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Improved particle swarm optimization applied to reactive power reserve maximization
Electrical Power and Energy Systems 32 (2010) 368–374
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved particle swarm optimization applied to reactive power reserve maximization L.D. Arya a,*, L.S. Titare b,1, D.P. Kothari c a
Department of Electrical Engg., SGSITS, 23-Park Road, Indore, MP 452 003, India Department of Electrical Engg., Govt. Engineering College, Jabalpur, MP 456 010, India c Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India b
a r t i c l e
i n f o
Article history: Received 27 June 2006 Received in revised form 5 October 2009 Accepted 6 November 2009
Keywords: Voltage stability Particle swarm optimization (PSO) Reactive power reserve Schur’s inequality Proximity indicator
a b s t r a c t This paper presents a new approach for scheduling of reactive power control variables for voltage stability enhancement using particle swarm optimization (PSO). Cost function selected is maximization of reactive reserves of the system. To get desired stability margin a Schur’s inequality based proximity indicator has been selected whose threshold value along with reactive power reserve maximization assures desired static voltage stability margin. PSO has been selected because not only it gives global optimal solution but also its mechanization is very simple and computationally efficient. Reactive generation participation factors have been used to decide weights for reactive power reserve for each of generating bus. Developed algorithm has been implemented on 6-bus, 7-line and 25-bus 35-line standard test systems. Results have been compared with those obtained using Devidon–Fletcher–Powell’s (DFP) method. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Power system optimization problems including reactive power optimization (RPO) have complex and non-linear characteristics with large number of inequality constraints. Recently, as an alternative to the classical mathematical approaches, the non-traditional techniques [16] such as genetic algorithm, Tabu search, simulated annealing, and PSO are considered practical and powerful solution schemes to obtain the global or quasi-global optimum solution to engineering optimization problems. At times such schemes are termed as heuristic optimization techniques [1]. PSO has been selected as an optimization methodology in this paper because its mechanization is extremely simple, robustness to control parameters and computational efficiency when compared with mathematical programs and other non-traditional algorithms. Reactive power and voltage control [2], power system stabilizer design [3], and dynamic security [4] studies are the areas to which PSO has been successfully applied. Yoshida et al. [2] suggested a modified PSO to control reactive power flow and alleviating voltage limit violations. The problem was a mixed-integer, non-linear optimization problem with inequality constraints.
* Corresponding author. Tel.: +91 0731 2321016; fax: +91 07312432540. E-mail addresses: [email protected] (L.D. Arya), [email protected] (L.S. Titare), [email protected] (D.P. Kothari). 1 Tel.: +91 0734 2514246. 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.11.007
Availability of reactive power at sources and network transfer capability are two important aspects, which should be considered while rescheduling of reactive power control variables. Nedwick et al. [5] have presented a reactive management program for a practical power system. They have discussed a planning goal of supplying system reactive demand by installation of adequately sized and adequately located capacitor banks which will permit the generating unit near to unity power factor. Vaahedi et al. [6] developed a hierarchical optimization scheme, which optimized a set of control of variables such that the solution satisfied a specified voltage stability margin. Menezes et al. [7] introduced a methodology for rescheduling reactive power generation of plants and synchronous condenser for maintaining desired level of stability margin. In Ref. [8] an algorithm for voltage stability enhancement has been presented for rescheduling of reactive power control variables for voltage stability margin improvement using linearized incremental model. Dong et al. [9] developed an optimized reactive reserve management scheme using Bender’s decomposition technique. Ant colony system algorithm is applied to the reactive control problem in order to minimize real power losses, subject to operating constraints over the whole planning period by Vlachogiannis et al. [10]. Yang et al. [23] presented a technique for reactive power planning based on chance constrained programming accounting uncertain factors. Generator outputs and load demands modeled as specified probability distribution. Monte Carlo simulation along with genetic algorithm has been used for solving the optimization problem. Wu et al. [24] described an
L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374
OPF based approach for assessing the minimal reactive power support for generators in deregulated power systems. He et al. [25] proposed a method to optimize reactive power flow (ORPF) with respects to multiple objectives while maintaining voltage security. Varadarajan and Swarup [26] applied differential evolutionary algorithm for optimal reactive power dispatch and voltage control. Zhang and Liu [27] developed an algorithm for reactive power and voltage control using fuzzy adaptive particle swarm optimization (FAPSO). Zhang et al. [28] developed a computational method for reactive power market clearing. Reactive power reserve available at source is an important and necessary requirement for maintaining a desired level of voltage stability margin. Power network may have the transfer capability of reactive power but if reserve is not available and reactive power limit violation occurs than the static voltage stability limit may be inadequate. Further reactive reserves available at sources will not be of much help in maintaining desired level of stability margin, if network transfer capability is limited. This paper proposes a methodology for voltage stability enhancement accounting network loading constraint as well as optimizing reactive power reserves at various sources in proportion to their participation factors decided based on incremental load model. Voltage dependent reactive power model has been used for determining reactive power reserves, which utilizes field heating as well as armature heating limit. Section 2 explains problem formulation. Section 3 presents an overview of PSO technique. Section 4 presents implementation of the algorithm for optimizing reactive reserves. Section 5 gives results and discussions. Section 6 gives conclusions and highlights of main contributions of the paper. 2. Problem formulation 2.1. Reactive reserve The voltage stability enhancement problem is formulated as an optimal search problem whose objective is two fold: (i) maximize the reactive reserves based on the participation of reactive sources for increased loading condition and (ii) maintaining the desired stability margin with respect to current operating point. Reactive power reserve is the ability of the generators to support bus voltages under increased load or disturbance condition. Amount of reactive power, which can be fed to network, depends on present operating condition, location of the source, field and armature heating of the alternators. Nature of the change in load scenario also has impact on reactive reserves. Availability of reactive power reserve of a generator is calculated using capability curves. For a given real power output the reactive power generation is limited by both armature and field heating limit [11]. Maximum reactive power output with respect to field current limit is expressed as:
Q g;max ¼ ðV 2g =X d Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððV 2g E2max Þ=X 2d Þ P2g
ð1Þ
Maximum reactive power output Qg,max of the generator is determined by internal maximum voltage Emax corresponding to the maximum field current. Thus maximum reactive power output is determined not only on real power output Pg but also on terminal voltage Vg. Maximum reactive power output due to armature current limitation as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2g I2g;max P2g
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than Qg,max. However if Qg reaches its limit the reactive reserve is set to zero. The bus is treated as variable voltage and internal voltage of the generator behind synchronous reactance is assumed constant. This way voltage dependent reactive power limits have been accounted in a realistic way. In such situation ‘Qg’ moves on the capability curve governed by Eqs. (1) and (2). Hence if ‘Qg’ reaches the boundary the reactive reserve is set to zero and then ‘Qg’ varies as a function of terminal (bus) voltage. Such modeling (voltage dependent reactive power limit) has been adopted by many researchers [9,17–19]. 2.2. Proximity indicator based on Schur’s inequality It is assumed that load flow Jacobian at current solution point is known. Following relation can be written based on Schur’s inequality [12]:
Skmax 6
sffiffiffiffiffiffiffiffiffiffiffiffiffi X s2i;j
ð4Þ
i;j
where, Skmax is greatest eigenvalue of sensitivity matrix given, as the inverse of load flow Jacobian, si,j is ijth element of sensitivity matrix [S], which is inverse of load flow Jacobian. It is observed from matrix theory that minimum eigenvalue magnitude of load flow Jacobian is reciprocal of greatest eigenvalue of sensitivity matrix [S]. Hence following relation follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffi X Jkmin P 1= s2i;j
