Chaotic artificial bee colony algorithm based solution of security and transient stability constrained optimal power flow

Electrical Power and Energy Systems 64 (2015) 136–147

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Chaotic artificial bee colony algorithm based solution of security and transient stability constrained optimal power flow Kürsßat Ayan a, Ulasß Kılıç b,⇑, Burhan Baraklı c a

Department of Computer Engineering, Sakarya University, Sakarya, Turkey Department of Mechatronics Engineering, Celal Bayar University, Manisa, Turkey c Department of Electrical and Electronics Engineering, Sakarya University, Sakarya, Turkey b

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 7 October 2013 Received in revised form 25 May 2014 Accepted 6 July 2014

Due to the increase rapidly of electricity demand and the deregulation of electricity markets, the energy networks are usually run close to their maximum capacity to transmit the needed power. Furthermore, the operators have to run the system to ensure its security and transient stability constraints under credible contingencies. Security and transient stability constrained optimal power flow (STSCOPF) problem can be illustrated as an extended OPF problem with additional line loading and rotor angle inequality constraints. This paper presents a new approach for STSCOPF solution by a chaotic artificial bee colony (CABC) algorithm based on chaos theory. The proposed algorithm is tested on IEEE 30-bus test system and New England 39-bus test system. The obtained results are compared to those obtained from previous studies in literature and the comparative results are given to show validity and effectiveness of proposed method. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Artificial bee colony algorithm Chaos theory Optimal power flow Security Transient stability

Introduction Optimal power flow (OPF) is an important tool for power system operators both in planning and operating stages. The OPF problem solution aims to optimize a selected objective function such as production cost via optimal adjustment of the power system control variables, while at the same time satisfying various equality and inequality constraints. The equality constraints are the power flow equations, while the inequality constraints are the limits on control variables and the operating limits of power system dependent variables [1]. The problem control variables include the generator active powers, the generator bus voltages, the transformer tap ratios, and the reactive power of switchable VAR sources, while the problem dependent variables include the load bus voltages, the generator reactive powers. Mathematically, the transient stability constrained OPF (TSCOPF) is OPF problem extended with additional rotor angle inequality constraints [2]. It means that after the disturbances the power system must be able to surviving and moving into an acceptable steady-state condition that meet all established limits [3]. Moreover, in Ref. [4] the authors indicate that TSCOPF is a nonlinear optimization problem with both algebraic and differential equations, which is difficult ⇑ Corresponding author. E-mail addresses: [email protected] (U. Kılıç), [email protected] (B. Baraklı).

(K.

http://dx.doi.org/10.1016/j.ijepes.2014.07.018 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

Ayan),

[email protected]

to be solved even for small power systems. The main difficulties of solving this problem include [4]: (1) how to deal with the differential equations that represent the dynamic behavior of the system; and (2) how to deal with the disruption of conventional optimization algorithms in highly nonlinear and non-convex solution landscapes. In the past, classical optimization techniques such as interior point method [5], and linear programming (LP) [6] were employed for TSCOPF solution. These techniques have many limitations and some drawbacks. They need an acceptable starting point that is close to the solution in order not to be stuck in local optimum and have poor convergence. As the numbers of parameters in the problem increase, the quality of solutions substantially depends on the initial conditions. Additionally, as they have extremely limited capability to solve realistic power system problems, the mathematical relationships should be mostly simplified to obtain the solutions of the problem. They are also weak in processing qualitative constraints. [7,8]. In Ref. [6], a linear programming (LP) based computational procedure was developed to solve an algebraic NP problem. Therefore, many heuristic optimization techniques have recently become more and more attractive in OPF and TSCOPF solution for researchers. Some of them are particle swarm optimization (PSO) [9], genetic algorithm (GA) [10], and differential evolution (DE) [4]. In the literature, many different heuristic methods have been used together with the chaos theory so far. However, there are

K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147

no works in the literature on STSCOPF solution based on the ABC and the CABC algorithm. At first place, this paper represents an approach for solving STSCOPF problem of the power systems using the ABC algorithm that was originally proposed by Karaboga in 2005 [11] and then this approach is developed by using the chaos theory. The ABC algorithm is a new population based meta-heuristic approach inspired by the food pursuit behavior of honey bees. After this introduction, the formulation of STSCOPF problem and the representation of security and transient stability constraints are reviewed in Section ‘Formulation of STSCOPF problem’. The ABC and CABC algorithms are introduced in Section ‘Illustration of the algorithms used in this study’. In Section ‘Simulation results’, the simulation results obtained from studies on IEEE 30-bus test system and the New England 39-bus test system are shown by comparing to those obtained by other methods in literature. Finally, the conclusions are discussed in last section.

Ng X ðai þ bi Pgi þ ci P2gi Þ

F cos t ¼

137

ð6Þ

i¼1

where Fcost is the total production cost, Pgi is the active power output of ith generator; Ng is the total number of generators; ai, bi and ci are the production cost coefficients of ith generator. Constraints in OPF problem with security and transient stability Security and transient stability constrained OPF can be considered as a conventional OPF with additional inequality constraints imposed by the transmission line loading limits and the rotor angle limits. The power flow solution should meet the steady-state constraints related to solution of the conventional OPF problem and the dynamic constraints imposed on the rotor angles during the transient period under undesirable conditions.

