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Optimal allocation of capacitor banks in radial distribution systems for minimization of real power loss and maximization of network savings using bio-inspired optimization algorithms
Electrical Power and Energy Systems 69 (2015) 441–455
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Optimal allocation of capacitor banks in radial distribution systems for minimization of real power loss and maximization of network savings using bio-inspired optimization algorithms Satish Kumar Injeti ⇑, Vinod Kumar Thunuguntla 1, Meera Shareef 2 Department of Electrical and Electronics Engineering, Sir C R Reddy College of Engineering, Eluru, West Godavari, Andhra Pradesh 534007, India
a r t i c l e
i n f o
Article history: Received 27 November 2014 Received in revised form 13 January 2015 Accepted 31 January 2015
Keywords: Bio-inspired algorithms Capacitor banks Real power loss minimization Network savings maximization Radial distribution systems
a b s t r a c t In this paper, two new algorithms are implemented to solve optimal placement of capacitors in radial distribution systems in two ways that is, optimal placement of fixed size of capacitor banks (Variable Locations Fixed Capacitor banks-VLFQ) and optimal sizing and placement of capacitors (Variable Locations Variable sizing of Capacitors-VLVQ) for real power loss minimization and network savings maximization. The two bio-inspired algorithms Bat Algorithm (BA) and Cuckoo Search (CS): search for all possible locations in the system along with the different sizes of capacitors, in which the optimal sizes of capacitor are chosen to be standard sizes that are available in the market. To check the feasibility, the proposed algorithms are applied on standard 34 and 85 bus radial distribution systems. And the results are compared with results of other methods like Particle Swarm Optimization (PSO), Harmonic Search (HS), Genetic Algorithm (GA), Artificial Bee Colony (ABC), Teaching Learning Based Optimization (TLBO) and Plant Growth Simulation Algorithm (PGSA), as available in the literature. The proposed approaches are capable of producing high-quality solutions with good performance of convergence. The entire simulation has been developed in MATLAB R2010a software. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The analysis of power distribution systems is an important area of research due to the fact that it is the final link between the bulk power system and consumers. However, reactive power flow in a distribution network always cause high power losses. The reactive power support is one of the well-recognized methods for the reduction of power losses together with other benefits; such as loss reduction, power factor correction, voltage profile improvement to the utmost extent under various operating constraints. The shunt capacitor is one of the basic equipment to fulfil these objectives. Therefore, it is important to find optimal location and sizes of capacitors in the system to achieve the above mentioned objectives. Numerous methods for solution to the optimal placement of capacitor with a view to minimizing losses have been suggested in the literature based on both traditional mathematical methods ⇑ Corresponding author. Mobile: +91 9581371537. E-mail addresses: [email protected] (S.K. Injeti), [email protected] (V.K. Thunuguntla), [email protected] (M. Shareef). 1 Mobile: +91 9032146188. 2 Mobile: +91 7207229663. http://dx.doi.org/10.1016/j.ijepes.2015.01.040 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
and more recent heuristic approaches. Over the last two decades, the studies on meta-heuristic techniques have shown that the most of the difficulties of classical methods can be eliminated by applying these techniques. Several heuristic methods have been developed in the last decade for optimal capacitor placement. Prakash et.al presented PSO [1] approach for finding the optimal sizes of capacitors with an objective of reduction of power loss in a radial distribution system. In this paper author used concept of loss sensitivity factors to determine the locations before sizing of capacitors using PSO. Kalyuzhny et al. proposed GA [2] as an optimization tool to place shunt capacitor on distribution system under capacitor switching constraints. Rao et al. [3] presented plant growth simulation algorithm (PGSA) presents two step solution methodology for optimal capacitors placement in radial distribution system with an objective to improve the voltage profile and reduction of power losses. In the first part, determination of optimal locations concept of loss sensitivity factors has been used, later PGSA is used for sizing of capacitors at the optimal locations determined in part one. Sizes of capacitors obtained in PSO & PGSA methods are continues & these are to be rounding off to nearest discrete capacitor sizes available in the market results in changes in the solution: active
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power loss and it may not be optimal. Raju et al. presented direct search algorithm (DSA) [4] to find the optimal size and location of fixed (discrete) and switched capacitors in a radial distribution system to maximize the savings by minimizing the active power loss. DSA uses step by step procedure for finding locations and capacitor sizes resembles the numerical method and it consumes much time to find the optimal solution. Some other heuristic methods such as tabu search [5], the harmony search algorithm [6], ant colony optimization-based algorithm [7] and a simulated annealing technique [8], and Teaching Learning Based Optimization (TLBO) [9] to solve the discrete size capacitor placement optimization problems. A comprehensive survey on the various heuristic optimization techniques applied to determine the optimal capacitor placement and size is presented in [10]. El-Fergany et al., used Artificial Bee Colony (ABC) [11] for optimal capacitor placement problem with an objective to maximize the net savings per year and to improve the voltage profile. In this work, potential location for capacitor connection has been determined by Loss Sensitive Factors (LSF) and voltage sensitive indices (VSI). Later, sizing of capacitor has been done using ABC algorithm. Obtained results are encouraging but, there is no guarantee for locations obtained by LSF & VSI method are optimal or not. The good results reported by others to various engineering optimization problems motivated us to apply novel meta-heuristic Bat and Cuckoo Search algorithms which are proposed by Xin-She-Xang [12,13]. However, from the literature review it is seen that the application of Bat and Cuckoo Search algorithms for optimal capacitor placement problem of distribution system has not been explored in previous works. This motivates the authors to use bio-inspired algorithm such as Bat and Cuckoo Search algorithms to locate optimal position and rating of capacitor in radial distribution system to maximize the annual network savings as an objective by minimizing real power loss. The present work describes Bat and Cuckoo Search algorithms methodologies for optimal capacitor allocation and sizing. Finding both locations and ratings of discrete or fixed size capacitors available in market in the optimal way named as VLFQ-case & finding both locations and ratings continues size of capacitors named as VLVQ case. In this paper, both Bat and Cuckoo Search (CS) optimization algorithms are applied to determine the optimal sizes and locations of capacitors for both VLVQ and VLFQ cases. This problem is formulated as a nonlinear constrained mixed discrete–continuous optimization problem. In order to show the effectiveness of proposed approaches: they have tested on the IEEE 34 and IEEE 85 bus radial distribution networks and the results are compared with the well existing methods available in the literature.
Mathematical modeling of radial distribution system Two bus model for distribution system analysis
V i ¼ jV i jL di V iþ1 ¼ jV iþ1 jL diþ1 Ii ¼ jIi jL hi Z i ¼ jZ i jL ;i
Sending end voltage Receiving end voltage Branch current Branch impedance, where Z i ¼ Ri þ jX i
Form Fig. 1
V iþ1 ¼ V i Ii Z i
ð1Þ
jV iþ1 jL diþ1 ¼ jV i jL di jIi jL hi jZ i jL ;i
ð2Þ
jV iþ1 j cos diþ1 þ jV iþ1 j sin diþ1 ¼ jV i j cos di þ jV i j sin di jIi jðcos hi j sin hi ÞðR2i þ X 2i Þ
ð3Þ
By separating real and imaginary terms
jV iþ1 j cos diþ1 ¼ jV i j cos di jIi jðRi cos hi þ X i sin hi Þ
ð4Þ
jV iþ1 j sin diþ1 ¼ jV i j sin di jIi jðX i cos hi Ri sin hi Þ
ð5Þ
By squaring and adding Eqs. (4) and (5)
jV iþ1 j2 ¼ jV i j2 2jV i jjIi j cos di fðRi cos hi þ X i sin hi Þg þ jIi j2 fðR2i þ X 2i Þg 2jV i jjIi j sin di fðX i cos hi Ri sin hi Þg ð6Þ After mathematical treatment Eq. (6) can be written as
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jfRðcosðdi hi ÞÞ þ X sinðdi hi Þg þ jIi j2 fðR2i þ X 2i Þg
ð7Þ
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jjZ i j cosðdi hi ;i Þ þ jIi j2 fðR2i þ X 2i Þg
ð8Þ
Since di hi ;i is very small hence, cosðdi hi ;i Þ ffi 1. Because in radial distribution system, voltage angle variations from source bus to the tail end of the feeder are only a few degrees. Therefore Eq. (8) can be directly written as
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jjZ i j þ jIi j2 fðR2i þ X 2i Þg
ð9Þ
jV iþ1 j2 ¼ ½jV i j jIi jjZ i j2
ð10Þ
jV iþ1 j ¼ jV i j jIi jjZ i j
ð11Þ
where 1=2
jIi j ¼
ðP2i þ Q 2i Þ jV i j
ð12Þ
or it is also written as Eq. (13) 1=2
jIi j ¼
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j
Single line diagram of a two bus radial distribution system is depicted in Fig. 1.
ð13Þ 1=2
jV iþ1 j ¼ jV i j
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j
jZ i j
ð14Þ
1=2
ðR2i þ X 2i Þ
ð15Þ
1=2
ðR2i þ X 2i Þ ¼ 0
ð16Þ
jV iþ1 j2 ¼ jV i jjV iþ1 j ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2 jV i jjV iþ1 j þ ðP2iþ1 þ Q 2iþ1 Þ Positive root for Eq. (16)
1=2
jV iþ1 j ¼ Fig. 1. Two bus of radial distribution system.
jV i j ðjV i j2 4ððP2iþ1 þ Q 2iþ1 Þ 2
ÞðR2i þ X 2i ÞÞ
1=2
From Eq. (17) receiving end voltage can be directly found.
ð17Þ
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The real power loss in the branch connected between i and i + 1 bus may be computed as
PLoss ði; i þ 1Þ ¼ jIi j2 Ri
ð18Þ
The real power loss in the branch connected between i and i + 1 bus may be computed as
PLoss ði; i þ 1Þ ¼
Q Loss ði; i þ 1Þ ¼
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2
Ri
ð19Þ Fig. 2. Single line diagram for integration of shunt capacitor in radial distribution system.
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2
Xi
ð20Þ
Total real power loss of the feeder, PT, Loss, may determined by summing up the losses of all branches of the feeder, which is given as
PT;Loss ¼
n1 X PLoss ði; i þ 1Þ
ð21Þ
i¼1
where n = number of buses in the distribution system.