ð5Þ
i;j
Jkmin is minimum eigenvalue of Jacobian. Right hand side of the above expression is lower bound on the minimum eigenvalue and termed in further application of this paper as proximity indicator (s). Under low loading condition elements of sensitivity matrix are smaller and value of proximity indicator is large. As the load on the system increases the value of proximity indicator decreases since element of sensitivity matrix (si,j) increases in magnitude. In the vicinity of collapse point the value of proximity indicator practically becomes zero. Hence magnitude of ‘s’ has been used for voltage stability assessment and control in this paper. For secure operation a threshold value of proximity indicator must be maintained. Variation of proximity indicator can be co-related with load on the system with the help of power flow run. A magnitude of ‘s’ which provides adequate voltage stability margin (distance to voltage collapse point from current total load) is selected as threshold value. This value is system dependent. Computation of proximity indicator requires Jacobian inversion, which is available directly at the end of current load flow solution. Further computational efficiency is achieved using sparsity and inversion using LU factorization [21,22]. Developed algorithm is for base point setting of reactive power control variables. 2.3. Mathematical formulation
Ig,maxis maximum armature current of the generator. The reactive power reserve of the gth generator is then represented as:
The reactive reserve optimization problem is formulated as search problem whose objective is to maximize the effective reactive reserve subject to various operating and stability constraints. Objective function is given as follows:
Q g;max;res ¼ Q g;max Q g
J ¼ Max
Q g;max ¼
ð2Þ
ð3Þ
where, Qg,max is the smaller of the two values obtained from Eqs. (1) and (2). Reactive reserve is calculated using relation (3) if Qg is less
X
W i Q i;res
ð6Þ
Above objective function is optimized subject to following constraints:
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Step-2 A current velocity for ith agent in Kth iteration is given as:
(i) Power flow equations:
P ¼ f ðV; dÞ
ð7Þ
Q ¼ gðV; dÞ (ii) Inequality constraints on load bus voltages:
V min 6 V i 6 V max i i
i 2 NL
ð8Þ
ð9Þ
sth is threshold value of ‘s’ proximity indicator (iv) Reactive power generation constraint: Q min 6 Q p 6 Q max p p
p ¼ 1; 2; . . . ; NG
U min 6 U i 6 U max i i
i 2 NC
ðKþ1Þ
ð10Þ
ð11Þ
NL and NC denotes set of load buses and number of control variables. In objective function (6) Qi,res denotes reactive power reserve available at ith generation bus and Wi is the weighting factor for ith generator bus. Weighting factor Wi is selected based on the relative participation of each generator to reactive load increase in specific direction in load parameter space. The bus, which participates to a smaller extent, is given higher weight and the bus participating to a greater extent should be given lesser weight such that reserve at such bus is reduced and increased respectively. Such weighting factors can be obtained by incremental power flow equations with some algebraic manipulation. Following sensitivity relation is obtained using incremental power flow equation:
SQDi ¼ @Q i =@Q d
ð12Þ
Qi and Qd represent reactive power injection at ith generator bus and total reactive power demand of the system. This sensitivity represents change in reactive power injection at ith bus with respect to change in total reactive power demand of the system. Then weighting factor for ith generator bus is given as:
Wi ¼
!, SQDp
SQDi
ð13Þ
p
The search optimization problem is solved using particle swarm optimization (PSO). Next section presents an overview of PSO technique. 3. Particle swarm optimization technique: an overview The PSO is a population-based optimization algorithm. Its population is called swarm and each individual is called a particle [13]. Each particle flies through the solution space to search for global optimal solution. The mechanization of the PSO procedure is explained in following steps: Step-1 A current position is an n-dimensional search space, which represents a potential solution (particle or agent):
U Ki ¼ ðuKi;1 ; uKi;2 ; . . . ; uKi;n Þ i denotes ith particle and K denotes the iteration.
ð14Þ
Ui
It is stressed here that Q max is voltages dependent and p obtained using Eqs. (1) and (2). It is further clarified here that based on minimum rotor current limiter, the purpose of Q min p which is to avoid very small rotor current (these may cause problem for excitation systems) are of interest for synchronous compensators not for synchronous generators [17]. (v) Inequality constraint on control variables:
X
Step-3 At each iteration, the particle is updated by the following relation:
qðKþ1Þ ¼ W qKi þ c1 r1 ðÞðP KbestðiÞ U Ki Þ þ c2 r2 ðÞðGKbest U Ki Þ i
(iii) Voltage stability constraint:
s P sth
qðKÞ ¼ ðqKi1 ; qKi2 ; . . . ; qKin Þ i
¼ U Ki þ qKþ1 i
ð15Þ
P KbestðiÞ
where, is the best previous position of ith particle, GKbest is the global best position among all the participation in the swarm from objective function viewpoint, r1() and r2() are random digit generated from uniform distribution [0, 1], W is an inertia weight that is typically chosen in the range [0, 1]. A large inertia weight facilitates global exploration and a smaller inertia weight tends to facilitate local exploration to fine-tune the current search area. Therefore the inertia weight W is an important parameter for the PSO’s convergence behavior. A suitable value for the inertia weight usually provides balance between global and local exploration abilities and consequently results in a better optimum solution. In view of this it has been suggested that iteration wise the weight W is varied according to following relation [1]:
W ¼ W max ððW max W min Þ=NITÞ NIT max where NITmax is the maximum number of iteration supplied and NIT denotes current number of iteration. Wmax and Wmin denote maximum and minimum values of inertia weights. Thus as iteration increases inertia weight varies from Wmax say 2.0 to Wmin say 0.5, c1 and c2 are acceleration constant selected in the range 1–2. 4. Implementation of PSO for reactive reserve optimization problem In this section PSO implementation to reactive reserve optimization as formulated in Section 2 has been explained. The process of optimization is summarized as follows: Step-1 Determination of initial population: Each particle in the population consists of NC component as reactive power control variables. Each reactive power control variables for an agent is selected from a uniform distribution between Ui,min and Ui,max satisfying the inequality constraint. ‘M’ such set of initial population may be selected. The following procedure is adopted for initialization of an agent: (a) Set j = 1 (b) Select a random digit from [0, 1]. (c) Using the random digits create the value of Uj, satisfying the inequality constraint (11). (d) If j P NC (total number of control variables) then go to step (e) otherwise j = j + 1 and repeat from (b) (e) Stop the initialization. Step-2 This step consists of initialization of velocities of each particle selected in step-1 above. Initial velocity of each particle is also created at random. The velocity of element ‘j’ of individual is generated at random within the boundary given as follows:
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U min U 0ij 6 q0ij 6 U max U 0ij j j
7
¼
ðKþ1Þ Ui ;
if
ðKþ1Þ Ji
>
ðKÞ Ji
¼
ðKÞ PbestðiÞ ;
if
ðKþ1Þ Ji
6
Step-6 Further a search reduction strategy [1] has been employed to accelerate the convergence. Space reduction strategy is introduced at a stage, when the enhancement in the objective function has not been taken place during a pre-specified iteration period. In such situation the search space is dynamically reduced according to following relation: ðKþ1Þ
ðKÞ
ðKÞ
ðKÞ
U j;max ¼ U j;max ðU j;max GbestðjÞ
9 Þ D=
U j;min ¼ U j;min þ ðU j;min GbestðjÞ Þ D ; ðKþ1Þ
ðKÞ
ðKÞ
ðKÞ
ð16Þ
where, D is known as step size, which is pre-specified. In fact magnitude of D will decide how search space is reduced. Step-7 The computational algorithm is stopped if maximum numbers of iteration have been executed. PSO is a heuristic search procedure and as such no analytical convergence criterion exists, except that PSO is terminated when a pre-specified maximum numbers of iterations are executed [1]. This maximum number of iterations may be specified based on experience on the system. Further the convergence is seen by plotting a graph between objective function and number of iterations (see Figs. 1 and 2). 5. Results and discussions The developed algorithm has been implemented on 6-bus and 25-bus systems [14]. 6-bus system consists of two generator buses and four load buses. This system has in all 6-reactive power control variables. Two generators buses and shunt compensation at bus Nos. 4 and 6. OLTCs provided at line numbers 4 and 7. Maximum internal voltages and synchronous reactances were assumed as Emax1 = 2.20 pu, Emax2 = 2.05 pu, Xd1 = 1.0 pu, and Xd2 = 1.15 pu. The limits of PV-bus voltages have been assumed as 0.95 and 1.15 pu. Shunt compensation limits were assumed as between 0.00 pu and 0.055 pu. OLTC limits have been assumed between
4 3 2 1
0
20
40
60
No of iterations Fig. 1. Plot of objective function with respect to number of iteration for 6-bus system.
