be chosen for power system applications.
6 References
1 HSIAO, Y.T., CHIANG, H.D., LIU, C.C.,,and CHEN, Y.L.: ‘A computer package for optimal multi-objectlve VAR planning in large scale power systems’, IEEE Trans., 1994, PS-9, (2), pp.
HEYDT, G.T., and GRADY, W.M.: ‘Optimal VAR siting using linear load flow formulation’, IEEE Trans., 1983, PAS-102, (5),
SACHDEVA, S.S., and BILLINGTON, R.: ‘Optimum network VAR planning by nonlinear programming’, IEEE Truns., 1973, PAS-92, pp. 1217-1225 AOKI, K., FAN, M., and NISHIKORI, A.: ‘Optimal VAR plan- ning by approximation method for recursive mixed integer linear programming’, IEEE Truns., 1988, PWRS-3, (4), pp. 1741-1747
5 MANGOLI, M.K., LEE, K.Y., and PARK, Y.M.: ‘Optimal long-term reactive power planning using decomposition tech- niques’, Electric Power Syst. Res., 1993, 26, pp. 41-52 LAI, L.L.: ‘Intelligent system applications in power engineering: evolutionary programming and neural networks’ (John Wiley, 1998) MA, J.T., LAI, L.L., and YANG, Y.H.: ‘Application of genetic algorithms in reactive power optimization’, Proc. Chin. Soc. Elect. Eng., 1995, 15, (5), pp. 347-353 (in Chinese) MA, J.T., and LAI, L.L.: ‘Evolutionary programming approach to reactive power planning’, IEE Proc., GTD, 1996, 143, (4), pp. 365-370 LAI, L.L., and MA. J.T.: ‘Application of evolutionary program- ming to reactive power planning: comparison with non-linear pro- gramming approach’, IEEE Truns., 1997, PS-12, (l), pp. 198-206
I O MA, J.T., and LAI, L.L.: ‘Improved genetic algorithm for reac- tive power planning’. 12th Power systems coniputution conference, Dresden, Germany, 1996, pp. 499-505
11 LEE, K.Y., and YANG, F.F.: ‘Optimal reactive power planning using evolutionary algorithms: a comparative study for evolution- ary programming, evolutionary strategy, genetic algorithm and linear programming’. IEEE power engineering society paper PE- 958-PWRS- 1-04- 1997
12 NEMHAUSER, G.L., RINNOOY KAN, A.H.G., and TODD, M.J. (Ed.): ‘Optimization’ (Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1989), Vol. 1, p. 25, p. 181
668-676 2
pp. 121&1222 3
4
6
7
8
9
7 Appendix
The network parameters are given in Tables 8 and 9. All the variables are per unit quantities based on SB = 100MVA. The schematic diagram of the simplified ver-
sion of the South-West power network in the UK is shown in Fig. 1. Case study parameters are shown in Table 10.
(i) The voltage limits are changed as +6% for super- grids of 275kV and 400kV and +lo% for 132kV and terminal buses of equivalent generators. (ii) Buses 6, 13, 26, 29 and 34 are selected as potential VAR source installation sites. (iii) The generic range from -75 to +150MVAR is used as the VAR source installation limits. (iv) The 400 kV transmission line outages, lines (1 1-1 2) and (1 1-38) and line (12-15), have been considered.