QCAP = Reactive power supplied by shunt capacitor. Hence, net injected reactive power at i + 1 bus can written as
ð22Þ
By substituting Eq. (22) in Eq. (16) then
jV iþ1 j2 jV i jjV iþ1 j þ ðP2iþ1 þ Q 2inj Þ
1=2
ðR2i þ X 2i Þ ¼ 0
ð23Þ
Possible feasible solution for Eq. (23) is
jV i j ðjV i j2 4ððP2iþ1 þ Q 2inj Þ jV iþ1 j ¼ 2
1=2
1=2
ÞðR2i þ X 2i ÞÞ
ð24Þ
Installation of shunt capacitor units in optimal locations of a distribution system results in several benefits. These include line loss reduction, improvement of voltage profile, power factor improvement etc. The power loss when capacitor is connected at i + 1 bus in the network as shown in Fig. 2, is given by
PCAP Lossði;iþ1Þ ¼
PCAP Lossði;iþ1Þ ¼
ðP 2iþ1 þ Q 2inj Þ jV iþ1 j2
Ri
ð25Þ
ððPiþ1 Þ2 þ ðQ iþ1 Q CAP Þ2 Þ
PCAP Lossði;iþ1Þ ¼ Ri
jV iþ1 j
2
ðP2iþ1 þ Q 2iþ1 Þ 2
jV iþ1 j
Q 2CAP 2Q iþ! Q CAP jV iþ1 j2
The load flow of a power network provides the steady state solution through which various parameters of interest like currents, voltages, losses etc can be calculated. The load flow is important tool for the analysis of power distribution system. Conventional load flow methods failed to give good results for distribution system analysis, due to special features like high R/X ratio and radial structure etc. Various distribution load flow techniques have been proposed in the literature [14–18]. Methods based on forward/backward sweep processes using Kirchhoff’s Laws or making use of the well-known bi-quadratic equation have gained popularity for distribution systems load flow analysis due to its low memory requirements, computational efficiency and robust convergence characteristics. In this paper, method which can find the load flow solution of radial distribution system directly by using topological characteristic of distribution network [19] is used. Single line diagram of radial distribution system with integration of shunt capacitor is shown in Fig. 2. Node one or first bus is considered as root bus or sub-station bus or source bus. At this bus generally load value is zero. Detailed flow chart for backward forward sweep load flow of radial distribution system has been given in Fig. 3. Objective function formulation
Ri
ð26Þ Objective function
þ
Q 2CAP 2Q iþ1 Q CAP jV iþ1 j2
Ri
ð27Þ
Net power loss reduction, DPCAP Loss in the system is the difference of power loss before and after installation of capacitor unit and is given by
DPCAP Loss ¼
In a distribution system, the voltage angle is not so important because, the variation of voltage angle from source to the tail end of the distribution feeder is only a few degrees. Forward/backward sweep based distribution load flow
Integration of shunt capacitor into the distribution system
Q inj ¼ Q iþ1 Q CAP
3. All lines are represented as short lines and half line charging susceptance of distribution line is negligible because of low level voltages unlike transmission.
Ri
ð28Þ
The positive sign of the DPCAP Loss indicates that the losses in the system reduces with installation of Capacitor.
Optimal capacitor placement in radial distribution system reduces the real power losses and improves the voltage profile. Reduction in power loss leads to the reduction in energy loss cost. However, the capacitor placement increases the installation and investment cost. Therefore the objective of the capacitor placement is to maximize the annual network savings by minimizing the total real power losses in the distribution system subjected to specific operational constraints. Mathematically, objective functions of the problem has been formulated as,
Total real power loss P la ¼
n1 X I2i ðRi Þ
ð29-aÞ
i¼1
Assumptions made in the above analysis are 1. System is a balanced. 2. In general, in radial distribution system first bus is considered as source bus or sub-station bus. Hence it is assumed that no load or Capacitor unit is connected at source bus.
Maximize S ¼ fK e ðPlb Pla ÞT / ½C i N c þ C p
Nc X
Q c ðiÞ C 0 Nc g
ð29-bÞ
i¼1
where n1 is the number of branches, n is the number of buses, I is respective branch current, R is the respective resistance of the
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Fig. 3. Flow chart of backward/forward sweep radial distribution load flow.
branch, S is the annual net savings, Plb & Pla are the total active power loss before and after compensation respectively, Ke is the energy cost per kW h, a is the depreciation factor, T is the time period in hours, Ci is the installation cost of capacitor per location, Nc is the number of compensated buses where capacitors are to be placed, Cp is the purchase cost of capacitor per kVAR, C0 is the capacitor operating cost per location and Qc(i) is the amount of reactive power of installed capacitor at bus i. Constraints The objective function is subjected to following constraints:
i¼1
Q Cmin and Q Cmax are minimum and maximum range of injected reactive power at ith bus For VLFC case capacitors are available in discrete sizes. So, shunt capacitors to be dealt with multiple integers of the smallest capacitor size available and it may be mathematically expressed as
ð34Þ
Maximum compensation with capacitor bank is limited to the total reactive power demand
ð31Þ
j¼1 Nc n X X Q c ðiÞ 6 Q D ðjÞ
Voltage limits
V min 6 V i 6 V max
ð33Þ
ð30Þ
j¼1
Nc n n1 X X X Q Slack þ Q ci ¼ PDi þ PLj i¼1
Q Cmin 6 Q Ci 6 Q Cmax
where Qs is the smallest capacitor size available and L is an integer multiple.
n n1 X X ¼ PDi þ PLj i¼1
Reactive power compensation limits for VLVQ case
Q c ðiÞ 6 LQ s
Power balance constraints
PSlack
Vmin and Vmax is minimum and maximum voltage limits of ith bus.
i¼1
ð32Þ
ð35Þ
j¼1
where n is the number of buses and QD(j) is the reactive power demand of load at bus j. Nc is number of capacitors.
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The apparent power flow through the line must be less than the maximum apparent power admissible for the line and it is expressed as
Si 6 Simax
i ¼ 1; 2; . . . nb
ð36Þ
where nb is the number of branches, Si is the apparent power flow of the ith branch and Simax is the maximum apparent power flow of the ith branch. Proposed algorithms Bat Algorithm The majority of heuristic and meta-heuristic algorithms have been derived from the behaviour of biological systems and/or physical systems in nature. The Bat Algorithm (BA) is based on the echolocation behaviour of bats, proposed by Xin-She-Xang for engineering optimization in [11]. If we idealize some of the echolocation characteristics of micro bats, we can develop various bat-inspired algorithms or Bat Algorithms. For simplicity, we now use the following approximate or idealized rules: (1) All bats use echolocation to sense distance, and they also ‘know’ the difference between food/prey and background barriers in some magical way. (2) Bats fly randomly with velocity Vi at position Xi with a fixed frequency fmin, varying wavelength k and loudness A0 to search for prey. They can automatically adjust the wavelength (or frequency) of their emitted pulses and adjust the rate of pulse emission r 2 [0, 1], depending on the proximity of their target. (3) Although the loudness can vary in many ways, we assume that the loudness varies from a large (positive) A0 to a minimum constant value Amin. Another obvious simplification is that no ray tracing is used in estimating the time delay and three dimensional topography. Though this might be a good feature for the application in computational geometry, however, we will not use this as it is more computationally extensive in multidimensional cases. In addition to these simplified assumptions, we also use the following approximations for simplicity. In general, the frequency f in a range [f min ; f max ] corresponds to a range of wavelengths [kmin, kmax]. For example a frequency range of [20 kHz, 500 kHz] corresponds to a range of wavelengths from 0.7 mm to 17 mm. For a given problem, we can also use any wavelength for the ease of implementation. In the actual implementation, we can adjust the range by adjusting the wavelengths (or frequencies), and the detectable range (or the largest wavelength) should be chosen such that it is comparable to the size of the domain of interest, and then toning down to smaller ranges. Furthermore, we do not necessarily have to use the wavelengths themselves; instead, we can also vary the frequency while fixing the wavelength k. This is because k and f are related due to the fact k/f is constant. For simplicity, we can assume f 2 [0, f max ]. We know that higher frequencies have short wavelengths and travel a shorter distance. For bats, the typical ranges are a few metres. The rate of pulse can simply be in the range of [0, 1] where 0 means no pulses at all, and 1 means the maximum rate of pulse emission. In simulations, we use virtual bats naturally. We have to define the rules how their positions X i and velocities V i in a d-dimensional search space are updated. The new solutions X ti and velocities V ti at time step t are given by
f i ¼ f min þ ðf max f min Þb
ð37Þ
V ti ¼ V t1 þ ðX ti X Þf i i
ð38Þ
X ti ¼ X t1 þ V ti i
ð39Þ
where b 2 [0, 1] is a random vector drawn from a uniform distribution. Here x⁄ is the current global best location (solution) which is located after comparing all the solutions among all the n bats. As the product ki f i is the velocity increment, we can use either f i (or ki) to adjust the velocity change while fixing the other factor ki (or f i ), depending on the type of the problem of interest. In our implementation, we will use f min ¼ 0 and f max ¼ 100, depending the domain size of the problem of interest. Initially, each bat is randomly assigned a frequency which is drawn uniformly from [f min ; f max ]. The update of the velocities and positions of bats have some similarity to the procedure in the standard particle swarm optimization as f i essentially controls the pace and range of the movement of the swarming particles. To a degree, BA can be considered as a balanced combination of the standard particle swarm optimization and the intensive local search controlled by the loudness and pulse rate. Simple pseudo code for Bat Algorithm (BA) has been given in Appendix A1. Steps for implementation of Bat Algorithm In this section, Bat Algorithm is described for solving the optimal placement of capacitors in radial distribution systems. Step 1: Initialization of problem and algorithm parameters In the first step, the algorithm parameters such as population size (POP), dimension of the problem and maximum number of iterations (itermax), limits of f, b and A are to be initialized. The problem parameters such as number of capacitors, limits of capacitor size, bus voltage limits and system data are to be initialized. Step 2: Random generation of locations and capacitor sizes
2
x11
6 x2 6 1 6 6 CAPLOC ¼ 6 ... 6 6 pop1 4 x1 xpop 1
2
y11 6 y2 6 1 6 6 CAPSIZE ¼ 6 ... 6 6 pop1 4 y1 ypop 1
x12
x1d1
x22 .. .
.. .
x2d1 .. .
xpop1 2 xpop 2
xpop1 d1 xpop d1
y12
y1d1
.. .
y2d1
ypop1 d1 ypop d1
y22 .. . ypop1 2 ypop 2
.. .
x1d
3
7 7 7 7 7 7 pop1 7 5 xd x2d .. .
ð40Þ
xpop d y1d
3
7 7 7 .. 7 . 7 7 7 5 ypop1 d y2d
ð41Þ
ypop d
xij ¼ xmin;i þ ðxmax;i xmin;i Þ randðÞ
ð42Þ
yij ¼ ymin;i þ ðymax;i ymin;i Þ randðÞ
ð43Þ
where d is the number of decision variables, xij , yij represents locations and capacitor sizes, i.e., jth population of ith capacitor location and size, which is generated randomly in between the limits as xmax;i and xmin;i are the ith capacitor location limits, ymax;i and ymin;i are the ith capacitor size limits and rand() is a random number in between 0 and 1. For VLFC case CAPSIZEi;d can be obtained by random generation of fixed size of capacitor banks between integral multiples of
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150 kVar. For VLVC case it can be obtained by random generation of capacitor sizes between 100 kVar and 1500 kVar.