10 9
ðKÞ Ji
where Ji is the objective function (6) evaluated at the poðKþ1Þ sition of individual ‘i’. Gbest at iteration is set as the best ðKþ1Þ value in term of objective function among PbestðiÞ .
5
0
Objective function (J)
ðKþ1Þ PbestðiÞ
6
Objective function (J)
The initial P0bestðiÞ of individual i is set as the initial position of individual i and the initial G0ðbestÞ is determined as the position of an individual with maximum value of objective function as obtained using relation (6). Step-3 Velocity of each individual is updated using relation (14). Step-4 The position of each individual is modified using relation (15). The resulting position of an individual does not guarantee the satisfaction of inequality constraints. Hence in this paper a modified constraint handling method has been adopted [13]. In the proposed procedure the intuitive idea to maintain a feasible population is for a particle to fly-back to its previous position, when it is outside the feasible region. This has been termed as fly-back mechanism. Since the population is initialized in the feasible region flying back to a previous position will guarantee the solution to be feasible. Flying back to its previous position when a particle violates the constraint will allow a new search closer to boundary. Step-5 This step consists of updating the Pbest(i) and Gbest. The Pbest at (K + 1)th iteration is updated as follows:
8 7 6 5 4 3 2 1 0
0
20
40
60
Number of iterations Fig. 2. Plot of objective function with respect to number of iteration for 25-bus system.
0.90 and 1.10. Total base case real and reactive power load on the system is 1.35 pu and 0.32 pu.Value of proximity indicator at base case condition is 0.4251. Table 1 shows PV-bus voltage and all other load bus voltages under base condition. Star marked buses are violating the load bus voltage limit. The desired range of load bus voltage is 0.95–1.05 pu. The static voltage stability limit is 2.199415 pu for base case setting of control variables. The static voltage stability limit is total load in pu as obtained using continuation power flow up to nose point of PV curve with voltage settings of PV-buses as specified above [20]. A threshold value of s as proximity indicator has been assumed as 0.55. Initially five particles were selected satisfying all inequality constraints by procedure explained in Section 4 and are given in Table 2. The values of c1 and c2 have been selected same as one. As in most power system problems Wmax = 1.0 and Wmin = 0.5 have been selected. A typical value of D selected as 0.12. Maximum numbers of iterations were set equal to 50. Table 3 gives maximum value of objective function ‘J’ has been obtained after 19 iterations. Fig. 1 shows variation of objective function (J) with respect to number of iteration. It is observed from this Fig. 1 that solution converged in 19 iterations. Maximum numbers of iterations were specified as 50. After 19th iteration no improvement is found in Gbest. Table 4 gives optimized set of control variables. In this situation all load bus
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Table 1 Load flow solution for 6-bus test system under base case condition. Sr. No.
Control variables
Control variables magnitudes (pu)
Load bus voltages
Load bus voltages magnitudes (pu)
1 2 3 4 5 6
V1 V2 BSH4 BSH6 TAP4 TAP7
1.0000 0.9500 0.0500 0.0550 1.0000 1.0000
V3 V4 V5 V6
0.8303* 0.8588* 0.7901* 0.8420*
Total load; Pd = 1.35 pu, Qd = 0.32pu. Proximity indicator s = 0.4251. * is the buses are violating the bus voltage limit.
Table 2 Initial solutions for 6-bus test system for PSO. Sr. No.
V1 (pu)
V2 (pu)
BSH4 (pu)
BSH6 (pu)
TAP4
TAP7
J
1 2 3 4 5
1.1480 1.1448 1.1445 1.1421 1.1478
1.0933 1.1249 1.1279 1.1405 1.1228
0.0232 0.0397 0.0450 0.0466 0.0497
0.0234 0.0491 0.0362 0.0007 0.0075
0.9481 1.0299 1.0495 0.9142 1.0682
0.9554 1.0676 1.0408 0.9842 1.0289
2.33852 2.41468 2.48728 2.58620 1.44843
Table 3 Various parameters in PSO for 6-bus test system. Case
Wmax
Wmin
C1
C2
No. of initial particles
Max J
No. of iterations for convergence
1
1.0
0.5
1.0
1.0
05
6.0529
19
voltages are also given in the same table. Total reactive reserve available 1.42294 pu. Continuation power flow was carried with optimized set of reactive power control variable and static voltage stability limit was obtained as 2.31484 pu. Best initial solution (particle) selected as V1 = 1.1421 pu, V2 = 1.1405 pu, BSH4 = 0.0466 pu, BSH6 = 0.0007 pu, TAP4 = 0.9142, TAP7 = 0.9842. Reactive reserves at bus Nos. 1 and 2 with 6 best initial solution were 0.2937 pu and 0.7839 pu. Where as with optimized solution these reactive reserves are obtained as 0.50874 pu and 0.9142 pu. Weighting factors are 3.1875 and 4.8472 obtained by sensitivity analysis. Magnitude of proximity indicator with optimized solution is s = 0.5523. Similar results have been obtained for 25-bus test system. This system consists of five generator buses and these are reactive power control variables. Remaining 20 buses are load buses. Maximum internal voltages and synchronous reactances were assumed as Emax1= 2.60 pu, Emax2 = 2.15 pu, Emax3 = 2.10 pu, Emax4 = 2.30 pu, Xd1 = 1.00 pu, Xd2 = 1.15 pu, Xd3 = 1.05 pu, Emax5 = 2.15 pu, Xd4 = 1.20 pu, and Xd5 = 1.15 pu. Total base case real and reactive power load on the system is 12.41 pu and 3.876 pu.Value of proximity indicator at base case condition is 0.30. Table 5 shows PV-bus voltage and all other load bus voltages under base condition. Star
marked buses are violating the bus voltage limit. The desired range of load bus voltage is 0.95 pu to 1.05 pu.The static voltage stability limit is 17.229396 pu in base case setting of control variables. A threshold value of s as proximity indicator has been assumed as 0.39. Initially five particles were selected satisfying all inequality constraints by procedure explained in Section 4 and are given in Table 6. The values of c1 and c2 have been selected same as 1.00. As in most power system problems Wmax = 1.0 and Wmin = 0.5 have been selected. A typical value of D selected as 0.12. Maximum numbers of iterations were set equal to 50. Table 7 gives maximum value of objective function ‘J’ has been obtained after 27 iterations. Fig. 2 shows variation of objective function (J) with respect to number of iteration. Solution gets converged in 27 iterations. Maximum numbers of iterations specified were 50. After twenty seven iterations no improvement in objective function was found. Table 8 gives optimized set of control variables. In this situation all load bus voltages are also given in the same table. Total reactive reserve available 2.4526 pu.Continuation power flow was carried with optimized set of reactive power control variable and static voltage stability limit was obtained as 20.132291 pu. Best initial solution (particle) selected as V1 = 1.1373 pu, V2 = 1.0740 pu, V3 = 1.0210 pu, V4 = 1.0488 pu, V5 = 1.1002 pu. Reactive reserves at bus Nos. 1–5 with 5
Table 4 Optimized set of control variables and all bus voltages for 6-bus system. Sr. No.
Control variables
1 2 3 4 5 6
V1 V2 BSH4 BSH6 TAP4 TAP7
Optimized control variables magnitudes (pu) PSO
DFP
1.0868 1.0661 0.0497 0.0526 0.9455 0.9872
1.1042 1.0711 0.0474 0.0542 0.9529 0.9870
Total load Pd = 1.35pu, Qd = 0.32 pu. Proximity indicator using (PSO) s = 0.5523. Proximity indicator using (DFP) s = 0.5590.