Formulation of STSCOPF problem

Equality constraints (power flow constraints) The power balance at ith bus can be expressed mathematically as follows [12,13]:

OPF formulation

Pgi  Pli  Pi ¼ 0

The power system is a complex network used for generating, transmitting, and distributing electric power. It is expected to operate by using minimum resources to satisfy maximum security and reliability. OPF problem is an important operator to try the implementation of these goals by optimizing whole controllable variables. Therefore, OPF is an optimization problem that has well known mathematical equations and is defined as follows [1,4–10]:

Q gi þ Q ci  Q li  Q i ¼ 0 i ¼ 1; . . . ; N

Minimize f ðx; uÞ

ð1Þ

Subject to gðx; uÞ ¼ 0

ð2Þ

hðx; uÞ 6 0

ð3Þ

where the objective function f(x, u) is taken as the total production cost, the equality constraints g(x, u) are taken as the power flow equations and the inequality constraints h(x, u) are taken as static and dynamic security constraints those are generation capacity constraints, security constraints (transmission line loading) and transient stability constraints. The vectors u and x what are the variables of the optimization problem are named as vector of control and state variables, respectively. The vector u is defined as follows:

uT ¼ ½Pg ; V g ; T; Q c 

ð4Þ

where the vector u consists of active power generation except the slack bus Pg, generator terminal voltage magnitude Vg, transformer tap ratio T and reactive power generation or absorption of compensation devices such as capacitor and reactor banks Qc. Moreover, the vector x is defined as follows:

xT ¼ ½Pgslack ; V L ; Q g 

ð5Þ

where the vector x includes slack bus active power Pgslack, load bus voltage magnitude VL and generator reactive power Qg.

i ¼ 1; . . . ; N

ð7Þ ð8Þ

where N is the total number of system buses; Pgi and Qgi are active and reactive power outputs of ith generator; Pli and Qli are total active and reactive power loads of ith bus; Pi and Qi are active and reactive power injections at bus i, Qci is shunt reactive source at ith bus. The active and reactive power injections at bus i can be also formulated as follows [14]: N X

Pi ¼ V i

V j ðGij cos hij þ Bij sin hij Þ  V i V k T i ðg ik cos hik þ bik sin hik Þ

j¼1 j–k j–i

þ V 2i ðGii þ T 2i g ik Þ Qi ¼ Vi

N X

ð9Þ

V j ðGij sin hij  Bij cos hij Þ  V i V k T i ðg ik sin hik  bik cos hik Þ

j¼1 j–k j–i

 V 2i ðBii þ T 2i bik Þ

ð10Þ

where Vi, Vj and Vk are the voltage magnitudes of ith, jth and kth buses; Gij and Bij are transfer conductance and susceptance between buses i and j of the bus admittance matrix (Ybus); Gii and Bii are self conductance and susceptance of bus i; hij is the voltage angle difference between buses i and j; i Ti is transformer tap ratio at ith bus; gik is transformer conductance between buses i and k; bik is transformer susceptance between buses and k. Inequality constraints (static and dynamic security constraints) Generation capacity constraints: For stable operation, the generator active and reactive power outputs, bus voltage magnitudes, transformer tap ratios and shunt reactive sources are restricted by their lower and upper limits as follows [1–10]: max Pmin gi 6 P gi 6 P gi

i ¼ 1; . . . ; Ng

ð11Þ

Objective function max Q min gi 6 Q gi 6 Q gi

The objective function can be described by the concepts such as production cost, social welfare, and fuel cost. In an interconnected energy system, the production cost is given by fuel cost curve approximated as a quadratic function of generator active power output. In this case, the total production cost minimization is taken into consideration as the objective of STSCOPF problem and it is expressed mathematically as follows [1,4–10]:

i ¼ 1; . . . ; N g

ð12Þ

V min 6 V i 6 V max i i

i ¼ 1; . . . ; N

ð13Þ

T min 6 T i 6 T max i i

i ¼ 1; . . . ; NT

ð14Þ

max Q min ci 6 Q ci 6 Q ci

i ¼ 1; . . . ; N c

ð15Þ

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K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147

where NT and Nc is the total number of transformers and shunt reactive sources, respectively. Security constraints: For secure operation, the apparent power injection at bus i (the transmission line loading), Si is restricted by its upper limits Smax as follows [15,16]: i

Si ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2i þ Q 2i 6 Smax i

i ¼ 1; . . . ; Nl

ð16Þ

where Nl is the number of transmission lines. Transient stability constraints: In this study, the classical generator model for transient stability analysis is used. This model denotes a synchronous machine with a constant voltage source behind a transient reactance. The mechanical power input is considered constant during the transient period. The transient stability problem of power systems is described with a set of differential– algebraic equations [17], which could be solved by time-domain simulation. The swing equation set for ith generator can be expressed as follows [4,5,9,18,19]:

d_ i ¼ xi  x0

i ¼ 1; . . . ; N g

ð17Þ

_ i ¼ x0 ðDi xi þ Pmi  Pgi Þ i ¼ 1; . . . ; Ng Mi x

ð18Þ

_ i is the rotor where d_ i is the rotor angle deviation of ith generator; x speed deviation of ith generator; xi and x0 are the rotor speed of ith generator and rated rotor speed; Mi is the moment of inertia of ith generator; Di is the damping constant of ith generator; Pmi and Pgi are the mechanical input power and electrical output power of ith generator. Pgi can be also expressed as follows:

Pgi ¼ E2i G0ii þ

Ng X

½Ei Ej B0ij sinðdi  dj Þ þ Ei Ej G0ij cosðdi  dj Þ i ¼ 1;. .. ;Ng

j¼1 j–i

ð19Þ where Ei and Ej are the constant voltages behind a transient reactance of the ith and jth generators; di and dj are the rotor angles of ith and jth generators; G0ij and B0ij are transfer conductance and susceptance between buses i and j of the reduced Ybus; G0ii is self conductance of bus i of the reduced Ybus. The reduced Ybus before, during and after a contingency can be obtained by eliminating all buses except for generator buses. All steps for building the reduced Ybus can be found in Ref. [17]. In this study, the state space model using generator angle as reference is used. Therefore, the generator of the slack bus is selected as the reference frame and the rotor angles of the rest generators are expressed with respect to the reference frame. Hence, the rotor angles of the generators except for slack bus are calculated by using modified Euler method that is a first-order numerical procedure for solving ordinary differential equations with a given initial value [17] and relative rotor angles can be expressed as follows:

direlativ e ¼ diþ1  d1

i ¼ 1; . . . ; ðNg  1Þ

Hence the relative rotor angle dmax as follows;

jdirelativ e jmax 6 dmax

ve drelati i

i ¼ 1; . . . ; ðNg  1Þ

ð20Þ is restricted by upper limit

ð21Þ

where dmax is the maximum allowable relative angle and its value is commonly based on experiences [5,9]. The value of dmax is determined by trial and error for both test systems in this study. These values are different for each system in the literature. Formulation of STSCOPF problem Formulation of STSCOPF problem is summarized according to the equality and inequality constraints which are defined in previous sections and can be given as follows;