Soln ¼ ½CAPLOC
CAPSIZE
ð44Þ
In Bat Algorithm, Soln. represents a group of Bats, where Bat is one position in search space. Bat is a solution that contains capacitor locations and sizes. Step 3: Fitness evaluation Run the load flow and calculate the real power loss of the system and corresponding network saving per year for each initial solution. Record the best solution. Step 4: Start evolution procedure of Bat Algorithm
ð45Þ
where b 2 [0, 1] is a random vector drawn from a uniform distribution. Initially each bat is randomly assigned a frequency which is drawn uniformly from [f min ; f max ]. Step 5: Random generation of Bat positions (locations and sizes of capacitors) t
t
locV i ¼ V t1 þ ðcaploci bestloc Þf i i t
t
sizeV i ¼ V it1 þ ðcapsizei bestsize Þf i t
CAPLOC ti ¼ CAPLOC t1 þ locV i i
ð46Þ ð47Þ ð48Þ
t
CAPSIZEti ¼ CAPSIZEit1 þ sizeV i
For simplicity, this last assumption can be approximated by a fraction pa of the n nests being replaced by new nests (with new random solutions at new locations). For a maximization problem, the quality or fitness of a solution can simply be proportional to the objective function. Other forms of fitness can be defined in a similar way to the fitness functions in genetic algorithms. Based on these three rules, the basic steps of the Cuckoo Search (CS) can be summarized as the pseudo code shown below. Pseudo code for CS algorithm is given in Appendix A1. When generating new solutions xtþ1 for, say cuckoo i, a L´evy flight is performed
xtþ1 ¼ xti þ / L0 ev yðkÞ i
Assign frequency for each Bat randomly
f i ¼ f min þ ðf max f min Þb
The number of available host nests is fixed, and a host can discover an alien egg with a probability p 2 [0, 1]. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.
ð49Þ
Step 6: Fitness evaluation (Objective function) Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new solution. Step 7: Compare each new bat solution with corresponding initial bat solution and replace better solution new bats to initial bat & find best bat, best solution among initial bats. Step 8: Stopping criterion. If the maximum number of iterations is reached, computation is terminated. otherwise, Step 4 to Step 7 is repeated. The steps of the Bat Algorithm (BA) are as follows can be summarized as the pseudo code. And the implementation flowchart for BA is given in Fig. 4.
where a > 0 is the step size which should be related to the scales of the problem of interest. In most cases, we can use a = O (1). The product means entry-wise multiplications. L´evy flights essentially provide a random walk while their random steps are drawn from a L´evy distribution for large steps
L0 ev y u ¼ tk ; ð1 < k 6 3Þ
Each cuckoo lays one egg at a time, and dumps it in a randomly chosen nest. The best nests with high quality of eggs (solutions) will carry over to the next generations.
ð51Þ
This has an infinite variance with an infinite mean. Here, the consecutive jumps/steps of a cuckoo essentially form a random walk process which obeys a power-law step length distribution with a heavy tail. It is worth pointing out that, in the real world, if a cuckoo’s egg is very similar to a host’s eggs, then this cuckoo’s egg is less likely to be discovered, thus the fitness should be related to the difference in solutions. Therefore, it is a good idea to do a random walk in a biased way with some random step sizes. Simple pseudo code for CS algorithm has been given in Appendix A2. Steps for implementation of CS algorithm In this section, CS algorithm is described for solving the optimal placement of capacitors in radial distribution systems. Step 1: Initialization of problem and algorithm parameters In the first step, the algorithm parameters such as population size (POP), dimension of the problem and maximum number of iterations (itermax) are to be initialized. The problem parameters such as number of capacitors, limits of capacitor size, bus voltage limits and system data are to be initialized. Step 2: Random generation of locations and capacitor sizes
2
x11
6 x2 6 1 6 6 CAPLOC ¼ 6 ... 6 6 pop1 4 x1 xpop 1
Cuckoo Search (CS) Algorithm A new meta-heuristic optimization algorithm, called Cuckoo Search (CS), was developed recently by Yang and Deb [12]. For simplicity in describing Cuckoo Search algorithm (Yang and Deb [13]), three idealized rules have to be followed:
ð50Þ
2
y11 6 y2 6 1 6 6 CAPSIZE ¼ 6 ... 6 6 pop1 4 y1 ypop 1
x12
x1d1
x22 .. .
.. .
x2d1 .. .
xpop1 2 xpop 2
xpop1 d1 xpop d1
y12
y1d1
y22 .. .
.. .
y2d1 .. .
ypop1 d1 ypop d1
ypop1 2 ypop 2
xij ¼ xmin;i þ ðxmax;i xmin;i Þ randðÞ
x1d
3
7 7 7 7 7 7 pop1 7 5 xd x2d .. .
ð52Þ
xpop d y1d
3
7 7 7 7 7 7 pop1 7 5 yd y2d .. .
ð53Þ
ypop d
ð54Þ
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447
Fig. 4. Flowchart for implementation of Bat Algorithm.
yij ¼ ymin;i þ ðymax;i ymin;i Þ randðÞ
ð55Þ
where d is the number of decision variables, xij , yij represents locations and capacitor sizes, i.e., jth population of ith capacitor location and size, which is generated randomly in between the limits as xmax;i and xmin;i are the ith capacitor location limits, ymax;i andymin;i are the ith capacitor size limits and rand() is a random number in between 0 and 1. For VLFC case CAPSIZEi;d can be obtained by random generation of fixed size of capacitor banks between integral multiples of 150 kVar. For VLVC case it can be obtained by random generation of capacitor sizes between 100 kVar and 1500 kVar.
nest ¼ ½CAPLOC
CAPSIZE
ð56Þ
Step 3: Fitness evaluation Run the load flow and calculate the real power loss of the system and corresponding network saving per year for each initial nest. Record the best solution vector.
Step 4: Start evolution procedure of CS algorithm
new caploci;d ¼ CAPLOC i;d þ / L0 ev yðkÞ
ð57Þ
new capsizei;d ¼ CAPSIZEi;d þ / L0 ev yðkÞ
ð58Þ
new nesti;d ¼ ½new caploci;d new capsizei;d
ð59Þ
Step 5: Fitness evaluation (Objective function) Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new nest. Step 6: Compare each new nest solution with corresponding initial nest solution and replace better solution new nests to initial nests & find best nest, best solution among initial solution.
448
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Fig. 5. Flowchart for implementation of CS algorithm.
Step 7: Abandon fraction of worst nests among new nests and built new ones at those nests using L0 ev y flight. Step 8: Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new nest. Step 9: Compare each new nest solution with corresponding initial nest solution and replace better solution new nests to initial nests & find best solution am better solution new nests to initial nests & find best nest, best solution among initial solution.ong initial nests. Step 10: Stopping criterion.
If the maximum number of iterations is reached, computation is terminated. otherwise, Step 4 to Step 9 is repeated. And for easy understanding of CS algorithm implementation flowchart is given in Fig. 5. Implementation results and discussions The performance and effectiveness of the proposed algorithms have been tested on 34-bus and 85-bus radial distribution systems for real power loss minimization and maximization of network savings. In this approach, optimal locations for connection of capacitors
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the test systems to determine the most suitable parameters. In this work, the tuned parameters of BAT and CS are given in Table 2. The entire simulation is developed in MATLAB R2010a software and the simulations are carried on a computer with Intel(R) Core(TM) i52450 M CPU @2.50 GHz, 4 GB RAM.
Table 1 Constants used in computation of net saving for the test cases. S.No
Parameter description
Value
1 2 3 4 5 6
Average energy cost (Ke) Depreciation factor (a) Purchase cost (Cp) Installation cost (Ci) Operating cost (Co) Hours per year (T)
$0.06/kW h 20% $25/kVAR $1600/location $300/year/location 8760
34-Bus test system numerical results and simulations
Table 2 Description of algorithm parameters. Algorithm
Parameter
Description
Value
CS
Pop d Pa maxitr
Population of host nests Dimension of a host nest Fraction of worst nests (Probability) Maximum number of generations
150 3&8 0.25 150
BAT
pop d A r fmin fmax maxitr
Population of bats Dimensional search space of a bat Loudness Pulse rate Minimum frequency Maximum frequency Maximum number of generations
150 3&8 0.50 0.50 0.00 2.00 150
Fig. 6. Single line diagram of 34-bus radial distribution system.
and optimal size of capacitors have been treated as a single problem, unlike other approaches quoted in the literature. The potential locations for capacitor placement are decided by algorithm itself along with optimal sizes of capacitors. In all calculations; for both test systems, the following constants are assumed and applied [11] as shown in Table 1. Problem of optimal capacitor placement is solved by two scenarios i.e., VLFQ and VLVQ. Backward/Forward sweep based power flow method has been used for radial distribution system load flow solution. All loads are assumed as constant power loads (peak load) and tap changing transformers are not considered in the present work to avoid complexity. A number of trails on the performance of the applied algorithms have been carried out on
The 34-bus test case consists of a main feeder and 4 sub-feeders (laterals) radial distribution system as shown in Fig. 6. The data of the system is obtained from [20]. The total load of the system is 4636.5 kW and 2873.5 kVAR. The rated voltage of the system is 11 kV. After an initial load flow run using Backward/Forward Sweep method for an uncompensated system, the active power loss is 221.7235 kW and maximum & minimum voltages are 0.9941 p.u and 0.9417 p.u, respectively. To observe the effectiveness of the proposed algorithms, obtained results are compared with the other techniques like PSO [1], ABC [11], HS [20], GA [21] and EA [22]. Table 3 shows, the optimal locations and capacitor sizes obtained by proposed algorithms along with existing algorithms for 34-bus radial distribution system. Comparison of technical and economic benefits of optimal capacitor placement in a 34bus radial distribution system with proposed and existing algorithms has been given in Table 4. From Table 4, it is observed that, both proposed algorithms yields to reduce peak losses to approximately 160 kW with 2250 kVar installed. Among all algorithms CSA-VLVQ has yielded best results i.e., real power loss is 160.61 kW, percentage of loss reduction is 27.56, the system overall power factor is 0.9904; minimum bus voltage is 0.9500 p.u., maximum value of stability index is 0.9804 and annual network saving is $19,006. These results clarify that, CS algorithm based approach possesses lower system losses and higher annual network saving compared to other heuristic methods. The convergence characteristics of proposed algorithms based on objectives for 34 bus test system are depicted in Figs. 7 and 8 respectively. From Fig. 7, it is observed that, objective function (minimization of real power loss of the system) has reached to best lowest value at 130th iteration and the other objective (annual network savings) has reached to best highest value at 130th iteration. Voltage profile characteristics of the test system with and without compensation for proposed algorithms have been shown in Fig. 9. From Fig. 9, it is noticed that proposed algorithms are succeeded in the improvement of voltage and minimum bus voltage is improved from 0.9416 p.u to 0.9501 p.u. and voltage magnitude has been improved at all buses of the test system. Coming to simulation time CS has consumed relatively more time than BA. Among VLVQ and VLFQ scenarios, combination of CSA-VLVQ yields best solution than other methods. Best results obtained are presented bold in Table 4 for easy comparison.