Load bus voltages
Load bus voltages magnitudes (pu) PSO
DFP
V3 V4 V5 V6
0.9586 0.9699 0.9530 0.9500
0.9608 0.9714 0.9500 0.9509
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best initial solution were 0.2841 pu, 0.4094 pu, 0.3092 pu, 0.2527 pu, and 0.1723 pu. Whereas with optimized solution these reactive reserves are obtained as 0.5826 pu, 0.8121 pu, 0.4609 pu, 0.2969 pu, and 0.3001 pu respectively. Weighting factors for five generations are 3.9328, 1.2648, 5.5761, 6.2756 and 5.8215 obtained by sensitivity analysis. Magnitude of proximity indicator with optimized solution is s = 0.3920. The reactive power reserve maximization problem as formulated in Section 2 has been solved using one of the most popular non-linear optimization techniques known as Devidon–Fletcher– Powell’s (DFP) method and accounting inequality constraints by an exterior penalty function method [15]. This gradient based method has been extensively used for non-linear optimization problem solution in power system studies. Mechanization of DFP method is much more involved than PSO technique. Various sensitivities are required to evaluate the gradient vector. A penalty parameter is selected at a low initial value say 0.5 and if violation persists than this is increased in step size by 10% each time till all inequality constraints are satisfied. Results obtained with this method for 6-bus test system are given in Table 4. This table gives value of control variable as obtained by DFP method. It is seen; control variables as obtained by both the methods are in close agreement. Value of Max J obtained using DFP method is 5.9619 as against 6.0529 obtained using PSO. Reactive power reserve obtained as 0.4551 pu, and 0.9307 pu. Similar results have been obtained for 25-bus system. Control variables as obtained using DFP method has shown in Table 8 along with those obtained using PSO. Value of objective function obtained using DFP method is 9.1749 as against those obtained using PSO as 9.4987 (Table 7). Results obtained by both the method are in close agreement. Reactive reserves obtained at all five buses obtained for 25bus test system are 0.5973 pu, 0.8035 pu, 0.4816 pu, 0.3011 pu, and 0.2677 pu.
Table 5 Load flow solution for 25-bus test system under base case condition. Sr. No.
Control variables
Control variables magnitudes (pu)
Load bus voltages
Load bus voltages magnitudes (pu)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
V1 V2 V3 V4 V5
1.0000 1.0000 1.0000 1.0000 1.0000
V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25
0.9467* 0.9421* 0.9374* 0.9192* 0.9398* 0.9312* 0.9333* 0.9472* 0.8845* 0.8809* 0.9065* 0.9309* 0.9305* 0.9764 0.9378* 0.8903* 0.8425* 0.8991* 0.8130* 0.8282*
Total load Pd = 12.41 pu, Qd = 3.876 pu. Proximity indicator s = 0.30. * is the buses are violating the bus voltage limit.
Table 6 Initial solutions for 25-bus test system for PSO. Sr. No.
V1 (pu)
V2 (pu)
V3 (pu)
V4 (pu)
V5 (pu)
J
1 2 3 4 5
1.1396 1.1406 1.1373 1.1427 1.1362
1.0903 1.1021 1.0740 1.0628 1.1028
1.0318 1.0195 1.0210 1.0363 1.0861
1.0527 1.0481 1.0488 1.0511 1.0137
1.1043 1.1105 1.1002 1.0993 1.0987
2.93852 3.11468 3.29528 3.04620 2.87043
Table 7 various parameters in PSO for 25-bus test system. Case
Wmax
Wmin
C1
C2
No. of initial particles
Max J
No. of iterations for convergence
1
1.0
0.5
1.0
1.0
05
9.4987
27
Table 8 Optimized set of control variables and all bus voltages for 25-bus system. Sr. No.
Control variables
Optimized control variables magnitudes (pu) PSO
DFP
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
V1 V2 V3 V4 V5
1.1354 1.0719 1.0161 1.0488 1.0954
1.1394 1.0813 1.0268 1.0475 1.0942
Total load Pd = 12.41 pu, Qd = 3.876 pu. Proximity indicator using (PSO) s = 0.3920. Proximity indicator using (DFP) s = 0.3997.
Load bus voltages
V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25
Load bus voltages magnitudes (pu) PSO
DFP
0.9897 1.0326 1.0306 1.0171 1.0402 1.0345 1.0309 0.9823 0.9820 0.9940 1.0297 1.0368 1.0226 1.0488 1.0017 0.9751 0.9539 1.0358 0.9500 0.9741
1.0026 1.0376 1.0350 1.0199 1.0412 1.0355 1.0341 0.9949 0.9871 0.9974 1.0326 1.0378 1.0231 1.0487 1.0004 0.9745 0.9531 1.0371 0.9500 0.9750
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Results obtained by both the methods are in close agreement and this justifies the use of PSO technique for its ease of implementation and computational efficiency, which is an established advantage for such evolutionary algorithm [16].
6. Conclusions This paper discusses the management of reactive power reserves in order to improve static voltage stability. This has been achieved via a modified PSO algorithm. Advantage of PSO algorithm is that its mechanization is simple without much mathematical complexity. Moreover global optimal solution is obtained and local optimal solution is avoided via search procedure. Important about the methodology is that not only reactive reserve is optimized but inequality constraint on proximity indicator guarantee required static voltage stability margin. Network as well as source capabilities are important from voltage instability viewpoint. Further contribution to reactive reserve has been considered from participation viewpoint by weighting factors. This is important aspect, which has been considered since large reactive reserve available at a generator bus, which is not utilized in a load increased scenario, is not of great significance. Hence a generator participating to a larger extent has been given lesser weight in the PSO algorithm. The algorithm has been implemented on 6-bus and 25-bus sample test systems.
References [1] Park JB, Lee KS, Shin JR, Lee KY. A particle swarm optimization for economic dispatch with non-smooth cost functions. IEEE Trans Power Syst 2005;20(1):34–42. [2] Yoshida H, Kawata K, Fukuyama Y, Takayama S, Nakanishi Y. A particle swarm optimization for reactive power and voltage control considering voltage security assessment. IEEE Trans Power Syst 2000;15(November):1232–9. [3] Abido MA. Optimal design of power system stabilizers using particle swarm optimization. IEEE Trans Energy Convers 2002;17(3):406–13. [4] Kasabadis IN, El-Sharkawi MA, Marks RJ, Moulin LS, da Silva APA. Dynamic security border identification using enhanced particle swarm optimization. IEEE Trans Power Syst 2002;17(August):723–9. [5] Nedwick P, Mistr Jr AF, Croasdale EB. Reactive management a key to survival in the 1990s. IEEE Trans Power Syst 1995;10(2):1036–43.