Minimize Eq. (6) Subject to the equality constraints 7, 8, 17 and 18 the inequality constraints (11)–(16) and (21) Enforcement of STSCOPF constraints For the equality constraints, the power balance based on Newton–Raphson method are met by the power flow constraints in Eqs. (7) and (8), while the swing equations in Eqs. (17) and (18) are satisfied by time-domain simulation based on the modified Euler method. The full details for handling the swing equations were described in subsection ‘Inequality constraints (static and dynamic security constraints)’. For the inequality constraints, the penalty function is adopted to deal with all operating limits in Eqs. (11)–(16) and (21). The penalty function for limit violation of any variable can be defined as follows [19]:

8 max 2 > > < ðki  ki Þ hðki Þ ¼ ðkmin  ki Þ2 i > > : 0

ki > kmax i

if if

ð22Þ

ki < kmin i

if

kmin i

6 ki 6

kmax i

where hðki Þ represents the penalty function of variable ki ; kmin and i kmax are the lower and upper limits of the variable k . k is represent i i i the variables expressed in Eqs. (11)–(16), but the penalty value for Eq. (21) is kept constant. The penalty functions reflect the violation of the inequality constraints and assign a high cost of penalty function to the candidate point far from the feasible region. In order to enforce all inequality constraints mentioned above, the objective function in Eq. (6) is added by the penalty functions of active power generation of slack bus, reactive power generation, load bus voltage magnitude, transient stability limit, and transmission line loading and so the fitness value of a food source in ith position is calculated as follows:

F i ¼ F cos t þ K P ½hðPslack Þ þ K Q

Ng Nl X X hðQ gi Þ þ K V hðV Li Þ i¼1

i¼1

Ng Nline X X hðdirelativ e Þ þ K S hðSi Þ þ KT i¼1

ð23Þ

i¼1

ve where h(Pslack), h(Qgi), h(VLi), hðdrelati Þ and h(Si) are penalty funci tions of active power output of slack bus, the reactive power output of the generator, load bus voltage magnitude, relative rotor angle and transmission line loading, respectively, and K s are the corresponding penalty weights. Nl is total number of the load buses. Note that the active power generation limits of all the generator buses except for slack bus, voltage magnitude limits of all the generators and transformer tap ratio limits will not be included in the extended objective function shown in Eq. (23). Because these control variables are randomly created inside their feasible limits during the proposed algorithm process.

Illustration of the algorithms used in this study ABC algorithm ABC algorithm that was proposed by Karaboga is a population based algorithm created by the inspiration from the food pursuit of honey bees [11]. The foraging and dance behaviors of honey bees are simulated by the ABC algorithm to get optimal solutions of different optimization problems. Even though the honey bees collect nectar from food sources around the hive in nature, artificial bees in ABC algorithm search the solution space and evaluate solution parameters found [11]. The solution (population) number of a problem is same as the number of onlooker or worker bees in the population. The position of a food source represents an optimal

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K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147

solution of the problem and its quality (fitness value) is also evaluated through the nectar amount of the corresponding food source. There are the food sources as the number of the worker bees. In the ABC algorithm, the bees in the hive dance to share information about the positions of food sources between worker and onlooker bees as well as in nature [11]. The other type of bee within a bee colony is scout bee and also works for foraging. A scout bee searches around the hive for obtaining new food sources. If a worker bee cannot improve the self solution in a period of time, it becomes scout bee to increase search ability of the ABC algorithm. Scout bees also provide to avoid local minimums by restraining laziness of the bee population. Herein, the number of scout bees and the time of occurrence of scout bee (called as limit) in the ABC algorithm is important. When this number is a great value and this time is short, utilization ability from food sources of ABC algorithm decreased. When this number is a very small value and this time is long, probability of avoiding local minimum of the ABC algorithm decreased. Therefore, it is necessary to balance this number and time for each problem. In this study, the number of scout bees in ABC algorithm is taken as 1 and so its percentage in colony is being 5%. There are two kinds of foragers in the hive of ABC. They are worker and non-worker foragers. The worker foragers utilize from food sources (the possible solution of the optimization problem) continually and they carry information to the hive about positions of food sources. There are two types of non-worker foragers. They are onlooker and scout bees. The onlooker bees fly to food source by taking information shared by worker foragers. The scout bees search new food source around the hive and average number of scout bees is about 5–10% [11]. Even though 5–10% of the bee population in the nature is scout bee, there is only one scout bee in the ABC hive. Besides, the half of population in the basic ABC is worker bees and other half is onlooker bees. The worker bees try to improve self solutions by using the following equation.

wijimprov ed ¼ wcurrent þ /ij ðwcurrent  wcurrent Þ i ¼ 1; . . . ; SN ij ij kj ¼ 1; . . . ; SN

j ¼ 1; 2; . . . ; D i–k

ð24Þ

ð25Þ

where wji is a parameter to be optimized of the ith worker bee for jth dimension of the D-dimensional solution space, wjmax and wjmin are the upper and lower limits of wji , respectively and rand(0, 1) is a number produced randomly in between 0 and 1. After the generating a new solution, the scout bee becomes the worker bee.Every onlooker bee memorizes the solution of one of n worker bees based on fitness values of the worker bees to generate a new (improved) food position by using Eq. (24). Onlooker bees select a worker bee by using roulette wheel selection mechanism and Eq. (26) [20,21]. The statement spi is a selection probability of ith worker bee and calculated as follows:

fit spi ¼ PSN i j¼1 fit j

fiti ¼

1 Fi

ð27Þ

where Fi is the object function to be minimized of the problem. Eq. (25) is not only used for generating new solution for scout bee, but also used for producing initial solutions for the worker bees at the beginning of the ABC algorithm. The onlooker and worker bees use to utilize from food sources the same equation. Primarily, the ABC algorithm produces the initial population (the positions of the food sources) randomly. After producing of the initial population, the worker, onlooker, and scout bees continuously search for whole food sources throughout a predetermined number of iterations. The each iteration of the ABC algorithm consists of five steps after initialization step [11]: Step 1: The worker bees fly to the food sources and calculate their nectar amounts (fitness values or qualities). Step 2: The onlooker bees take information about nectar amounts (fitness values or qualities) of the food sources from the worker bees and then they place to the new food sources and calculate their nectar amounts. Step 3: If a worker bee abandon its food source, it becomes a scout bee. The determined scout bee flies to new possible food source. Step 4: The new food sources are ranked according to their nectar amounts (fitness values or qualities) and the finest quality food sources are memorized. Step 5: Once the stopping criterion is ensured the algorithm is ended. The ABC algorithm has been applied successfully to the distinct fields of science such as mathematics, civil and electrical engineering, and design and manufacturing [20–23]. Principles of CABC algorithm

k

where wi is ith worker bee, wcurrent is the current solution for wi, i wiimprov ed is the improved solution for wi, wk is a neighbor worker bee of wi, / is a number produced randomly in between 1 and 1, SN is the number of worker bees, D is the dimension of the problem and j = 1, . . ., D and k = 1, . . ., SN are selected randomly. Besides, only one parameter of worker bee is improved at the each iteration. Whole solution parameters of worker bee are also copied to the improved solution produced by this bee for evaluating objective function. If a worker bee cannot improve self solution in a period of time, it becomes a scout bee and a new (improved) solution is generated for it by the following equation:

wji ¼ wjmin þ randð0; 1Þðwjmax  wjmin Þ j ¼ 1; 2; . . . ; D

where fiti is the fitness value (quality of a food source) of ith worker bee (position of food source) and is calculated as follows:

ð26Þ

Overview of chaos theory Chaos occurs commonly in non-linear dynamic systems and extremely is an unstable behavior of deterministic systems in finite state space. Chaos theory can be illustrated by the entitled ‘butterfly effect’ explained by Lorenz [24]. While Lorenz simulates numerically a global weather system, he found out that tiny changes in initial conditions caused the following simulations toward absolutely different finals and illustrated that its long-term prediction is generally impossible. Chaotic behavior of various systems such as power transmission, nonlinear electrical circuits, fluid dynamics, DNA computing, and image processing has already been observed in the laboratories and their models have been realized by the computers so far. For simulation of a complex process, numerical analysis, decision making and particularly heuristic optimization requires random sequences having a good uniformity along a long period [25]. Chaos is a random-like process appeared in a non-linear dynamical system. Furthermore, it depends on its initial condition and parameter [25]. Although there are randomness and unpredictability in the nature of chaos, it also has an element of regularity. Mathematically, chaos is randomness of a simple deterministic dynamical system and chaotic system can be taken into account as sources of randomness [25]. A chaotic structure can be expressed as a discrete-time dynamical system operating in chaotic medium as follows:

wkþ1 ¼ f ðwk Þ;

0 < wk < 1;

k ¼ 0; 1; 2; . . .

ð28Þ

A chaotic sequence {wk:k = 0, 1, 2, . . .} can be utilized as a sequence having both the spread-spectrum and the random number.

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K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147

As the generation and storage of chaotic sequences is easy and fast, it is unnecessary to store the long sequences [26]. Even if very long sequences, there is only need for several functions (chaotic structures) and parameters (initial conditions). Furthermore, we can produce easily too many distinct sequences by varying its initial condition. Moreover these sequences are deterministic and reproducible. In recent times, chaotic sequences instead of random sequences have been used for many scientific studies by applying to the distinct disciplines such as secure transmission, nonlinear circuits, DNA computing, image processing, system modeling, electrical engineering, and aerospace science and good results were obtained [26–33]. In this paper, a new approach introducing chaotic structure with irregularity and the stochastic property in ABC is proposed to improve the global convergence by escaping the local solutions. The usage of chaotic sequence in ABC can be assist to escape more easily from local minima in comparison with the classical ABC. Application of CABC algorithm to the problem The food sources produced by chaos process are called as chaotic food sources and the bees which determine the nectar amount in these sources are called as the chaotic bees. In this study, the chaos process is applied to the best food source at the end of the each iteration. Thus the parameters of the best food source are normalized by a value between 0 and 1 as follows [34]:

wj;normalized ¼ 1

wj1  wjmin wjmax  wjmin

j ¼ 1; . . . ; D

ð29Þ

In chaos process, the parameter value of a new food source is calculated by that of the previous food source through Eq. (30) which is the simplest logistic equation. This process continues until the number of chaotic food source is reached to the predetermined value [25,31,33].

wj;normalized ¼ 4wj;normalized  ð1  wj;normalized Þ i ðiþ1Þ i i ¼ 1; . . . ; CBN  1 J ¼ 1; . . . ; D

ð30Þ

where CBN is the number of chaotic food sources. The parameter values of all the normalized chaotic food sources are converted to their real values by using Eq. (31), the fitness values of chaotic food sources are calculated and then they are included to the current food sources [35].

wji ¼ wj;normalized  ðwjmax  wjmin Þ þ wjmin i ¼ 1; . . . ; D

i ¼ 1; . . . ; CBN

Fig. 1. Flowchart of the CABC algorithm for STSCOPF problem.

j ð31Þ

Flowchart of CABC algorithm for STSCOPF can also be shown in Fig. 1. Here, ‘‘bas’’ is a better food source search number around a food source, ‘‘baslimit’’ is the maximum search number, ‘‘ite’’ is the iteration number, and ‘‘itemax’’ is the maximum iteration number. Here, the value of ‘‘baslimit’’ is considered as being 5. Simulation results The proposed CABC based solution of STSCOPF is tested on the IEEE 30-bus test system and the New England 39-bus test system. Trials made for different population sizes show that although the ABC and CABC algorithms reach the better solutions for high population sizes, in this case, it is observed that the CPU times of both the algorithms increase. For both the algorithms, therefore, the best population sizes are selected as being 20 and 25, respectively. The limit value of the population size for both algorithms is also selected as being 5. Iteration number of simulation is considered as 100. For two test systems, the population size related to the ABC and CABC algorithms is given in Table 1. Furthermore, the

results of the ABC and CABC algorithm are obtained after carrying out 30 independent runs for different cases. In other words, the initial population was randomly generated in each run by using different seeds. The software used for transient stability analysis is in Ref. [36]. The prototype software developed on MATLAB was run on two different personal computers so that the results obtained using the ABC and CABC algorithms can be compared to those obtained using other algorithms as realistic. First one is a personal computer with Intel Core 2 Duo CPU 2.67 GHz processor and 1.87 GB memory used in this study. Second one is a personal computer with Pentium IV CPU 2.66 GHz processor and 512 MB memory used in