Table 3 Optimal locations and sizes of capacitors in kVar for 34 bus system. S.No of capacitors
Proposed methods VLFQ
BAT
1 2 3 4 5
Existing methods
VLVQ CUCKOO
BAT
CUCKOO
VLVQ
VLFQ
PSO [1]
EA [22]
ABC [19]
GA [21]
HS [20]
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
10 19 26
766 750 734
9 20 25
767.04 834.50 648.45
10 19 24
750 750 750
9 20 25
750 900 600
19 20 22
781 479 803
8 18 25
1050 750 750
8 18 25
900 900 800
5 9 12 22 26
300 300 300 600 300
4 11 17 26
250 750 300 1400
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Table 4 Results analysis for 34 bus system with and without optimal capacitors. Parameter of comparison
WOC
WC – proposed methods VLVQ
Total KVAR placed (kVAR) Real power losses (kW) Loss reduction (%) Overall P.F Vmin (p.u) Vmax (p.u) VSImin VSImax Network savings/year ($) Elapsed time (s)
– 221.73 – 0.8557 0.9416 0.9941 0.7860 0.9765 – –
WC – existing methods VLVQ
VLFQ
BAT
CUCKOO
VLFQ BAT
CUCKOO
PSO [1]
EA [22]
ABC [19]
GA [21]
HS [20]
2250 160.99 27.56 0.9904 0.9501 0.9951 0.8149 0.9804 18,809 19.92
2250 160.61 27.56 0.9904 0.9500 0.9951 0.8149 0.9804 19,006 36.83
2250 160.69 27.52 0.9904 0.9500 0.9951 0.8149 0.9804 18,967 17.11
2250 160.65 27.54 0.9904 0.9500 0.9951 0.8149 0.9804 18,985 34.16
2063 169.35 23.62 0.9970 0.9486 0.9950 0.8097 0.9800 15,348 N/A
2550 161.26 27.26 0.9837 0.9501 0.9952 0.8149 0.9808 17,165 N/A
2600 161.08 27.34 0.9798 0.9496 0.9949 0.8163 0.9810 17,018 N/A
1800 164.95 25.61 0.9825 0.9478 0.9949 0.8071 0.9796 17,740 N/A
2700 168.48 24.02 0.9989 0.9522 0.9953 0.8219 0.9811 11,991 N/A
Fig. 7. Convergence characteristics of Bat Algorithm based on real power loss of 34-bus test system.
Fig. 8. Convergence characteristics of Bat Algorithm based on annual network savings of 34-bus test system.
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Fig. 9. Voltage profile analysis of proposed algorithms for 34-bus test system.
Fig. 10. Single line diagram for 85-bus radial distribution system.
85-Bus test system numerical results and simulations The 85-bus test case consists of a main feeder, 9 sub-feeders (laterals) and sub-laterals radial distribution system as shown in Fig. 10. The data of the system is obtained from [23]. The total load of the system is 2574.3 kW and 2622.6 kVAR. The rated voltage of the system is 11 kV. After an initial load flow run using Backward/ Forward Sweep method for an uncompensated system, the active power loss is 316.8497 kW and maximum & minimum voltages are 0.9973 p.u and 0.8712 p.u, respectively. In order to evaluate the effectiveness of the proposed algorithms, their results are
compared with the other techniques like PSO [1], PGSA [3], DSA [4], TLBO [9] and GA [21]. Table 5 shows, the optimal locations and sizes of capacitors with proposed and existing algorithms. A comparative analysis of total kVar compensated, total real power loss, percentage of loss reduction and annual network savings of proposed and existing algorithms has been furnished in Table 6. For this test system eight capacitors are optimally placed and sized simultaneously. The locations and sizes of capacitors obtained by Bat Algorithm in case of VLVQ scenario are 7, 25, 28, 45, 48, 60, 65, 85 and 181, 323, 288, 138, 369, 400, 328, 224 kVAR respectively, with an active power loss
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Table 5 Optimal locations and sizes of capacitors in kVar for 85 bus system. S.No of capacitors
Proposed methods VLVQ
VLFQ
BAT
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Existing methods
CUCKOO
VLVQ
BAT
CUCKOO
VLFQ
PSO [1]
PGSA [3]
GA [20]
TLBO [9]
DSA [4]
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
7 25 28 45 48 60 65 85
181 323 288 138 369 400 328 224
8 68 32 63 12 44 48 21
367 356 220 313 337 175 347 134
8 12 31 40 56 63 71 79
600 300 300 150 300 300 150 150
18 27 29 42 48 60 69 80
150 150 300 150 300 450 300 450
8 58 7 27
796 453 324 901
8 58 7
1200 908 200
26 28 37 38 39 51 54 55 59 60 61 62 66 69 72 74 76 80 82
48.437 214.06 103.12 120.31 178.12 100 212.5 101.56 4.687 157.81 112.5 104.68 9.375 100 67.18 112.5 71.87 356.25 31.25
4 7 9 21 26 30 31 45 49 55 61 68 83 85
300 150 300 150 150 0 300 150 150 150 300 300 150 150
6 8 14 17 20 26 30 36 57 61 66 69 80
150 150 150 150 150 150 150 450 150 150 150 300 150
Table 6 Results analysis for 85 bus system with and without optimal capacitors. Parameter of comparison
WOC
WC – proposed methods VLVQ
Total KVAR placed (kVAR) Real power losses (kW) Overall P.F Vmin (p.u) Vmax (p.u) VSImin VSImax Loss reduction (%) Network savings/year ($) Elapsed time (s)
0.0 316.84 0.8649 0.8712 0.9973 0.6524 0.9806 – – –
WC – existing methods VLFQ
VLVQ
VLFQ
BAT
CUCKOO
BAT
CUCKOO
PSO [1]
PGSA [3]
GA [20]
TLBO [9]
DSA [4]
2250 147.8715 0.9858 0.9212 0.9973 0.7625 0.9806 53.33 72605 101.79
2250 145.743 0.9869 0.9215 0.9973 0.7614 0.9806 54 73700 199.32
2250 146.9557 0.9858 0.9218 0.9973 0.7625 0.9806 53.61 73086 86.09
2250 146.62 0.9858 0.9200 0.9973 0.7625 0.9806 53.72 73260 174.86
2473 163.32 0.9404 0.8774 0.9973 0.6625 0.9769 48.27 65850.21 N/A
2308 161.40 0.9475 0.8865 0.9973 0.7605 0.9685 48.88 68304.36 N/A
2206.25 146.06 0.9869 0.9205 0.9973 0.7625 0.9806 53.72 66950.71 N/A
2700 143.18 0.9885 0.9387 0.9973 0.7863 0.9806 54.8 69100.79 N/A
2550 144.00 0.9878 0.9315 0.9973 0.7987 0.9806 54.54 69414.54 N/A
Fig. 11. Convergence characteristics of Bat Algorithm based on real power loss of the 85-bus test system.
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453
Fig. 12. Convergence characteristics of Bat Algorithm based on annual network savings of 85-bus test system.
of 147.87 kW, minimum voltage of 0.9212 p.u and annual network savings of $72,605. The location and sizes of capacitors obtained by Cuckoo Search algorithm in case of VLVQ scenario are 8, 68, 32, 63, 12, 44, 48, 21 and 367, 356, 220, 313, 337, 175, 347, 134 kVar respectively, with an active power loss of 145.73 kW, minimum voltage of 0.9218 p.u and annual network savings of $73,700. The optimal locations and sizes of capacitors obtained by Bat Algorithm in case of VLFQ scenario are 8, 12, 31, 40, 56, 63, 71, 79 and 600, 300, 300, 150, 300, 300, 150, 150 kVAR respectively, with an active power loss of 146.95 kW, minimum voltage of 0.9218 p.u and annual network savings of $73,086. The location and sizes of capacitors obtained by CS algorithm in case of VLFQ scenario are 18, 27, 29, 42, 48, 60, 69, 80 and 150, 150, 300, 150, 300, 450, 300, 450 kVar
respectively, with an active power loss of 146.62 kW, minimum voltage of 0.9200 p.u and annual network savings of $73,260. From Table 6, it is noted that annual network savings for proposed algorithms is magnificent when compared with other algorithms, among which the best annual network savings and loss reduction is noticed in the results of Cuckoo Search algorithm. The convergence characteristics of proposed algorithms for simultaneous objectives have been shown in Figs. 11 and 12 respectively. It is observed that both objectives reached to their best values at 137th iteration for the case of VLVQ scenario with CS algorithm. Voltage profile characteristics of the test system with and without compensation for proposed algorithms have been shown in Fig. 13. Form Fig. 13, it is noticed that proposed algorithms are succeeded in the improvement of voltage at
Fig. 13. Voltage profile analysis of proposed algorithms for 85-bus test system.
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Table 7 Statistical analysis of numerical results for 20 runs. Metric
Bus system
Algorithm-method
Best
Worst
Mean
STD
Real power loss (kW)
34-Bus
CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ
160.6599 160.6100 160.6642 160.7648 146.5825 145.7057 146.9557 147.5487
160.7605 160.8700 161.8436 161.5664 146.9103 145.8626 148.4659 148.9768
160.6773 160.6500 161.2615 161.0709 146.7384 145.7626 147.7423 148.0433
0.0325 0.0029 0.4633 0.2643 0.1335 0.0457 0.4263 0.4482
CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ
18,932 19,006 18,363 18,509 73,110 73,661 72,293 72,024
18,985 19,009 18,983 18,930 73,282 73,743 73,086 72,775
18,976 19,006 18,669 18,769 73,200 73,706 72,673 72,519
17.1039 5.2460 243.5191 138.7858 69.8821 28.6800 223.9296 237.0653
85-Bus
Annual network savings ($)
34-Bus
85-Bus
each and every bus of the system. Particularly minimum bus voltage is improved from 0.8712 p.u. to 0.9215 p.u. Coming to simulation time CS has consumed relatively more time than BA. Among VLVQ and VLFQ methods, combination of VLVQ with CS yields best solution than other methods. Best results obtained are presented bold in Table 6 for easy comparison and understanding purpose.
Appendix Appendix. A1 Simple pseudo code for BA
Performance analysis of Bat and CS algorithms In order to analyze the performance of the proposed algorithms with existing algorithms, both algorithms were executed for 20 times by considering population size of 150 and maximum number of iterations is 150. From the obtained results, the best, worst, mean and the standard deviation of two metrics (real power loss and annual network savings) has been presented in Table 7 for the two test systems. From Table 7, it is observed that two metrics which are related to objective function seems to be better in case of CSA-VLVQ method than other methods for both test systems. The value of standard deviation has significance in the analysis of optimization algorithms i.e., lesser standard deviation indicates that efficiency of the algorithm is high. Hence, it is clear from the statistical analysis that CSA-VLVQ has been succeeded in achieving the desired objective with quality solution than other proposed and existing algorithms and the best results are presented in bold. Appendix Appendix. A2 Conclusions Simple pseudo code for CS In this paper, application of Bat and CS algorithms to the optimal placement of capacitors banks in radial distribution systems has been discussed. The practical application and efficiency of this method is evaluated using two test systems (34 and 85 bus). From the comparative analysis it is concluded that, CS algorithm gives better results than Bat and other existing algorithms, in terms of solution quality. Both CS and Bat generate solutions which satisfy all the constraints. According to convergence Bat converged very quickly due to simple evolution process. However, CS convergence is slower than Bat, the reason is rigorous evolution process in CS. In fact, for any optimization algorithm parameter tuning plays an important role in the performance of the algorithm. From the results; CS and Bat are proved to be promising tools to solve such type of constrained objective optimization problems. So, it may be concluded that the solution given by CS to the specific problem is best so far. Thus the results obtained pave the way for new and promising research area, utilizing CS and Bat Algorithms with proper modifications, may give better results with high convergence speed.