[6] Vaahedi E, Masour Y, Fuchs C, Granville S, Latore MDL, Hamadoni Zadheh H. Dynamic security constrained optimal power flow/Var planning. IEEE Trans on Power Syst 2001;16(1):38–43. [7] Menezes T, da Silva LC, da Costa VF. Dynamic Var sources scheduling for improving voltage stability margin. IEEE Trans Power Syst 2003(May):469–71. [8] Arya LD, Sakravadia DK, Kothari DP. Corrective rescheduling for static voltage stability control. Int J Elect Power Energy Syst 2005;27(January):3–12. [9] Dong F, Chowdhury BH, Crow ML, Acar L. Improving voltage stability by reactive power reserve management. IEEE Trans PS 2005;20(1):338–44. [10] Vlachogiannis JG, Hatziargyriou ND, Lee KY. Ant colony system-based algorithm for constrained load flow problem. IEEE Trans PS 2005;20(3): 1241–9. [11] Kundur P. Power System Stability and Control. Mc Graw-Hill Inc.; 1994. [12] Kreyszig EW. Advanced Engineering Mathematics. John Wiley; 2001. [13] HE S, Prempain E, Wu QH. An improved particle swarm optimization for mechanical design optimization problems. J Eng Optim 2004;36(5):585–605. [14] Bijwe PR, Kothari DP, Arya LD. Alleviation of line over loads and voltage violations by corrective rescheduling. Proc IEE Part – C 1993;140(4):249–55. [15] arya LD, Choube SC, Kothari DP. Emission constrained secure economic dispatch. Int J Elect Power Energy Sys 1997;19(5):279–85. [16] Deb K. Multi Objective Optimization Using Evolutionary Algorithms. John Wiley and Sons; 2003. [17] Lof PA, Andersson G, Hill DJ. Voltage dependent reactive power limits for voltage stability studies. IEEE Trans Power Syst 1995;10(1):220–8. [18] Van Cutsem T. A method to compute reactive power margins with respect to voltage collapse. IEEE Trans Power Syst 1991;6(1):145–56. [19] Johansson SG, Daalder JE, Popovic D, Hill DJ. Avoiding voltage collapse by fast active power rescheduling. Int J Elect Power Energy Syst 1997;19(8):501–9. [20] Ajjarapu V, Christy C. The continuation power flow: a tool for steady state voltage stability analysis. IEEE Trans PS 1992;7(1):416–22. [21] Tinney WF, Walker JW. Direct solution of sparse network equations by optimally ordered factorization. Proc IEE 1967;55:1801–5. [22] Wallach Y. Calculations and Program for Power System network. Engle Wood Cliffs (NJ): Prentice-Hall, Inc.; 1986. [23] Yang N, Yu CW, Wen F, Chung CY. An investigation of reactive power planning based on chance constrained programming. Int J Elect Power Energy Sys 2007;29(9):650–6. [24] Wu H, Yu CW, Xu N, Lin XJ. An OPF based approach for assessing the minimal reactive power support for generators in deregulated power systems. Int J Elect Power Energy Sys 2008;30(1):23–30. [25] He R, Taylor GA, Song YH. Multi-objective optimal reactive power flow including voltage security and demand profile classification. Int J Elect Power Energy Syst 2008;30(5):327–36. [26] Varadarajan M, Swarup KS. Differential evolutionary algorithm for optimal reactive power dispatch. Int J Elect Power Energy Syst 2008;30(8):435–41. [27] Zhang W, Liu Y. Multi-objective reactive power and voltage control based on fuzzy optimization strategy and fuzzy adaptive particle swarm. Int J Elect Power Energy Syst 2008;30(9):525–32. [28] Zhang T, Elkasrawy A, Venkatesh B. A new computational method for reactive power marked clearing. Int J Elect Power Energy Syst 2009;31(6):285–93.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved particle swarm optimization applied to reactive power reserve maximization L.D. Arya a,*, L.S. Titare b,1, D.P. Kothari c a
Department of Electrical Engg., SGSITS, 23-Park Road, Indore, MP 452 003, India Department of Electrical Engg., Govt. Engineering College, Jabalpur, MP 456 010, India c Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India b
a r t i c l e
i n f o
Article history: Received 27 June 2006 Received in revised form 5 October 2009 Accepted 6 November 2009
Keywords: Voltage stability Particle swarm optimization (PSO) Reactive power reserve Schur’s inequality Proximity indicator
a b s t r a c t This paper presents a new approach for scheduling of reactive power control variables for voltage stability enhancement using particle swarm optimization (PSO). Cost function selected is maximization of reactive reserves of the system. To get desired stability margin a Schur’s inequality based proximity indicator has been selected whose threshold value along with reactive power reserve maximization assures desired static voltage stability margin. PSO has been selected because not only it gives global optimal solution but also its mechanization is very simple and computationally efficient. Reactive generation participation factors have been used to decide weights for reactive power reserve for each of generating bus. Developed algorithm has been implemented on 6-bus, 7-line and 25-bus 35-line standard test systems. Results have been compared with those obtained using Devidon–Fletcher–Powell’s (DFP) method. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Power system optimization problems including reactive power optimization (RPO) have complex and non-linear characteristics with large number of inequality constraints. Recently, as an alternative to the classical mathematical approaches, the non-traditional techniques [16] such as genetic algorithm, Tabu search, simulated annealing, and PSO are considered practical and powerful solution schemes to obtain the global or quasi-global optimum solution to engineering optimization problems. At times such schemes are termed as heuristic optimization techniques [1]. PSO has been selected as an optimization methodology in this paper because its mechanization is extremely simple, robustness to control parameters and computational efficiency when compared with mathematical programs and other non-traditional algorithms. Reactive power and voltage control [2], power system stabilizer design [3], and dynamic security [4] studies are the areas to which PSO has been successfully applied. Yoshida et al. [2] suggested a modified PSO to control reactive power flow and alleviating voltage limit violations. The problem was a mixed-integer, non-linear optimization problem with inequality constraints.
* Corresponding author. Tel.: +91 0731 2321016; fax: +91 07312432540. E-mail addresses: [email protected] (L.D. Arya), [email protected] (L.S. Titare), [email protected] (D.P. Kothari). 1 Tel.: +91 0734 2514246. 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.11.007
Availability of reactive power at sources and network transfer capability are two important aspects, which should be considered while rescheduling of reactive power control variables. Nedwick et al. [5] have presented a reactive management program for a practical power system. They have discussed a planning goal of supplying system reactive demand by installation of adequately sized and adequately located capacitor banks which will permit the generating unit near to unity power factor. Vaahedi et al. [6] developed a hierarchical optimization scheme, which optimized a set of control of variables such that the solution satisfied a specified voltage stability margin. Menezes et al. [7] introduced a methodology for rescheduling reactive power generation of plants and synchronous condenser for maintaining desired level of stability margin. In Ref. [8] an algorithm for voltage stability enhancement has been presented for rescheduling of reactive power control variables for voltage stability margin improvement using linearized incremental model. Dong et al. [9] developed an optimized reactive reserve management scheme using Bender’s decomposition technique. Ant colony system algorithm is applied to the reactive control problem in order to minimize real power losses, subject to operating constraints over the whole planning period by Vlachogiannis et al. [10]. Yang et al. [23] presented a technique for reactive power planning based on chance constrained programming accounting uncertain factors. Generator outputs and load demands modeled as specified probability distribution. Monte Carlo simulation along with genetic algorithm has been used for solving the optimization problem. Wu et al. [24] described an
L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374