Table 1 Population sizes for the ABC and CABC algorithms. Algorithm

Worker bees

Non-worker bees

Chaotic bees

ABC CABC

10 10

10 10

– 5

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K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147 Table 4 Comparative results of IEEE 30-bus test system for case A. Fault at bus 2

Minimum cost [$/h] Maximum cost [$/h] Average cost [$/h] Time (s) Timea (s) a

Case A GA Ref. [9]

PSO Ref. [9]

ABC

CABC

585.62 585.71 585.66 – 752.3

585.17 585.69 585.34 – 576.4

577.78 583.90 580.84 46.59 115

577.63 580.83 579.23 75.50 182

The computational times by using the personal computer mentioned in Ref. [9].

Table 5 Comparative results of IEEE 30-bus test system for case B. Fault at bus 2

Minimum cost [$/h] Maximum cost [$/h] Average cost [$/h] Time (s) Timea (s)

Case B EP Ref. [19]

EPNN Ref. [19]

ABC

CABC

585.15 586.86 585.83 – 451.13

585.12 586.73 585.84 – 220.17

577.71 583.26 580.21 49.97 125

577.47 580.74 579.10 77.80 195

a The computational times by using the personal computer mentioned in Ref. [19].

Table 6 The apparent powers of transmission lines for IEEE 30-bus test system. From bus

To bus

S (MVA)

Smax (MVA)

Case A

Fig. 2. The IEEE 30-bus test system.

Table 2 Active power generations and voltages of IEEE 30-bus test system for case A. Bus number

1 2 13 22 23 27

ABC

CABC

Pg (MW)

V (p.u.)

Pg (MW)

V (p.u.)

40.5512 51.9248 18.9168 23.8110 16.8010 40.0000

0.9858 0.9780 1.0601 1.0191 1.0400 1.0639

41.4823 55.3017 17.0909 20.8952 17.0019 40.4145

0.9723 0.9738 1.0785 1.0241 1.0340 1.0610

Table 3 Active power generations and voltages of IEEE 30-bus test system for case B. Bus number

1 2 13 22 23 27

ABC

CABC

Pg (MW)

V (p.u.)

Pg (MW)

V (p.u.)

40.6796 55.0000 15.4663 22.1919 19.9917 38.8263

0.9700 0.9715 1.0752 1.0187 1.0264 1.0643

42.7411 54.7034 14.8287 24.3374 17.9055 37.6754

0.9733 0.9653 1.0620 1.0106 1.0242 1.0685

Refs. [9,19]. All the computational times and the production costs obtained by the ABC and CABC algorithms were given for comparing to those obtained using other algorithms in literature. IEEE 30-bus test system The IEEE 30-bus system consists of 41 lines, 6 generators, and 4 tap-changing transformers as shown in Fig. 2. The bus and line data

1 1 2 3 2 2 4 5 6 6 6 6 9 9 4 12 12 12 12 14 16 15 18 19 10 10 10 10 21 15 22 23 24 25 25 28 27 27 29 8 6

2 3 4 4 5 6 6 7 7 8 9 10 11 10 12 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 27 29 30 30 28 28

Case B

ABC

CABC

ABC

CABC

21.6685 19.4269 18.1381 16.7792 13.9201 20.7614 18.9336 13.8063 11.6109 31.9902 13.6369 6.9284 0 9.4572 20.2104 33.6452 4.7785 5.9268 7.4701 1.6939 3.7142 8.5059 5.1947 5.4412 7.7732 7.5963 11.7644 9.558 31.6267 12.9532 5.236 6.2164 11.7494 4.2578 15.6769 24.3544 6.3759 7.288 3.7279 10.3218 12.7275

23.5495 21.247 19.4754 18.5095 14.5279 22.1683 18.7699 14.3484 11.4004 31.9852 16.8973 8.5849 0 11.7874 25.2114 40.114 5.2345 7.2436 7.999 1.7175 4.3213 8.4836 5.1382 5.3414 7.6943 7.3138 12.7849 9.9712 31.8153 11.5589 7.3049 4.0953 12.1407 4.2581 15.9311 24.4786 6.3769 7.2891 3.7281 10.3333 12.7798

23.1133 20.8515 19.1179 18.1175 14.3772 21.8232 18.5814 14.2169 11.4578 31.9795 15.3954 7.8218 0 10.7184 24.0223 40.4843 5.1576 7.1635 7.8841 2.1871 4.3206 8.4823 5.1164 5.2833 7.6482 7.5982 12.4379 9.8808 31.9675 13.3135 7.377 3.7319 11.3102 4.258 15.4399 25.0629 6.3758 7.2878 3.7278 10.3974 13.3411

22.4209 20.8783 19.8417 18.1878 14.637 22.4921 19.2382 14.4066 11.5775 31.9953 14.5401 7.3873 0 10.104 23.141 36.9048 4.9401 6.1156 7.5252 1.9548 4.128 8.245 4.9624 5.7262 8.0655 8.109 11.9265 9.667 31.8432 12.6041 5.1254 3.7142 11.6215 4.2579 15.9898 26.1041 6.3744 7.2861 3.7275 10.4951 14.2996

130 130 65 130 130 65 90 70 130 32 65 32 65 65 65 65 32 32 32 16 16 16 16 32 32 32 32 32 32 16 16 16 16 16 16 65 16 16 16 32 32

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Fitness value

ABC

Phase angle difference (fault cleared at 0.18s)

CABC

30

8000 6000

20

4000

10

0

10

20

30

40

50

60

70

80

90

100

Iteration number Fig. 3. Variation of fitness value against iteration for case A of IEEE 30-bus test system.