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Optimal allocation of capacitor banks in radial distribution systems for minimization of real power loss and maximization of network savings using bio-inspired optimization algorithms Satish Kumar Injeti ⇑, Vinod Kumar Thunuguntla 1, Meera Shareef 2 Department of Electrical and Electronics Engineering, Sir C R Reddy College of Engineering, Eluru, West Godavari, Andhra Pradesh 534007, India
a r t i c l e
i n f o
Article history: Received 27 November 2014 Received in revised form 13 January 2015 Accepted 31 January 2015
Keywords: Bio-inspired algorithms Capacitor banks Real power loss minimization Network savings maximization Radial distribution systems
a b s t r a c t In this paper, two new algorithms are implemented to solve optimal placement of capacitors in radial distribution systems in two ways that is, optimal placement of fixed size of capacitor banks (Variable Locations Fixed Capacitor banks-VLFQ) and optimal sizing and placement of capacitors (Variable Locations Variable sizing of Capacitors-VLVQ) for real power loss minimization and network savings maximization. The two bio-inspired algorithms Bat Algorithm (BA) and Cuckoo Search (CS): search for all possible locations in the system along with the different sizes of capacitors, in which the optimal sizes of capacitor are chosen to be standard sizes that are available in the market. To check the feasibility, the proposed algorithms are applied on standard 34 and 85 bus radial distribution systems. And the results are compared with results of other methods like Particle Swarm Optimization (PSO), Harmonic Search (HS), Genetic Algorithm (GA), Artificial Bee Colony (ABC), Teaching Learning Based Optimization (TLBO) and Plant Growth Simulation Algorithm (PGSA), as available in the literature. The proposed approaches are capable of producing high-quality solutions with good performance of convergence. The entire simulation has been developed in MATLAB R2010a software. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The analysis of power distribution systems is an important area of research due to the fact that it is the final link between the bulk power system and consumers. However, reactive power flow in a distribution network always cause high power losses. The reactive power support is one of the well-recognized methods for the reduction of power losses together with other benefits; such as loss reduction, power factor correction, voltage profile improvement to the utmost extent under various operating constraints. The shunt capacitor is one of the basic equipment to fulfil these objectives. Therefore, it is important to find optimal location and sizes of capacitors in the system to achieve the above mentioned objectives. Numerous methods for solution to the optimal placement of capacitor with a view to minimizing losses have been suggested in the literature based on both traditional mathematical methods ⇑ Corresponding author. Mobile: +91 9581371537. E-mail addresses: [email protected] (S.K. Injeti), [email protected] (V.K. Thunuguntla), [email protected] (M. Shareef). 1 Mobile: +91 9032146188. 2 Mobile: +91 7207229663. http://dx.doi.org/10.1016/j.ijepes.2015.01.040 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
and more recent heuristic approaches. Over the last two decades, the studies on meta-heuristic techniques have shown that the most of the difficulties of classical methods can be eliminated by applying these techniques. Several heuristic methods have been developed in the last decade for optimal capacitor placement. Prakash et.al presented PSO [1] approach for finding the optimal sizes of capacitors with an objective of reduction of power loss in a radial distribution system. In this paper author used concept of loss sensitivity factors to determine the locations before sizing of capacitors using PSO. Kalyuzhny et al. proposed GA [2] as an optimization tool to place shunt capacitor on distribution system under capacitor switching constraints. Rao et al. [3] presented plant growth simulation algorithm (PGSA) presents two step solution methodology for optimal capacitors placement in radial distribution system with an objective to improve the voltage profile and reduction of power losses. In the first part, determination of optimal locations concept of loss sensitivity factors has been used, later PGSA is used for sizing of capacitors at the optimal locations determined in part one. Sizes of capacitors obtained in PSO & PGSA methods are continues & these are to be rounding off to nearest discrete capacitor sizes available in the market results in changes in the solution: active
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power loss and it may not be optimal. Raju et al. presented direct search algorithm (DSA) [4] to find the optimal size and location of fixed (discrete) and switched capacitors in a radial distribution system to maximize the savings by minimizing the active power loss. DSA uses step by step procedure for finding locations and capacitor sizes resembles the numerical method and it consumes much time to find the optimal solution. Some other heuristic methods such as tabu search [5], the harmony search algorithm [6], ant colony optimization-based algorithm [7] and a simulated annealing technique [8], and Teaching Learning Based Optimization (TLBO) [9] to solve the discrete size capacitor placement optimization problems. A comprehensive survey on the various heuristic optimization techniques applied to determine the optimal capacitor placement and size is presented in [10]. El-Fergany et al., used Artificial Bee Colony (ABC) [11] for optimal capacitor placement problem with an objective to maximize the net savings per year and to improve the voltage profile. In this work, potential location for capacitor connection has been determined by Loss Sensitive Factors (LSF) and voltage sensitive indices (VSI). Later, sizing of capacitor has been done using ABC algorithm. Obtained results are encouraging but, there is no guarantee for locations obtained by LSF & VSI method are optimal or not. The good results reported by others to various engineering optimization problems motivated us to apply novel meta-heuristic Bat and Cuckoo Search algorithms which are proposed by Xin-She-Xang [12,13]. However, from the literature review it is seen that the application of Bat and Cuckoo Search algorithms for optimal capacitor placement problem of distribution system has not been explored in previous works. This motivates the authors to use bio-inspired algorithm such as Bat and Cuckoo Search algorithms to locate optimal position and rating of capacitor in radial distribution system to maximize the annual network savings as an objective by minimizing real power loss. The present work describes Bat and Cuckoo Search algorithms methodologies for optimal capacitor allocation and sizing. Finding both locations and ratings of discrete or fixed size capacitors available in market in the optimal way named as VLFQ-case & finding both locations and ratings continues size of capacitors named as VLVQ case. In this paper, both Bat and Cuckoo Search (CS) optimization algorithms are applied to determine the optimal sizes and locations of capacitors for both VLVQ and VLFQ cases. This problem is formulated as a nonlinear constrained mixed discrete–continuous optimization problem. In order to show the effectiveness of proposed approaches: they have tested on the IEEE 34 and IEEE 85 bus radial distribution networks and the results are compared with the well existing methods available in the literature.
Mathematical modeling of radial distribution system Two bus model for distribution system analysis
V i ¼ jV i jL di V iþ1 ¼ jV iþ1 jL diþ1 Ii ¼ jIi jL hi Z i ¼ jZ i jL ;i
Sending end voltage Receiving end voltage Branch current Branch impedance, where Z i ¼ Ri þ jX i
Form Fig. 1
V iþ1 ¼ V i Ii Z i
ð1Þ
jV iþ1 jL diþ1 ¼ jV i jL di jIi jL hi jZ i jL ;i
ð2Þ
jV iþ1 j cos diþ1 þ jV iþ1 j sin diþ1 ¼ jV i j cos di þ jV i j sin di jIi jðcos hi j sin hi ÞðR2i þ X 2i Þ
ð3Þ
By separating real and imaginary terms
jV iþ1 j cos diþ1 ¼ jV i j cos di jIi jðRi cos hi þ X i sin hi Þ
ð4Þ
jV iþ1 j sin diþ1 ¼ jV i j sin di jIi jðX i cos hi Ri sin hi Þ
ð5Þ
By squaring and adding Eqs. (4) and (5)
jV iþ1 j2 ¼ jV i j2 2jV i jjIi j cos di fðRi cos hi þ X i sin hi Þg þ jIi j2 fðR2i þ X 2i Þg 2jV i jjIi j sin di fðX i cos hi Ri sin hi Þg ð6Þ After mathematical treatment Eq. (6) can be written as
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jfRðcosðdi hi ÞÞ þ X sinðdi hi Þg þ jIi j2 fðR2i þ X 2i Þg
ð7Þ
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jjZ i j cosðdi hi ;i Þ þ jIi j2 fðR2i þ X 2i Þg
ð8Þ
Since di hi ;i is very small hence, cosðdi hi ;i Þ ffi 1. Because in radial distribution system, voltage angle variations from source bus to the tail end of the feeder are only a few degrees. Therefore Eq. (8) can be directly written as
jV iþ1 j2 ¼ jV i j2 2jV i jjIi jjZ i j þ jIi j2 fðR2i þ X 2i Þg
ð9Þ
jV iþ1 j2 ¼ ½jV i j jIi jjZ i j2
ð10Þ
jV iþ1 j ¼ jV i j jIi jjZ i j
ð11Þ
where 1=2
jIi j ¼
ðP2i þ Q 2i Þ jV i j
ð12Þ
or it is also written as Eq. (13) 1=2
jIi j ¼
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j
Single line diagram of a two bus radial distribution system is depicted in Fig. 1.
ð13Þ 1=2
jV iþ1 j ¼ jV i j
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j
jZ i j
ð14Þ
1=2
ðR2i þ X 2i Þ
ð15Þ
1=2
ðR2i þ X 2i Þ ¼ 0
ð16Þ
jV iþ1 j2 ¼ jV i jjV iþ1 j ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2 jV i jjV iþ1 j þ ðP2iþ1 þ Q 2iþ1 Þ Positive root for Eq. (16)
1=2
jV iþ1 j ¼ Fig. 1. Two bus of radial distribution system.
jV i j ðjV i j2 4ððP2iþ1 þ Q 2iþ1 Þ 2
ÞðR2i þ X 2i ÞÞ
1=2
From Eq. (17) receiving end voltage can be directly found.
ð17Þ
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The real power loss in the branch connected between i and i + 1 bus may be computed as
PLoss ði; i þ 1Þ ¼ jIi j2 Ri
ð18Þ
The real power loss in the branch connected between i and i + 1 bus may be computed as
PLoss ði; i þ 1Þ ¼
Q Loss ði; i þ 1Þ ¼
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2
Ri
ð19Þ Fig. 2. Single line diagram for integration of shunt capacitor in radial distribution system.
ðP2iþ1 þ Q 2iþ1 Þ jV iþ1 j2
Xi
ð20Þ
Total real power loss of the feeder, PT, Loss, may determined by summing up the losses of all branches of the feeder, which is given as
PT;Loss ¼
n1 X PLoss ði; i þ 1Þ
ð21Þ
i¼1
where n = number of buses in the distribution system.