OPF based approach for assessing the minimal reactive power support for generators in deregulated power systems. He et al. [25] proposed a method to optimize reactive power flow (ORPF) with respects to multiple objectives while maintaining voltage security. Varadarajan and Swarup [26] applied differential evolutionary algorithm for optimal reactive power dispatch and voltage control. Zhang and Liu [27] developed an algorithm for reactive power and voltage control using fuzzy adaptive particle swarm optimization (FAPSO). Zhang et al. [28] developed a computational method for reactive power market clearing. Reactive power reserve available at source is an important and necessary requirement for maintaining a desired level of voltage stability margin. Power network may have the transfer capability of reactive power but if reserve is not available and reactive power limit violation occurs than the static voltage stability limit may be inadequate. Further reactive reserves available at sources will not be of much help in maintaining desired level of stability margin, if network transfer capability is limited. This paper proposes a methodology for voltage stability enhancement accounting network loading constraint as well as optimizing reactive power reserves at various sources in proportion to their participation factors decided based on incremental load model. Voltage dependent reactive power model has been used for determining reactive power reserves, which utilizes field heating as well as armature heating limit. Section 2 explains problem formulation. Section 3 presents an overview of PSO technique. Section 4 presents implementation of the algorithm for optimizing reactive reserves. Section 5 gives results and discussions. Section 6 gives conclusions and highlights of main contributions of the paper. 2. Problem formulation 2.1. Reactive reserve The voltage stability enhancement problem is formulated as an optimal search problem whose objective is two fold: (i) maximize the reactive reserves based on the participation of reactive sources for increased loading condition and (ii) maintaining the desired stability margin with respect to current operating point. Reactive power reserve is the ability of the generators to support bus voltages under increased load or disturbance condition. Amount of reactive power, which can be fed to network, depends on present operating condition, location of the source, field and armature heating of the alternators. Nature of the change in load scenario also has impact on reactive reserves. Availability of reactive power reserve of a generator is calculated using capability curves. For a given real power output the reactive power generation is limited by both armature and field heating limit [11]. Maximum reactive power output with respect to field current limit is expressed as:
Q g;max ¼ ðV 2g =X d Þ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððV 2g E2max Þ=X 2d Þ P2g
ð1Þ
Maximum reactive power output Qg,max of the generator is determined by internal maximum voltage Emax corresponding to the maximum field current. Thus maximum reactive power output is determined not only on real power output Pg but also on terminal voltage Vg. Maximum reactive power output due to armature current limitation as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2g I2g;max P2g
369
than Qg,max. However if Qg reaches its limit the reactive reserve is set to zero. The bus is treated as variable voltage and internal voltage of the generator behind synchronous reactance is assumed constant. This way voltage dependent reactive power limits have been accounted in a realistic way. In such situation ‘Qg’ moves on the capability curve governed by Eqs. (1) and (2). Hence if ‘Qg’ reaches the boundary the reactive reserve is set to zero and then ‘Qg’ varies as a function of terminal (bus) voltage. Such modeling (voltage dependent reactive power limit) has been adopted by many researchers [9,17–19]. 2.2. Proximity indicator based on Schur’s inequality It is assumed that load flow Jacobian at current solution point is known. Following relation can be written based on Schur’s inequality [12]:
Skmax 6
sffiffiffiffiffiffiffiffiffiffiffiffiffi X s2i;j
ð4Þ
i;j
where, Skmax is greatest eigenvalue of sensitivity matrix given, as the inverse of load flow Jacobian, si,j is ijth element of sensitivity matrix [S], which is inverse of load flow Jacobian. It is observed from matrix theory that minimum eigenvalue magnitude of load flow Jacobian is reciprocal of greatest eigenvalue of sensitivity matrix [S]. Hence following relation follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffi X Jkmin P 1= s2i;j
ð5Þ
i;j
Jkmin is minimum eigenvalue of Jacobian. Right hand side of the above expression is lower bound on the minimum eigenvalue and termed in further application of this paper as proximity indicator (s). Under low loading condition elements of sensitivity matrix are smaller and value of proximity indicator is large. As the load on the system increases the value of proximity indicator decreases since element of sensitivity matrix (si,j) increases in magnitude. In the vicinity of collapse point the value of proximity indicator practically becomes zero. Hence magnitude of ‘s’ has been used for voltage stability assessment and control in this paper. For secure operation a threshold value of proximity indicator must be maintained. Variation of proximity indicator can be co-related with load on the system with the help of power flow run. A magnitude of ‘s’ which provides adequate voltage stability margin (distance to voltage collapse point from current total load) is selected as threshold value. This value is system dependent. Computation of proximity indicator requires Jacobian inversion, which is available directly at the end of current load flow solution. Further computational efficiency is achieved using sparsity and inversion using LU factorization [21,22]. Developed algorithm is for base point setting of reactive power control variables. 2.3. Mathematical formulation
Ig,maxis maximum armature current of the generator. The reactive power reserve of the gth generator is then represented as:
The reactive reserve optimization problem is formulated as search problem whose objective is to maximize the effective reactive reserve subject to various operating and stability constraints. Objective function is given as follows:
Q g;max;res ¼ Q g;max Q g
J ¼ Max
Q g;max ¼
ð2Þ
ð3Þ
where, Qg,max is the smaller of the two values obtained from Eqs. (1) and (2). Reactive reserve is calculated using relation (3) if Qg is less
X
W i Q i;res
ð6Þ
Above objective function is optimized subject to following constraints:
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Step-2 A current velocity for ith agent in Kth iteration is given as:
(i) Power flow equations:
P ¼ f ðV; dÞ
ð7Þ
Q ¼ gðV; dÞ (ii) Inequality constraints on load bus voltages:
V min 6 V i 6 V max i i
i 2 NL
ð8Þ
ð9Þ
sth is threshold value of ‘s’ proximity indicator (iv) Reactive power generation constraint: Q min 6 Q p 6 Q max p p
p ¼ 1; 2; . . . ; NG
U min 6 U i 6 U max i i
i 2 NC
ðKþ1Þ
ð10Þ
ð11Þ
NL and NC denotes set of load buses and number of control variables. In objective function (6) Qi,res denotes reactive power reserve available at ith generation bus and Wi is the weighting factor for ith generator bus. Weighting factor Wi is selected based on the relative participation of each generator to reactive load increase in specific direction in load parameter space. The bus, which participates to a smaller extent, is given higher weight and the bus participating to a greater extent should be given lesser weight such that reserve at such bus is reduced and increased respectively. Such weighting factors can be obtained by incremental power flow equations with some algebraic manipulation. Following sensitivity relation is obtained using incremental power flow equation:
SQDi ¼ @Q i =@Q d
ð12Þ
Qi and Qd represent reactive power injection at ith generator bus and total reactive power demand of the system. This sensitivity represents change in reactive power injection at ith bus with respect to change in total reactive power demand of the system. Then weighting factor for ith generator bus is given as:
Wi ¼
!, SQDp
SQDi
ð13Þ
p
The search optimization problem is solved using particle swarm optimization (PSO). Next section presents an overview of PSO technique. 3. Particle swarm optimization technique: an overview The PSO is a population-based optimization algorithm. Its population is called swarm and each individual is called a particle [13]. Each particle flies through the solution space to search for global optimal solution. The mechanization of the PSO procedure is explained in following steps: Step-1 A current position is an n-dimensional search space, which represents a potential solution (particle or agent):
U Ki ¼ ðuKi;1 ; uKi;2 ; . . . ; uKi;n Þ i denotes ith particle and K denotes the iteration.