Delta, degree

2000

0

-10

-20

8000

Fitness value

ABC

-30

-40

CABC

6000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, sec Fig. 5. Relative rotor angles obtained by the ABC algorithm for case A.

Phase angle difference (fault cleared at 0.18s) 30 20

Delta, degree

10 0 -10 -20 -30 -40

-50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, sec Fig. 6. Relative rotor angles obtained by the CABC algorithm for case A.

Phase angle difference (fault cleared at 0.35s) 100

50

Delta, degree

of the test system are given in Ref. [37]. The total load demand is 189.2 MW and 107.2 MVAr. A single contingency consisting of a three-phase fault occurred near at bus 2 of line 2–5 at t = 0 s is considered. The study is implemented for two different critical clearing times (CCT) given in Ref. [9] and Ref. [19] of the test system. Case A: The fault is subsequently cleared at t = 0.18 s. To observe the rotor oscillation, the maximum integration period is 1.0 s. Here the integration time step is 0.1 s. and dmax is set to 120°. Case B: The fault is subsequently cleared at t = 0.35 s. To observe the rotor oscillation, the maximum integration period is 1.5 s. Here the integration time step is 0.01 s. and dmax is set to 120°. Active power generations and voltages obtained by the ABC and CABC algorithm for both case A and B are shown in Tables 2 and 3, respectively. For case A, comparative results obtained by GA, PSO, the ABC and CABC algorithms are shown in Table 4. The times (sec) in Table 4 correspond to the average production costs. As seen from table, the times for the ABC and CABC algorithms are 46.59 s (115 s by using PC mentioned in Ref. [9]) and 75.50 s (182 s by using PC mentioned in Ref. [9]), respectively. So it means that these algorithms are faster than GA and PSO in any case. Again, the average production costs by the ABC and CABC algorithms are 580.84 $/h and 579.23 $/h, respectively. These values give the reduction ratios of 0.82% and 1.09% in comparison with GA and PSO, respectively. For case B, comparative results obtained by EP, EPNN, the ABC and CABC algorithms are shown in Table 5. The times in Table 5 also correspond to the average production costs. As seen from table, the times for the ABC and CABC algorithms are 49.97 s (125 s by using PC mentioned in Ref. [19]) and 77.80 s (195 s by using PC mentioned in Ref. [19]), respectively. So it means that these algorithms are also faster than EP and EPNN in any case. Again, the average production costs by the ABC and CABC algorithms are 580.21 $/h and 579.10 $/h, respectively. These values give the reduction ratios of 0.96% and 1.15% in comparison with EP and EPNN, respectively. For cases A and B, the apparent powers obtained by the ABC and CABC algorithms and the maximum apparent powers of transmission lines are shown in Table 6.

0

-50

4000

-100

2000 0

10

20

30

40

50

60

70

80

90

100

Iteration number

-150 0

0.5

1

t, sec Fig. 4. Variation of fitness value against iteration for case B of IEEE 30-bus test system.

Fig. 7. Relative rotor angles obtained by the ABC algorithm for case B.

1.5

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Table 7 Active power generations and bus voltages of New England 39-bus test system for case C.

Phase angle difference (fault cleared at 0.35s) 80 60

Bus number 40

30 31 32 33 34 35 36 37 38 39

Delta, degree

20 0 -20 -40 -60 -80

ABC

CABC

Pg (MW)

V (p.u.)

Pg (MW)

V (p.u.)

313.03 616.15 610.23 617.57 455.23 602.95 507.08 600.00 702.48 1110.00

1.0008 1.0057 1.0193 1.0261 1.0103 1.0344 1.0285 1.0163 1.0327 0.9514

300.00 570.30 653.26 600.00 443.87 645.14 503.38 600.00 719.36 1100.00

1.0081 1.0027 1.0102 1.0270 1.0163 1.0336 1.0280 1.0200 1.0372 0.9587

-100 -120 0

0.5

1

1.5

t, sec Fig. 8. Relative rotor angles obtained by the CABC algorithm for case B.

Variation of fitness value against iteration for cases A and B of IEEE 30-bus test system are shown in Figs. 3 and 4, respectively. It can be seen clearly that both the ABC and CABC algorithms reach to the optimum point in minimum iteration for both cases, although their starting point is worse one. For cases A and B, the relative rotor angels obtained by the ABC and CABC algorithms of all the generators are shown in Figs. 5–8, respectively. As seen from figures, all the generators are stable and the rotor angels of all the generators do not exceed the value dmax. Furthermore the apparent powers of all the transmission lines do not exceed their upper limits Smax. New England 39-bus test system New England 39-bus test system comprises 10-generator, 39 bus and 46-line as shown in Fig. 9. The data for this test system

Table 8 Active power generations and bus voltages of New England 39-bus test system for case D. Bus number

30 31 32 33 34 35 36 37 38 39

ABC

CABC

Pg (MW)

V (p.u.)

Pg (MW)

V (p.u.)

336.96 562.59 549.17 627.34 489.38 535.59 577.91 580.68 759.98 1119.7

1.0324 1.0495 0.9919 1.0518 1.0207 1.0516 1.0291 1.0110 0.9532 1.0334

350 564.26 577.14 600.00 491.62 556.22 564.63 568.49 763.34 1100.0

0.9971 0.9825 1.0062 1.0600 1.0600 1.0568 1.0599 1.0109 1.0552 0.9954

is taken from [38]. The total load for the operating condition considered is 6098 MW and 1409 MVAr. There is no shunt capacity in the test system. A single contingency consisting of a three-phase fault occurred near at bus 17 of line 17–18 at t = 0 s is considered. The fault is subsequently cleared at t = 0.20 s. To observe the rotor oscillation, the maximum integration period is 5 s. Here the integration time step is 0.01 s and dmax is set 170°.

Fig. 9. New England 39-bus test system.