QCAP = Reactive power supplied by shunt capacitor. Hence, net injected reactive power at i + 1 bus can written as
ð22Þ
By substituting Eq. (22) in Eq. (16) then
jV iþ1 j2 jV i jjV iþ1 j þ ðP2iþ1 þ Q 2inj Þ
1=2
ðR2i þ X 2i Þ ¼ 0
ð23Þ
Possible feasible solution for Eq. (23) is
jV i j ðjV i j2 4ððP2iþ1 þ Q 2inj Þ jV iþ1 j ¼ 2
1=2
1=2
ÞðR2i þ X 2i ÞÞ
ð24Þ
Installation of shunt capacitor units in optimal locations of a distribution system results in several benefits. These include line loss reduction, improvement of voltage profile, power factor improvement etc. The power loss when capacitor is connected at i + 1 bus in the network as shown in Fig. 2, is given by
PCAP Lossði;iþ1Þ ¼
PCAP Lossði;iþ1Þ ¼
ðP 2iþ1 þ Q 2inj Þ jV iþ1 j2
Ri
ð25Þ
ððPiþ1 Þ2 þ ðQ iþ1 Q CAP Þ2 Þ
PCAP Lossði;iþ1Þ ¼ Ri
jV iþ1 j
2
ðP2iþ1 þ Q 2iþ1 Þ 2
jV iþ1 j
Q 2CAP 2Q iþ! Q CAP jV iþ1 j2
The load flow of a power network provides the steady state solution through which various parameters of interest like currents, voltages, losses etc can be calculated. The load flow is important tool for the analysis of power distribution system. Conventional load flow methods failed to give good results for distribution system analysis, due to special features like high R/X ratio and radial structure etc. Various distribution load flow techniques have been proposed in the literature [14–18]. Methods based on forward/backward sweep processes using Kirchhoff’s Laws or making use of the well-known bi-quadratic equation have gained popularity for distribution systems load flow analysis due to its low memory requirements, computational efficiency and robust convergence characteristics. In this paper, method which can find the load flow solution of radial distribution system directly by using topological characteristic of distribution network [19] is used. Single line diagram of radial distribution system with integration of shunt capacitor is shown in Fig. 2. Node one or first bus is considered as root bus or sub-station bus or source bus. At this bus generally load value is zero. Detailed flow chart for backward forward sweep load flow of radial distribution system has been given in Fig. 3. Objective function formulation
Ri
ð26Þ Objective function
þ
Q 2CAP 2Q iþ1 Q CAP jV iþ1 j2
Ri
ð27Þ
Net power loss reduction, DPCAP Loss in the system is the difference of power loss before and after installation of capacitor unit and is given by
DPCAP Loss ¼
In a distribution system, the voltage angle is not so important because, the variation of voltage angle from source to the tail end of the distribution feeder is only a few degrees. Forward/backward sweep based distribution load flow
Integration of shunt capacitor into the distribution system
Q inj ¼ Q iþ1 Q CAP
3. All lines are represented as short lines and half line charging susceptance of distribution line is negligible because of low level voltages unlike transmission.
Ri
ð28Þ
The positive sign of the DPCAP Loss indicates that the losses in the system reduces with installation of Capacitor.
Optimal capacitor placement in radial distribution system reduces the real power losses and improves the voltage profile. Reduction in power loss leads to the reduction in energy loss cost. However, the capacitor placement increases the installation and investment cost. Therefore the objective of the capacitor placement is to maximize the annual network savings by minimizing the total real power losses in the distribution system subjected to specific operational constraints. Mathematically, objective functions of the problem has been formulated as,
Total real power loss P la ¼
n1 X I2i ðRi Þ
ð29-aÞ
i¼1
Assumptions made in the above analysis are 1. System is a balanced. 2. In general, in radial distribution system first bus is considered as source bus or sub-station bus. Hence it is assumed that no load or Capacitor unit is connected at source bus.
Maximize S ¼ fK e ðPlb Pla ÞT / ½C i N c þ C p
Nc X
Q c ðiÞ C 0 Nc g
ð29-bÞ
i¼1
where n1 is the number of branches, n is the number of buses, I is respective branch current, R is the respective resistance of the
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Fig. 3. Flow chart of backward/forward sweep radial distribution load flow.
branch, S is the annual net savings, Plb & Pla are the total active power loss before and after compensation respectively, Ke is the energy cost per kW h, a is the depreciation factor, T is the time period in hours, Ci is the installation cost of capacitor per location, Nc is the number of compensated buses where capacitors are to be placed, Cp is the purchase cost of capacitor per kVAR, C0 is the capacitor operating cost per location and Qc(i) is the amount of reactive power of installed capacitor at bus i. Constraints The objective function is subjected to following constraints:
i¼1
Q Cmin and Q Cmax are minimum and maximum range of injected reactive power at ith bus For VLFC case capacitors are available in discrete sizes. So, shunt capacitors to be dealt with multiple integers of the smallest capacitor size available and it may be mathematically expressed as
ð34Þ
Maximum compensation with capacitor bank is limited to the total reactive power demand
ð31Þ
j¼1 Nc n X X Q c ðiÞ 6 Q D ðjÞ
Voltage limits
V min 6 V i 6 V max
ð33Þ
ð30Þ
j¼1
Nc n n1 X X X Q Slack þ Q ci ¼ PDi þ PLj i¼1
Q Cmin 6 Q Ci 6 Q Cmax
where Qs is the smallest capacitor size available and L is an integer multiple.
n n1 X X ¼ PDi þ PLj i¼1
Reactive power compensation limits for VLVQ case
Q c ðiÞ 6 LQ s
Power balance constraints
PSlack
Vmin and Vmax is minimum and maximum voltage limits of ith bus.
i¼1
ð32Þ
ð35Þ
j¼1
where n is the number of buses and QD(j) is the reactive power demand of load at bus j. Nc is number of capacitors.
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The apparent power flow through the line must be less than the maximum apparent power admissible for the line and it is expressed as
Si 6 Simax
i ¼ 1; 2; . . . nb
ð36Þ
where nb is the number of branches, Si is the apparent power flow of the ith branch and Simax is the maximum apparent power flow of the ith branch. Proposed algorithms Bat Algorithm The majority of heuristic and meta-heuristic algorithms have been derived from the behaviour of biological systems and/or physical systems in nature. The Bat Algorithm (BA) is based on the echolocation behaviour of bats, proposed by Xin-She-Xang for engineering optimization in [11]. If we idealize some of the echolocation characteristics of micro bats, we can develop various bat-inspired algorithms or Bat Algorithms. For simplicity, we now use the following approximate or idealized rules: (1) All bats use echolocation to sense distance, and they also ‘know’ the difference between food/prey and background barriers in some magical way. (2) Bats fly randomly with velocity Vi at position Xi with a fixed frequency fmin, varying wavelength k and loudness A0 to search for prey. They can automatically adjust the wavelength (or frequency) of their emitted pulses and adjust the rate of pulse emission r 2 [0, 1], depending on the proximity of their target. (3) Although the loudness can vary in many ways, we assume that the loudness varies from a large (positive) A0 to a minimum constant value Amin. Another obvious simplification is that no ray tracing is used in estimating the time delay and three dimensional topography. Though this might be a good feature for the application in computational geometry, however, we will not use this as it is more computationally extensive in multidimensional cases. In addition to these simplified assumptions, we also use the following approximations for simplicity. In general, the frequency f in a range [f min ; f max ] corresponds to a range of wavelengths [kmin, kmax]. For example a frequency range of [20 kHz, 500 kHz] corresponds to a range of wavelengths from 0.7 mm to 17 mm. For a given problem, we can also use any wavelength for the ease of implementation. In the actual implementation, we can adjust the range by adjusting the wavelengths (or frequencies), and the detectable range (or the largest wavelength) should be chosen such that it is comparable to the size of the domain of interest, and then toning down to smaller ranges. Furthermore, we do not necessarily have to use the wavelengths themselves; instead, we can also vary the frequency while fixing the wavelength k. This is because k and f are related due to the fact k/f is constant. For simplicity, we can assume f 2 [0, f max ]. We know that higher frequencies have short wavelengths and travel a shorter distance. For bats, the typical ranges are a few metres. The rate of pulse can simply be in the range of [0, 1] where 0 means no pulses at all, and 1 means the maximum rate of pulse emission. In simulations, we use virtual bats naturally. We have to define the rules how their positions X i and velocities V i in a d-dimensional search space are updated. The new solutions X ti and velocities V ti at time step t are given by
f i ¼ f min þ ðf max f min Þb
ð37Þ
V ti ¼ V t1 þ ðX ti X Þf i i
ð38Þ
X ti ¼ X t1 þ V ti i
ð39Þ
where b 2 [0, 1] is a random vector drawn from a uniform distribution. Here x⁄ is the current global best location (solution) which is located after comparing all the solutions among all the n bats. As the product ki f i is the velocity increment, we can use either f i (or ki) to adjust the velocity change while fixing the other factor ki (or f i ), depending on the type of the problem of interest. In our implementation, we will use f min ¼ 0 and f max ¼ 100, depending the domain size of the problem of interest. Initially, each bat is randomly assigned a frequency which is drawn uniformly from [f min ; f max ]. The update of the velocities and positions of bats have some similarity to the procedure in the standard particle swarm optimization as f i essentially controls the pace and range of the movement of the swarming particles. To a degree, BA can be considered as a balanced combination of the standard particle swarm optimization and the intensive local search controlled by the loudness and pulse rate. Simple pseudo code for Bat Algorithm (BA) has been given in Appendix A1. Steps for implementation of Bat Algorithm In this section, Bat Algorithm is described for solving the optimal placement of capacitors in radial distribution systems. Step 1: Initialization of problem and algorithm parameters In the first step, the algorithm parameters such as population size (POP), dimension of the problem and maximum number of iterations (itermax), limits of f, b and A are to be initialized. The problem parameters such as number of capacitors, limits of capacitor size, bus voltage limits and system data are to be initialized. Step 2: Random generation of locations and capacitor sizes
2
x11
6 x2 6 1 6 6 CAPLOC ¼ 6 ... 6 6 pop1 4 x1 xpop 1
2
y11 6 y2 6 1 6 6 CAPSIZE ¼ 6 ... 6 6 pop1 4 y1 ypop 1
x12
x1d1
x22 .. .
.. .
x2d1 .. .
xpop1 2 xpop 2
xpop1 d1 xpop d1
y12
y1d1
.. .
y2d1
ypop1 d1 ypop d1
y22 .. . ypop1 2 ypop 2
.. .
x1d
3
7 7 7 7 7 7 pop1 7 5 xd x2d .. .
ð40Þ
xpop d y1d
3
7 7 7 .. 7 . 7 7 7 5 ypop1 d y2d
ð41Þ
ypop d
xij ¼ xmin;i þ ðxmax;i xmin;i Þ randðÞ
ð42Þ
yij ¼ ymin;i þ ðymax;i ymin;i Þ randðÞ
ð43Þ
where d is the number of decision variables, xij , yij represents locations and capacitor sizes, i.e., jth population of ith capacitor location and size, which is generated randomly in between the limits as xmax;i and xmin;i are the ith capacitor location limits, ymax;i and ymin;i are the ith capacitor size limits and rand() is a random number in between 0 and 1. For VLFC case CAPSIZEi;d can be obtained by random generation of fixed size of capacitor banks between integral multiples of
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150 kVar. For VLVC case it can be obtained by random generation of capacitor sizes between 100 kVar and 1500 kVar.