ð14Þ
Ui
It is stressed here that Q max is voltages dependent and p obtained using Eqs. (1) and (2). It is further clarified here that based on minimum rotor current limiter, the purpose of Q min p which is to avoid very small rotor current (these may cause problem for excitation systems) are of interest for synchronous compensators not for synchronous generators [17]. (v) Inequality constraint on control variables:
X
Step-3 At each iteration, the particle is updated by the following relation:
qðKþ1Þ ¼ W qKi þ c1 r1 ðÞðP KbestðiÞ U Ki Þ þ c2 r2 ðÞðGKbest U Ki Þ i
(iii) Voltage stability constraint:
s P sth
qðKÞ ¼ ðqKi1 ; qKi2 ; . . . ; qKin Þ i
¼ U Ki þ qKþ1 i
ð15Þ
P KbestðiÞ
where, is the best previous position of ith particle, GKbest is the global best position among all the participation in the swarm from objective function viewpoint, r1() and r2() are random digit generated from uniform distribution [0, 1], W is an inertia weight that is typically chosen in the range [0, 1]. A large inertia weight facilitates global exploration and a smaller inertia weight tends to facilitate local exploration to fine-tune the current search area. Therefore the inertia weight W is an important parameter for the PSO’s convergence behavior. A suitable value for the inertia weight usually provides balance between global and local exploration abilities and consequently results in a better optimum solution. In view of this it has been suggested that iteration wise the weight W is varied according to following relation [1]:
W ¼ W max ððW max W min Þ=NITÞ NIT max where NITmax is the maximum number of iteration supplied and NIT denotes current number of iteration. Wmax and Wmin denote maximum and minimum values of inertia weights. Thus as iteration increases inertia weight varies from Wmax say 2.0 to Wmin say 0.5, c1 and c2 are acceleration constant selected in the range 1–2. 4. Implementation of PSO for reactive reserve optimization problem In this section PSO implementation to reactive reserve optimization as formulated in Section 2 has been explained. The process of optimization is summarized as follows: Step-1 Determination of initial population: Each particle in the population consists of NC component as reactive power control variables. Each reactive power control variables for an agent is selected from a uniform distribution between Ui,min and Ui,max satisfying the inequality constraint. ‘M’ such set of initial population may be selected. The following procedure is adopted for initialization of an agent: (a) Set j = 1 (b) Select a random digit from [0, 1]. (c) Using the random digits create the value of Uj, satisfying the inequality constraint (11). (d) If j P NC (total number of control variables) then go to step (e) otherwise j = j + 1 and repeat from (b) (e) Stop the initialization. Step-2 This step consists of initialization of velocities of each particle selected in step-1 above. Initial velocity of each particle is also created at random. The velocity of element ‘j’ of individual is generated at random within the boundary given as follows:
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U min U 0ij 6 q0ij 6 U max U 0ij j j
7
¼
ðKþ1Þ Ui ;
if
ðKþ1Þ Ji
>
ðKÞ Ji
¼
ðKÞ PbestðiÞ ;
if
ðKþ1Þ Ji
6
Step-6 Further a search reduction strategy [1] has been employed to accelerate the convergence. Space reduction strategy is introduced at a stage, when the enhancement in the objective function has not been taken place during a pre-specified iteration period. In such situation the search space is dynamically reduced according to following relation: ðKþ1Þ
ðKÞ
ðKÞ
ðKÞ
U j;max ¼ U j;max ðU j;max GbestðjÞ
9 Þ D=
U j;min ¼ U j;min þ ðU j;min GbestðjÞ Þ D ; ðKþ1Þ
ðKÞ
ðKÞ
ðKÞ
ð16Þ
where, D is known as step size, which is pre-specified. In fact magnitude of D will decide how search space is reduced. Step-7 The computational algorithm is stopped if maximum numbers of iteration have been executed. PSO is a heuristic search procedure and as such no analytical convergence criterion exists, except that PSO is terminated when a pre-specified maximum numbers of iterations are executed [1]. This maximum number of iterations may be specified based on experience on the system. Further the convergence is seen by plotting a graph between objective function and number of iterations (see Figs. 1 and 2). 5. Results and discussions The developed algorithm has been implemented on 6-bus and 25-bus systems [14]. 6-bus system consists of two generator buses and four load buses. This system has in all 6-reactive power control variables. Two generators buses and shunt compensation at bus Nos. 4 and 6. OLTCs provided at line numbers 4 and 7. Maximum internal voltages and synchronous reactances were assumed as Emax1 = 2.20 pu, Emax2 = 2.05 pu, Xd1 = 1.0 pu, and Xd2 = 1.15 pu. The limits of PV-bus voltages have been assumed as 0.95 and 1.15 pu. Shunt compensation limits were assumed as between 0.00 pu and 0.055 pu. OLTC limits have been assumed between
4 3 2 1
0
20
40
60
No of iterations Fig. 1. Plot of objective function with respect to number of iteration for 6-bus system.
10 9
ðKÞ Ji
where Ji is the objective function (6) evaluated at the poðKþ1Þ sition of individual ‘i’. Gbest at iteration is set as the best ðKþ1Þ value in term of objective function among PbestðiÞ .
5
0
Objective function (J)
ðKþ1Þ PbestðiÞ
6
Objective function (J)
The initial P0bestðiÞ of individual i is set as the initial position of individual i and the initial G0ðbestÞ is determined as the position of an individual with maximum value of objective function as obtained using relation (6). Step-3 Velocity of each individual is updated using relation (14). Step-4 The position of each individual is modified using relation (15). The resulting position of an individual does not guarantee the satisfaction of inequality constraints. Hence in this paper a modified constraint handling method has been adopted [13]. In the proposed procedure the intuitive idea to maintain a feasible population is for a particle to fly-back to its previous position, when it is outside the feasible region. This has been termed as fly-back mechanism. Since the population is initialized in the feasible region flying back to a previous position will guarantee the solution to be feasible. Flying back to its previous position when a particle violates the constraint will allow a new search closer to boundary. Step-5 This step consists of updating the Pbest(i) and Gbest. The Pbest at (K + 1)th iteration is updated as follows:
8 7 6 5 4 3 2 1 0
0
20
40
60
Number of iterations Fig. 2. Plot of objective function with respect to number of iteration for 25-bus system.
0.90 and 1.10. Total base case real and reactive power load on the system is 1.35 pu and 0.32 pu.Value of proximity indicator at base case condition is 0.4251. Table 1 shows PV-bus voltage and all other load bus voltages under base condition. Star marked buses are violating the load bus voltage limit. The desired range of load bus voltage is 0.95–1.05 pu. The static voltage stability limit is 2.199415 pu for base case setting of control variables. The static voltage stability limit is total load in pu as obtained using continuation power flow up to nose point of PV curve with voltage settings of PV-buses as specified above [20]. A threshold value of s as proximity indicator has been assumed as 0.55. Initially five particles were selected satisfying all inequality constraints by procedure explained in Section 4 and are given in Table 2. The values of c1 and c2 have been selected same as one. As in most power system problems Wmax = 1.0 and Wmin = 0.5 have been selected. A typical value of D selected as 0.12. Maximum numbers of iterations were set equal to 50. Table 3 gives maximum value of objective function ‘J’ has been obtained after 19 iterations. Fig. 1 shows variation of objective function (J) with respect to number of iteration. It is observed from this Fig. 1 that solution converged in 19 iterations. Maximum numbers of iterations were specified as 50. After 19th iteration no improvement is found in Gbest. Table 4 gives optimized set of control variables. In this situation all load bus
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Table 1 Load flow solution for 6-bus test system under base case condition. Sr. No.
Control variables
Control variables magnitudes (pu)
Load bus voltages
Load bus voltages magnitudes (pu)
1 2 3 4 5 6
V1 V2 BSH4 BSH6 TAP4 TAP7
1.0000 0.9500 0.0500 0.0550 1.0000 1.0000
V3 V4 V5 V6
0.8303* 0.8588* 0.7901* 0.8420*
Total load; Pd = 1.35 pu, Qd = 0.32pu. Proximity indicator s = 0.4251. * is the buses are violating the bus voltage limit.
Table 2 Initial solutions for 6-bus test system for PSO. Sr. No.
V1 (pu)
V2 (pu)
BSH4 (pu)
BSH6 (pu)
TAP4
TAP7
J
1 2 3 4 5
1.1480 1.1448 1.1445 1.1421 1.1478
1.0933 1.1249 1.1279 1.1405 1.1228
0.0232 0.0397 0.0450 0.0466 0.0497
0.0234 0.0491 0.0362 0.0007 0.0075
0.9481 1.0299 1.0495 0.9142 1.0682
0.9554 1.0676 1.0408 0.9842 1.0289
2.33852 2.41468 2.48728 2.58620 1.44843
Table 3 Various parameters in PSO for 6-bus test system. Case
Wmax
Wmin
C1
C2
No. of initial particles
Max J
No. of iterations for convergence
1
1.0
0.5
1.0
1.0
05
6.0529
19
voltages are also given in the same table. Total reactive reserve available 1.42294 pu. Continuation power flow was carried with optimized set of reactive power control variable and static voltage stability limit was obtained as 2.31484 pu. Best initial solution (particle) selected as V1 = 1.1421 pu, V2 = 1.1405 pu, BSH4 = 0.0466 pu, BSH6 = 0.0007 pu, TAP4 = 0.9142, TAP7 = 0.9842. Reactive reserves at bus Nos. 1 and 2 with 6 best initial solution were 0.2937 pu and 0.7839 pu. Where as with optimized solution these reactive reserves are obtained as 0.50874 pu and 0.9142 pu. Weighting factors are 3.1875 and 4.8472 obtained by sensitivity analysis. Magnitude of proximity indicator with optimized solution is s = 0.5523. Similar results have been obtained for 25-bus test system. This system consists of five generator buses and these are reactive power control variables. Remaining 20 buses are load buses. Maximum internal voltages and synchronous reactances were assumed as Emax1= 2.60 pu, Emax2 = 2.15 pu, Emax3 = 2.10 pu, Emax4 = 2.30 pu, Xd1 = 1.00 pu, Xd2 = 1.15 pu, Xd3 = 1.05 pu, Emax5 = 2.15 pu, Xd4 = 1.20 pu, and Xd5 = 1.15 pu. Total base case real and reactive power load on the system is 12.41 pu and 3.876 pu.Value of proximity indicator at base case condition is 0.30. Table 5 shows PV-bus voltage and all other load bus voltages under base condition. Star
marked buses are violating the bus voltage limit. The desired range of load bus voltage is 0.95 pu to 1.05 pu.The static voltage stability limit is 17.229396 pu in base case setting of control variables. A threshold value of s as proximity indicator has been assumed as 0.39. Initially five particles were selected satisfying all inequality constraints by procedure explained in Section 4 and are given in Table 6. The values of c1 and c2 have been selected same as 1.00. As in most power system problems Wmax = 1.0 and Wmin = 0.5 have been selected. A typical value of D selected as 0.12. Maximum numbers of iterations were set equal to 50. Table 7 gives maximum value of objective function ‘J’ has been obtained after 27 iterations. Fig. 2 shows variation of objective function (J) with respect to number of iteration. Solution gets converged in 27 iterations. Maximum numbers of iterations specified were 50. After twenty seven iterations no improvement in objective function was found. Table 8 gives optimized set of control variables. In this situation all load bus voltages are also given in the same table. Total reactive reserve available 2.4526 pu.Continuation power flow was carried with optimized set of reactive power control variable and static voltage stability limit was obtained as 20.132291 pu. Best initial solution (particle) selected as V1 = 1.1373 pu, V2 = 1.0740 pu, V3 = 1.0210 pu, V4 = 1.0488 pu, V5 = 1.1002 pu. Reactive reserves at bus Nos. 1–5 with 5
Table 4 Optimized set of control variables and all bus voltages for 6-bus system. Sr. No.