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Table 9 Comparative results of New England 39-bus test system for case C. Case C

Minimum cost [$/h] Maximum cost [$/h] Average cost [$/h] Time (s) Timea (s) a

TS Ref. [35]

ABC

CABC

62261.28 – – – –

61485.48 61703.42 61594.45 65.29 160

61369.19 61602.53 61485.86 100.00 251

50

The computational times by using the personal computer mentioned in Ref. [9].

0

Delta, degree

Fault at bus 17

Phase angle difference (fault cleared at 0.2s) 100

-50

-100

Table 10 Comparative results of New England 39-bus test system for case D. Fault at bus 17

Minimum cost [$/h] Maximum cost [$/h] Average cost [$/h] Time (s) Timea (s) a

-150

Case D PSO Ref. [9]

ABC

CABC

36,262 36,386 36,336 – 952.5

36058.69 36678.11 36368.4 66.50 165

35869.23 36258.53 36063.88 104.36 255

-200

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t, sec Fig. 12. Relative rotor angles obtained by the ABC algorithm for case C.

The computational times by using the personal computer mentioned in Ref. [9].

Phase angle difference (fault cleared at 0.2s) 100 4

8.5

x 10

CABC

50

7.5 7

0

6.5 6

10

20

30

40

50

60

70

80

90

100

Iteration number

Delta, degree

Fitness value

ABC 8

-50

-100

Fig. 10. Variation of fitness value against iteration for case C of New England 39-bus test system.

-150 4

5

x 10

Fitness value

ABC

-200

CABC

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t, sec

4.5

Fig. 13. Relative rotor angles obtained by the CABC algorithm for case C. 4

3.5

10

20

30

40

50

60

70

80

90

100

Phase angle difference (fault cleared at 0.2s) 100

Iteration number Fig. 11. Variation of fitness value against iteration for case D of New England 39bus test system.

Delta, degree

50

There are two different coefficient sets related to this test system for production cost in literature. The study is implemented for two different cases related to these coefficient sets. Case C: In this case, the production cost coefficients in Ref. [39] are used. Case D: In this case, the production cost coefficients in Ref. [37] are used. Active power generations and bus voltages obtained by the ABC and CABC algorithm for both cases C and D are shown in Tables 7 and 8, respectively. For case C, the comparative results obtained by TS method in Ref. [39], the ABC and CABC algorithms are shown in Table 9. The times in Table 9 correspond to the average production costs. As seen from table, the minimum production costs by the ABC and CABC algorithms are 61485.48 $/h and 61369.19 $/h, respectively.

0

-50

-100

-150 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t, sec Fig. 14. Relative rotor angles obtained by the ABC algorithm for case D.

5

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K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147

Phase angle difference (fault cleared at 0.2s)

Table 12 Load data of IEEE 30-bus test system.

100

Delta, degree

50

0

-50

-100

-150 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t, sec Fig. 15. Relative rotor angles obtained by the CABC algorithm for case D.

Table 11 AC transmission line characteristics of IEEE 30-bus test system. From bus

To bus

R (p.u.)

X (p.u.)

B/2 (p.u.)

1 1 2 3 2 2 4 5 6 6 6 6 9 9 4 12 12 12 12 14 16 15 18 19 10 10 10 10 21 15 22 23 24 25 25 28 27 27 29 8 6

2 3 4 4 5 6 6 7 7 8 9 10 11 10 12 13 14 15 16 15 17 18 19 20 20 17 21 22 22 23 24 24 25 26 27 27 29 30 30 28 28

0.02 0.05 0.06 0.01 0.05 0.06 0.01 0.05 0.03 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.07 0.09 0.22 0.08 0.11 0.06 0.03 0.09 0.03 0.03 0.07 0.01 0.10 0.12 0.13 0.19 0.25 0.11 0.00 0.22 0.32 0.24 0.06 0.02

0.06 0.19 0.17 0.04 0.20 0.18 0.04 0.12 0.08 0.04 0.21 0.56 0.21 0.11 0.26 0.14 0.26 0.13 0.20 0.20 0.19 0.22 0.13 0.07 0.21 0.08 0.07 0.15 0.02 0.20 0.18 0.27 0.33 0.38 0.21 0.40 0.42 0.60 0.45 0.20 0.06

0.03 0.02 0.02 0.00 0.02 0.02 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.01

So it means that these values give the reduction ratios of 1.24% and 1.43% in comparison with TS, respectively. For case D, comparative results obtained by PSO in Ref. [9], the ABC and CABC algorithms are shown in Table 10. The times in Table 10 correspond to the average production costs. As seen from

a

Bus number

Type of busa

Pload (MW)

Qload (MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1 2 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 2 2 3 3 3 2 3 3 3

0.00 21.70 2.40 7.60 0.00 0.00 22.80 30.00 0.00 5.80 0.00 11.20 0.00 6.20 8.20 3.50 9.00 3.20 9.50 2.20 17.50 0.00 3.20 8.70 0.00 3.50 0.00 0.00 2.40 10.60

0.00 12.70 1.20 1.60 0.00 0.00 10.90 30.00 0.00 2.00 0.00 7.50 0.00 1.60 2.50 1.80 5.80 0.90 3.40 0.70 11.20 0.00 1.60 6.70 0.00 2.30 0.00 0.00 0.90 1.90

Type of bus: 1 – Slack bus, 2 – Generator bus, 3 – Load bus.