Soln ¼ ½CAPLOC
CAPSIZE
ð44Þ
In Bat Algorithm, Soln. represents a group of Bats, where Bat is one position in search space. Bat is a solution that contains capacitor locations and sizes. Step 3: Fitness evaluation Run the load flow and calculate the real power loss of the system and corresponding network saving per year for each initial solution. Record the best solution. Step 4: Start evolution procedure of Bat Algorithm
ð45Þ
where b 2 [0, 1] is a random vector drawn from a uniform distribution. Initially each bat is randomly assigned a frequency which is drawn uniformly from [f min ; f max ]. Step 5: Random generation of Bat positions (locations and sizes of capacitors) t
t
locV i ¼ V t1 þ ðcaploci bestloc Þf i i t
t
sizeV i ¼ V it1 þ ðcapsizei bestsize Þf i t
CAPLOC ti ¼ CAPLOC t1 þ locV i i
ð46Þ ð47Þ ð48Þ
t
CAPSIZEti ¼ CAPSIZEit1 þ sizeV i
For simplicity, this last assumption can be approximated by a fraction pa of the n nests being replaced by new nests (with new random solutions at new locations). For a maximization problem, the quality or fitness of a solution can simply be proportional to the objective function. Other forms of fitness can be defined in a similar way to the fitness functions in genetic algorithms. Based on these three rules, the basic steps of the Cuckoo Search (CS) can be summarized as the pseudo code shown below. Pseudo code for CS algorithm is given in Appendix A1. When generating new solutions xtþ1 for, say cuckoo i, a L´evy flight is performed
xtþ1 ¼ xti þ / L0 ev yðkÞ i
Assign frequency for each Bat randomly
f i ¼ f min þ ðf max f min Þb
The number of available host nests is fixed, and a host can discover an alien egg with a probability p 2 [0, 1]. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.
ð49Þ
Step 6: Fitness evaluation (Objective function) Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new solution. Step 7: Compare each new bat solution with corresponding initial bat solution and replace better solution new bats to initial bat & find best bat, best solution among initial bats. Step 8: Stopping criterion. If the maximum number of iterations is reached, computation is terminated. otherwise, Step 4 to Step 7 is repeated. The steps of the Bat Algorithm (BA) are as follows can be summarized as the pseudo code. And the implementation flowchart for BA is given in Fig. 4.
where a > 0 is the step size which should be related to the scales of the problem of interest. In most cases, we can use a = O (1). The product means entry-wise multiplications. L´evy flights essentially provide a random walk while their random steps are drawn from a L´evy distribution for large steps
L0 ev y u ¼ tk ; ð1 < k 6 3Þ
Each cuckoo lays one egg at a time, and dumps it in a randomly chosen nest. The best nests with high quality of eggs (solutions) will carry over to the next generations.
ð51Þ
This has an infinite variance with an infinite mean. Here, the consecutive jumps/steps of a cuckoo essentially form a random walk process which obeys a power-law step length distribution with a heavy tail. It is worth pointing out that, in the real world, if a cuckoo’s egg is very similar to a host’s eggs, then this cuckoo’s egg is less likely to be discovered, thus the fitness should be related to the difference in solutions. Therefore, it is a good idea to do a random walk in a biased way with some random step sizes. Simple pseudo code for CS algorithm has been given in Appendix A2. Steps for implementation of CS algorithm In this section, CS algorithm is described for solving the optimal placement of capacitors in radial distribution systems. Step 1: Initialization of problem and algorithm parameters In the first step, the algorithm parameters such as population size (POP), dimension of the problem and maximum number of iterations (itermax) are to be initialized. The problem parameters such as number of capacitors, limits of capacitor size, bus voltage limits and system data are to be initialized. Step 2: Random generation of locations and capacitor sizes
2
x11
6 x2 6 1 6 6 CAPLOC ¼ 6 ... 6 6 pop1 4 x1 xpop 1
Cuckoo Search (CS) Algorithm A new meta-heuristic optimization algorithm, called Cuckoo Search (CS), was developed recently by Yang and Deb [12]. For simplicity in describing Cuckoo Search algorithm (Yang and Deb [13]), three idealized rules have to be followed:
ð50Þ
2
y11 6 y2 6 1 6 6 CAPSIZE ¼ 6 ... 6 6 pop1 4 y1 ypop 1
x12
x1d1
x22 .. .
.. .
x2d1 .. .
xpop1 2 xpop 2
xpop1 d1 xpop d1
y12
y1d1
y22 .. .
.. .
y2d1 .. .
ypop1 d1 ypop d1
ypop1 2 ypop 2
xij ¼ xmin;i þ ðxmax;i xmin;i Þ randðÞ
x1d
3
7 7 7 7 7 7 pop1 7 5 xd x2d .. .
ð52Þ
xpop d y1d
3
7 7 7 7 7 7 pop1 7 5 yd y2d .. .
ð53Þ
ypop d
ð54Þ
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447
Fig. 4. Flowchart for implementation of Bat Algorithm.
yij ¼ ymin;i þ ðymax;i ymin;i Þ randðÞ
ð55Þ
where d is the number of decision variables, xij , yij represents locations and capacitor sizes, i.e., jth population of ith capacitor location and size, which is generated randomly in between the limits as xmax;i and xmin;i are the ith capacitor location limits, ymax;i andymin;i are the ith capacitor size limits and rand() is a random number in between 0 and 1. For VLFC case CAPSIZEi;d can be obtained by random generation of fixed size of capacitor banks between integral multiples of 150 kVar. For VLVC case it can be obtained by random generation of capacitor sizes between 100 kVar and 1500 kVar.
nest ¼ ½CAPLOC
CAPSIZE
ð56Þ
Step 3: Fitness evaluation Run the load flow and calculate the real power loss of the system and corresponding network saving per year for each initial nest. Record the best solution vector.
Step 4: Start evolution procedure of CS algorithm
new caploci;d ¼ CAPLOC i;d þ / L0 ev yðkÞ
ð57Þ
new capsizei;d ¼ CAPSIZEi;d þ / L0 ev yðkÞ
ð58Þ
new nesti;d ¼ ½new caploci;d new capsizei;d
ð59Þ
Step 5: Fitness evaluation (Objective function) Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new nest. Step 6: Compare each new nest solution with corresponding initial nest solution and replace better solution new nests to initial nests & find best nest, best solution among initial solution.
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Fig. 5. Flowchart for implementation of CS algorithm.
Step 7: Abandon fraction of worst nests among new nests and built new ones at those nests using L0 ev y flight. Step 8: Run the load flow and calculate the real power loss of the system and corresponding network savings per year for each new nest. Step 9: Compare each new nest solution with corresponding initial nest solution and replace better solution new nests to initial nests & find best solution am better solution new nests to initial nests & find best nest, best solution among initial solution.ong initial nests. Step 10: Stopping criterion.
If the maximum number of iterations is reached, computation is terminated. otherwise, Step 4 to Step 9 is repeated. And for easy understanding of CS algorithm implementation flowchart is given in Fig. 5. Implementation results and discussions The performance and effectiveness of the proposed algorithms have been tested on 34-bus and 85-bus radial distribution systems for real power loss minimization and maximization of network savings. In this approach, optimal locations for connection of capacitors
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the test systems to determine the most suitable parameters. In this work, the tuned parameters of BAT and CS are given in Table 2. The entire simulation is developed in MATLAB R2010a software and the simulations are carried on a computer with Intel(R) Core(TM) i52450 M CPU @2.50 GHz, 4 GB RAM.
Table 1 Constants used in computation of net saving for the test cases. S.No
Parameter description
Value
1 2 3 4 5 6
Average energy cost (Ke) Depreciation factor (a) Purchase cost (Cp) Installation cost (Ci) Operating cost (Co) Hours per year (T)
$0.06/kW h 20% $25/kVAR $1600/location $300/year/location 8760
34-Bus test system numerical results and simulations
Table 2 Description of algorithm parameters. Algorithm
Parameter
Description
Value
CS
Pop d Pa maxitr
Population of host nests Dimension of a host nest Fraction of worst nests (Probability) Maximum number of generations
150 3&8 0.25 150
BAT
pop d A r fmin fmax maxitr
Population of bats Dimensional search space of a bat Loudness Pulse rate Minimum frequency Maximum frequency Maximum number of generations
150 3&8 0.50 0.50 0.00 2.00 150
Fig. 6. Single line diagram of 34-bus radial distribution system.
and optimal size of capacitors have been treated as a single problem, unlike other approaches quoted in the literature. The potential locations for capacitor placement are decided by algorithm itself along with optimal sizes of capacitors. In all calculations; for both test systems, the following constants are assumed and applied [11] as shown in Table 1. Problem of optimal capacitor placement is solved by two scenarios i.e., VLFQ and VLVQ. Backward/Forward sweep based power flow method has been used for radial distribution system load flow solution. All loads are assumed as constant power loads (peak load) and tap changing transformers are not considered in the present work to avoid complexity. A number of trails on the performance of the applied algorithms have been carried out on
The 34-bus test case consists of a main feeder and 4 sub-feeders (laterals) radial distribution system as shown in Fig. 6. The data of the system is obtained from [20]. The total load of the system is 4636.5 kW and 2873.5 kVAR. The rated voltage of the system is 11 kV. After an initial load flow run using Backward/Forward Sweep method for an uncompensated system, the active power loss is 221.7235 kW and maximum & minimum voltages are 0.9941 p.u and 0.9417 p.u, respectively. To observe the effectiveness of the proposed algorithms, obtained results are compared with the other techniques like PSO [1], ABC [11], HS [20], GA [21] and EA [22]. Table 3 shows, the optimal locations and capacitor sizes obtained by proposed algorithms along with existing algorithms for 34-bus radial distribution system. Comparison of technical and economic benefits of optimal capacitor placement in a 34bus radial distribution system with proposed and existing algorithms has been given in Table 4. From Table 4, it is observed that, both proposed algorithms yields to reduce peak losses to approximately 160 kW with 2250 kVar installed. Among all algorithms CSA-VLVQ has yielded best results i.e., real power loss is 160.61 kW, percentage of loss reduction is 27.56, the system overall power factor is 0.9904; minimum bus voltage is 0.9500 p.u., maximum value of stability index is 0.9804 and annual network saving is $19,006. These results clarify that, CS algorithm based approach possesses lower system losses and higher annual network saving compared to other heuristic methods. The convergence characteristics of proposed algorithms based on objectives for 34 bus test system are depicted in Figs. 7 and 8 respectively. From Fig. 7, it is observed that, objective function (minimization of real power loss of the system) has reached to best lowest value at 130th iteration and the other objective (annual network savings) has reached to best highest value at 130th iteration. Voltage profile characteristics of the test system with and without compensation for proposed algorithms have been shown in Fig. 9. From Fig. 9, it is noticed that proposed algorithms are succeeded in the improvement of voltage and minimum bus voltage is improved from 0.9416 p.u to 0.9501 p.u. and voltage magnitude has been improved at all buses of the test system. Coming to simulation time CS has consumed relatively more time than BA. Among VLVQ and VLFQ scenarios, combination of CSA-VLVQ yields best solution than other methods. Best results obtained are presented bold in Table 4 for easy comparison.