Control variables
1 2 3 4 5 6
V1 V2 BSH4 BSH6 TAP4 TAP7
Optimized control variables magnitudes (pu) PSO
DFP
1.0868 1.0661 0.0497 0.0526 0.9455 0.9872
1.1042 1.0711 0.0474 0.0542 0.9529 0.9870
Total load Pd = 1.35pu, Qd = 0.32 pu. Proximity indicator using (PSO) s = 0.5523. Proximity indicator using (DFP) s = 0.5590.
Load bus voltages
Load bus voltages magnitudes (pu) PSO
DFP
V3 V4 V5 V6
0.9586 0.9699 0.9530 0.9500
0.9608 0.9714 0.9500 0.9509
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best initial solution were 0.2841 pu, 0.4094 pu, 0.3092 pu, 0.2527 pu, and 0.1723 pu. Whereas with optimized solution these reactive reserves are obtained as 0.5826 pu, 0.8121 pu, 0.4609 pu, 0.2969 pu, and 0.3001 pu respectively. Weighting factors for five generations are 3.9328, 1.2648, 5.5761, 6.2756 and 5.8215 obtained by sensitivity analysis. Magnitude of proximity indicator with optimized solution is s = 0.3920. The reactive power reserve maximization problem as formulated in Section 2 has been solved using one of the most popular non-linear optimization techniques known as Devidon–Fletcher– Powell’s (DFP) method and accounting inequality constraints by an exterior penalty function method [15]. This gradient based method has been extensively used for non-linear optimization problem solution in power system studies. Mechanization of DFP method is much more involved than PSO technique. Various sensitivities are required to evaluate the gradient vector. A penalty parameter is selected at a low initial value say 0.5 and if violation persists than this is increased in step size by 10% each time till all inequality constraints are satisfied. Results obtained with this method for 6-bus test system are given in Table 4. This table gives value of control variable as obtained by DFP method. It is seen; control variables as obtained by both the methods are in close agreement. Value of Max J obtained using DFP method is 5.9619 as against 6.0529 obtained using PSO. Reactive power reserve obtained as 0.4551 pu, and 0.9307 pu. Similar results have been obtained for 25-bus system. Control variables as obtained using DFP method has shown in Table 8 along with those obtained using PSO. Value of objective function obtained using DFP method is 9.1749 as against those obtained using PSO as 9.4987 (Table 7). Results obtained by both the method are in close agreement. Reactive reserves obtained at all five buses obtained for 25bus test system are 0.5973 pu, 0.8035 pu, 0.4816 pu, 0.3011 pu, and 0.2677 pu.
Table 5 Load flow solution for 25-bus test system under base case condition. Sr. No.
Control variables
Control variables magnitudes (pu)
Load bus voltages
Load bus voltages magnitudes (pu)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
V1 V2 V3 V4 V5
1.0000 1.0000 1.0000 1.0000 1.0000
V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25
0.9467* 0.9421* 0.9374* 0.9192* 0.9398* 0.9312* 0.9333* 0.9472* 0.8845* 0.8809* 0.9065* 0.9309* 0.9305* 0.9764 0.9378* 0.8903* 0.8425* 0.8991* 0.8130* 0.8282*
Total load Pd = 12.41 pu, Qd = 3.876 pu. Proximity indicator s = 0.30. * is the buses are violating the bus voltage limit.
Table 6 Initial solutions for 25-bus test system for PSO. Sr. No.
V1 (pu)
V2 (pu)
V3 (pu)
V4 (pu)
V5 (pu)
J
1 2 3 4 5
1.1396 1.1406 1.1373 1.1427 1.1362
1.0903 1.1021 1.0740 1.0628 1.1028
1.0318 1.0195 1.0210 1.0363 1.0861
1.0527 1.0481 1.0488 1.0511 1.0137
1.1043 1.1105 1.1002 1.0993 1.0987
2.93852 3.11468 3.29528 3.04620 2.87043
Table 7 various parameters in PSO for 25-bus test system. Case
Wmax
Wmin
C1
C2
No. of initial particles
Max J
No. of iterations for convergence
1
1.0
0.5
1.0
1.0
05
9.4987
27
Table 8 Optimized set of control variables and all bus voltages for 25-bus system. Sr. No.
Control variables
Optimized control variables magnitudes (pu) PSO
DFP
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
V1 V2 V3 V4 V5
1.1354 1.0719 1.0161 1.0488 1.0954
1.1394 1.0813 1.0268 1.0475 1.0942
Total load Pd = 12.41 pu, Qd = 3.876 pu. Proximity indicator using (PSO) s = 0.3920. Proximity indicator using (DFP) s = 0.3997.
Load bus voltages
V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25
Load bus voltages magnitudes (pu) PSO
DFP
0.9897 1.0326 1.0306 1.0171 1.0402 1.0345 1.0309 0.9823 0.9820 0.9940 1.0297 1.0368 1.0226 1.0488 1.0017 0.9751 0.9539 1.0358 0.9500 0.9741
1.0026 1.0376 1.0350 1.0199 1.0412 1.0355 1.0341 0.9949 0.9871 0.9974 1.0326 1.0378 1.0231 1.0487 1.0004 0.9745 0.9531 1.0371 0.9500 0.9750
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Results obtained by both the methods are in close agreement and this justifies the use of PSO technique for its ease of implementation and computational efficiency, which is an established advantage for such evolutionary algorithm [16].
6. Conclusions This paper discusses the management of reactive power reserves in order to improve static voltage stability. This has been achieved via a modified PSO algorithm. Advantage of PSO algorithm is that its mechanization is simple without much mathematical complexity. Moreover global optimal solution is obtained and local optimal solution is avoided via search procedure. Important about the methodology is that not only reactive reserve is optimized but inequality constraint on proximity indicator guarantee required static voltage stability margin. Network as well as source capabilities are important from voltage instability viewpoint. Further contribution to reactive reserve has been considered from participation viewpoint by weighting factors. This is important aspect, which has been considered since large reactive reserve available at a generator bus, which is not utilized in a load increased scenario, is not of great significance. Hence a generator participating to a larger extent has been given lesser weight in the PSO algorithm. The algorithm has been implemented on 6-bus and 25-bus sample test systems.
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