Table 3 Optimal locations and sizes of capacitors in kVar for 34 bus system. S.No of capacitors
Proposed methods VLFQ
BAT
1 2 3 4 5
Existing methods
VLVQ CUCKOO
BAT
CUCKOO
VLVQ
VLFQ
PSO [1]
EA [22]
ABC [19]
GA [21]
HS [20]
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
10 19 26
766 750 734
9 20 25
767.04 834.50 648.45
10 19 24
750 750 750
9 20 25
750 900 600
19 20 22
781 479 803
8 18 25
1050 750 750
8 18 25
900 900 800
5 9 12 22 26
300 300 300 600 300
4 11 17 26
250 750 300 1400
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Table 4 Results analysis for 34 bus system with and without optimal capacitors. Parameter of comparison
WOC
WC – proposed methods VLVQ
Total KVAR placed (kVAR) Real power losses (kW) Loss reduction (%) Overall P.F Vmin (p.u) Vmax (p.u) VSImin VSImax Network savings/year ($) Elapsed time (s)
– 221.73 – 0.8557 0.9416 0.9941 0.7860 0.9765 – –
WC – existing methods VLVQ
VLFQ
BAT
CUCKOO
VLFQ BAT
CUCKOO
PSO [1]
EA [22]
ABC [19]
GA [21]
HS [20]
2250 160.99 27.56 0.9904 0.9501 0.9951 0.8149 0.9804 18,809 19.92
2250 160.61 27.56 0.9904 0.9500 0.9951 0.8149 0.9804 19,006 36.83
2250 160.69 27.52 0.9904 0.9500 0.9951 0.8149 0.9804 18,967 17.11
2250 160.65 27.54 0.9904 0.9500 0.9951 0.8149 0.9804 18,985 34.16
2063 169.35 23.62 0.9970 0.9486 0.9950 0.8097 0.9800 15,348 N/A
2550 161.26 27.26 0.9837 0.9501 0.9952 0.8149 0.9808 17,165 N/A
2600 161.08 27.34 0.9798 0.9496 0.9949 0.8163 0.9810 17,018 N/A
1800 164.95 25.61 0.9825 0.9478 0.9949 0.8071 0.9796 17,740 N/A
2700 168.48 24.02 0.9989 0.9522 0.9953 0.8219 0.9811 11,991 N/A
Fig. 7. Convergence characteristics of Bat Algorithm based on real power loss of 34-bus test system.
Fig. 8. Convergence characteristics of Bat Algorithm based on annual network savings of 34-bus test system.
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Fig. 9. Voltage profile analysis of proposed algorithms for 34-bus test system.
Fig. 10. Single line diagram for 85-bus radial distribution system.
85-Bus test system numerical results and simulations The 85-bus test case consists of a main feeder, 9 sub-feeders (laterals) and sub-laterals radial distribution system as shown in Fig. 10. The data of the system is obtained from [23]. The total load of the system is 2574.3 kW and 2622.6 kVAR. The rated voltage of the system is 11 kV. After an initial load flow run using Backward/ Forward Sweep method for an uncompensated system, the active power loss is 316.8497 kW and maximum & minimum voltages are 0.9973 p.u and 0.8712 p.u, respectively. In order to evaluate the effectiveness of the proposed algorithms, their results are
compared with the other techniques like PSO [1], PGSA [3], DSA [4], TLBO [9] and GA [21]. Table 5 shows, the optimal locations and sizes of capacitors with proposed and existing algorithms. A comparative analysis of total kVar compensated, total real power loss, percentage of loss reduction and annual network savings of proposed and existing algorithms has been furnished in Table 6. For this test system eight capacitors are optimally placed and sized simultaneously. The locations and sizes of capacitors obtained by Bat Algorithm in case of VLVQ scenario are 7, 25, 28, 45, 48, 60, 65, 85 and 181, 323, 288, 138, 369, 400, 328, 224 kVAR respectively, with an active power loss
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Table 5 Optimal locations and sizes of capacitors in kVar for 85 bus system. S.No of capacitors
Proposed methods VLVQ
VLFQ
BAT
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Existing methods
CUCKOO
VLVQ
BAT
CUCKOO
VLFQ
PSO [1]
PGSA [3]
GA [20]
TLBO [9]
DSA [4]
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
Bus No.
kVar
7 25 28 45 48 60 65 85
181 323 288 138 369 400 328 224
8 68 32 63 12 44 48 21
367 356 220 313 337 175 347 134
8 12 31 40 56 63 71 79
600 300 300 150 300 300 150 150
18 27 29 42 48 60 69 80
150 150 300 150 300 450 300 450
8 58 7 27
796 453 324 901
8 58 7
1200 908 200
26 28 37 38 39 51 54 55 59 60 61 62 66 69 72 74 76 80 82
48.437 214.06 103.12 120.31 178.12 100 212.5 101.56 4.687 157.81 112.5 104.68 9.375 100 67.18 112.5 71.87 356.25 31.25
4 7 9 21 26 30 31 45 49 55 61 68 83 85
300 150 300 150 150 0 300 150 150 150 300 300 150 150
6 8 14 17 20 26 30 36 57 61 66 69 80
150 150 150 150 150 150 150 450 150 150 150 300 150
Table 6 Results analysis for 85 bus system with and without optimal capacitors. Parameter of comparison
WOC
WC – proposed methods VLVQ
Total KVAR placed (kVAR) Real power losses (kW) Overall P.F Vmin (p.u) Vmax (p.u) VSImin VSImax Loss reduction (%) Network savings/year ($) Elapsed time (s)
0.0 316.84 0.8649 0.8712 0.9973 0.6524 0.9806 – – –
WC – existing methods VLFQ
VLVQ
VLFQ
BAT
CUCKOO
BAT
CUCKOO
PSO [1]
PGSA [3]
GA [20]
TLBO [9]
DSA [4]
2250 147.8715 0.9858 0.9212 0.9973 0.7625 0.9806 53.33 72605 101.79
2250 145.743 0.9869 0.9215 0.9973 0.7614 0.9806 54 73700 199.32
2250 146.9557 0.9858 0.9218 0.9973 0.7625 0.9806 53.61 73086 86.09
2250 146.62 0.9858 0.9200 0.9973 0.7625 0.9806 53.72 73260 174.86
2473 163.32 0.9404 0.8774 0.9973 0.6625 0.9769 48.27 65850.21 N/A
2308 161.40 0.9475 0.8865 0.9973 0.7605 0.9685 48.88 68304.36 N/A
2206.25 146.06 0.9869 0.9205 0.9973 0.7625 0.9806 53.72 66950.71 N/A
2700 143.18 0.9885 0.9387 0.9973 0.7863 0.9806 54.8 69100.79 N/A
2550 144.00 0.9878 0.9315 0.9973 0.7987 0.9806 54.54 69414.54 N/A
Fig. 11. Convergence characteristics of Bat Algorithm based on real power loss of the 85-bus test system.
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Fig. 12. Convergence characteristics of Bat Algorithm based on annual network savings of 85-bus test system.
of 147.87 kW, minimum voltage of 0.9212 p.u and annual network savings of $72,605. The location and sizes of capacitors obtained by Cuckoo Search algorithm in case of VLVQ scenario are 8, 68, 32, 63, 12, 44, 48, 21 and 367, 356, 220, 313, 337, 175, 347, 134 kVar respectively, with an active power loss of 145.73 kW, minimum voltage of 0.9218 p.u and annual network savings of $73,700. The optimal locations and sizes of capacitors obtained by Bat Algorithm in case of VLFQ scenario are 8, 12, 31, 40, 56, 63, 71, 79 and 600, 300, 300, 150, 300, 300, 150, 150 kVAR respectively, with an active power loss of 146.95 kW, minimum voltage of 0.9218 p.u and annual network savings of $73,086. The location and sizes of capacitors obtained by CS algorithm in case of VLFQ scenario are 18, 27, 29, 42, 48, 60, 69, 80 and 150, 150, 300, 150, 300, 450, 300, 450 kVar
respectively, with an active power loss of 146.62 kW, minimum voltage of 0.9200 p.u and annual network savings of $73,260. From Table 6, it is noted that annual network savings for proposed algorithms is magnificent when compared with other algorithms, among which the best annual network savings and loss reduction is noticed in the results of Cuckoo Search algorithm. The convergence characteristics of proposed algorithms for simultaneous objectives have been shown in Figs. 11 and 12 respectively. It is observed that both objectives reached to their best values at 137th iteration for the case of VLVQ scenario with CS algorithm. Voltage profile characteristics of the test system with and without compensation for proposed algorithms have been shown in Fig. 13. Form Fig. 13, it is noticed that proposed algorithms are succeeded in the improvement of voltage at
Fig. 13. Voltage profile analysis of proposed algorithms for 85-bus test system.
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Table 7 Statistical analysis of numerical results for 20 runs. Metric
Bus system
Algorithm-method
Best
Worst
Mean
STD
Real power loss (kW)
34-Bus
CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ
160.6599 160.6100 160.6642 160.7648 146.5825 145.7057 146.9557 147.5487
160.7605 160.8700 161.8436 161.5664 146.9103 145.8626 148.4659 148.9768
160.6773 160.6500 161.2615 161.0709 146.7384 145.7626 147.7423 148.0433
0.0325 0.0029 0.4633 0.2643 0.1335 0.0457 0.4263 0.4482
CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ CSA-VLFQ CSA-VLVQ BAT-VLFQ BAT-VLVQ
18,932 19,006 18,363 18,509 73,110 73,661 72,293 72,024
18,985 19,009 18,983 18,930 73,282 73,743 73,086 72,775
18,976 19,006 18,669 18,769 73,200 73,706 72,673 72,519
17.1039 5.2460 243.5191 138.7858 69.8821 28.6800 223.9296 237.0653
85-Bus
Annual network savings ($)
34-Bus
85-Bus
each and every bus of the system. Particularly minimum bus voltage is improved from 0.8712 p.u. to 0.9215 p.u. Coming to simulation time CS has consumed relatively more time than BA. Among VLVQ and VLFQ methods, combination of VLVQ with CS yields best solution than other methods. Best results obtained are presented bold in Table 6 for easy comparison and understanding purpose.
Appendix Appendix. A1 Simple pseudo code for BA
Performance analysis of Bat and CS algorithms In order to analyze the performance of the proposed algorithms with existing algorithms, both algorithms were executed for 20 times by considering population size of 150 and maximum number of iterations is 150. From the obtained results, the best, worst, mean and the standard deviation of two metrics (real power loss and annual network savings) has been presented in Table 7 for the two test systems. From Table 7, it is observed that two metrics which are related to objective function seems to be better in case of CSA-VLVQ method than other methods for both test systems. The value of standard deviation has significance in the analysis of optimization algorithms i.e., lesser standard deviation indicates that efficiency of the algorithm is high. Hence, it is clear from the statistical analysis that CSA-VLVQ has been succeeded in achieving the desired objective with quality solution than other proposed and existing algorithms and the best results are presented in bold. Appendix Appendix. A2 Conclusions Simple pseudo code for CS In this paper, application of Bat and CS algorithms to the optimal placement of capacitors banks in radial distribution systems has been discussed. The practical application and efficiency of this method is evaluated using two test systems (34 and 85 bus). From the comparative analysis it is concluded that, CS algorithm gives better results than Bat and other existing algorithms, in terms of solution quality. Both CS and Bat generate solutions which satisfy all the constraints. According to convergence Bat converged very quickly due to simple evolution process. However, CS convergence is slower than Bat, the reason is rigorous evolution process in CS. In fact, for any optimization algorithm parameter tuning plays an important role in the performance of the algorithm. From the results; CS and Bat are proved to be promising tools to solve such type of constrained objective optimization problems. So, it may be concluded that the solution given by CS to the specific problem is best so far. Thus the results obtained pave the way for new and promising research area, utilizing CS and Bat Algorithms with proper modifications, may give better results with high convergence speed.
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