Table 13 Generator data of IEEE 30-bus test system. Bus number

1 2 13 22 23 27

Active power limit

Reactive power limit

Cost coefficient

Min (MW)

Max (MW)

Min (MW)

Max (MW)

a ($/ h)

b ($/ MW h)

c ($/ MW2 h)

0.00 0.00 0.00 0.00 0.00 0.00

80.00 80.00 40.00 50.00 30.00 55.00

20.00 20.00 15.00 15.00 10.00 15.00

150.00 60.00 44.70 62.50 40.00 48.70

0.00 0.00 0.00 0.00 0.00 0.00

200 175 300 100 300 325

200 175 250 625 250 83.4

Table 14 Synchronous machine data of IEEE 30-bus test system. Bus number

Ra

X 0d

H

1 2 13 22 23 27

0.003 0.003 0.003 0.003 0.002 0.003

0.18 0.20 0.18 0.18 0.24 0.19

3.5 2.7 3.01 3.5 3.2 3.0

table, the times for the ABC and CABC algorithms are 66.50 s (165 s by using PC mentioned in Ref. [9]) and 104.36 s (255 s by using PC mentioned in Ref. [9]), respectively. So it means that these algorithms are faster than PSO in any case. Again, the average production costs by the ABC and CABC algorithms are 36368.4 $/h and 36063.88 $/h, respectively. These values give the increase ratio of 0.089% and the reduction ratio of 0.75% in comparison with PSO, respectively.

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Table 15 AC transmission line characteristics of New England 39-bus test system. From bus

To bus

R (p.u.)

X (p.u.)

B/2 (p.u.)

1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 10 10 13 14 15 16 16 16 16 17 17 21 22 23 25 26 26 26 28 12 12 6 10 19 20 22 23 25 2 29 19

2 39 3 25 4 18 5 14 6 8 7 11 8 9 39 11 13 14 15 16 17 19 21 24 18 27 22 23 24 26 27 28 29 29 11 13 31 32 33 34 35 36 37 30 38 20

0.0035 0.0010 0.0013 0.0070 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0.0004 0.0023 0.0010 0.0004 0.0004 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008 0.0003 0.0007 0.0013 0.0008 0.0006 0.0022 0.0032 0.0014 0.0043 0.0057 0.0014 0.0016 0.0016 0.0000 0.0000 0.0007 0.0009 0.0000 0.0005 0.0006 0.0000 0.0008 0.0007

0.0411 0.0250 0.0151 0.0086 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.0046 0.0363 0.0250 0.0043 0.0043 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135 0.0059 0.0082 0.0173 0.0140 0.0096 0.0350 0.0323 0.0147 0.0474 0.0625 0.0151 0.0435 0.0435 0.0250 0.0200 0.0142 0.0180 0.0143 0.0272 0.0232 0.0181 0.0156 0.0138

0.6987 0.7500 0.2572 0.1460 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.1130 0.1389 0.0780 0.3804 1.2000 0.0729 0.0729 0.1723 0.3660 0.1710 0.1342 0.3040 0.2548 0.0680 0.1319 0.3216 0.2565 0.1846 0.3610 0.5130 0.2396 0.7802 1.0290 0.2490 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

For cases C and D, variation of the fitness value against iteration for cases C and D of New England 39-bus test system are shown in Figs. 10 and 11, respectively. As seen from figures, the CABC algorithm reach to the optimum point in lesser iteration according to the ABC algorithm for both cases C and D. For cases C and D, the relative rotor angels of all the generators for the ABC and CABC algorithms are shown in Figs. 12–15, respectively. As seen from figures, all the generators are stable and the their rotor angels do not exceed the value dmax. Furthermore the apparent powers of all the transmission lines do not exceed their upper limits Smax. The data for IEEE 30-bus test system New England 39-bus test system are given by Tables 11–14 in Appendix A and by Tables 15–19 in Appendix B, respectively.

Conclusions and discussions In this paper, a robust and efficient method for solving STSCOPF problems based on the CABC algorithm has been developed to meet the pressing need of the modern power systems. The studies show that the CABC is useful as an optimization technique to solve the challenging STSCOPF problem. The validity and efficiency of the CABC algorithm proposed for solution of STSCOPF are

demonstrated by studies on IEEE 30-bus test system and New England 39-bus test system. The computational times obtained by these algorithms are better than those reported in a paper published several years ago, while all the production costs obtained by the ABC and CABC algorithms do not vary considerably. The results confirm the potential of the CABC algorithm for solving STSCOPF and show that the CABC algorithm is faster and superior than PSO, GA, EP, EPNN and TS for specially this study. Hence, the CABC algorithm can be applied to large-scale power systems in the later researches.

Table 16 Load data of New England 39-bus test system.

a

Bus number

Type of busa

Pload (MW)

Qload (MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1 2 2 2 2 2 2 2 2

0.00 0.00 322.00 500.00 0.00 0.00 233.80 522.00 0.00 0.00 0.00 7.50 0.00 0.00 320.00 329.00 0.00 158.00 0.00 628.00 274.00 0.00 247.50 308.60 224.00 139.00 281.00 206.00 283.50 0.00 9.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1104.00

0.00 0.00 2.40 184.00 0.00 0.00 84.00 176.00 0.00 0.00 0.00 88.00 0.00 0.00 153.00 32.30 0.00 30.00 0.00 103.00 115.00 0.00 84.60 -92.00 47.20 17.00 75.50 27.60 26.90 0.00 4.60 0.00 0.00 0.00 0.00 0.00 0.00 0.00 250.00

Type of bus: 1 – Slack bus, 2 – Generator bus, 3 – Load bus.

Table 17 Generator data of New England 39-bus test system. Bus number

30 31 32 33 34 35 36 37 38 39

Active power limit

Reactive power limit

Min (MW)

Max (MW)

Min (MW)

Max (MW)

100 200 300 300 250 300 250 250 400 600

350 650 800 750 650 750 750 700 900 1200

150 150 150 150 150 150 150 150 150 150

150 150 150 150 150 150 150 150 150 150

K. Ayan et al. / Electrical Power and Energy Systems 64 (2015) 136–147 Table 18 Cost coefficient of New England 39-bus test system. Bus number

30 31 32 33 34 35 36 37 38 39

Cost coefficient Case C

Case D

a ($/ h)

b ($/ MW h)

c ($/ MW2 h)

a ($/ h)

b ($/ MW h)

c ($/ MW2 h)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6.9 3.7 2.8 4.7 2.8 3.7 4.8 3.6 3.7 3.9

0.0193 0.0111 0.0104 0.0088 0.0128 0.0094 0.0099 0.0113 0.0071 0.0064

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.006 0.006

Table 19 Synchronous machine data of New England 39-bus test system. Bus number

Ra

X 0d

H

30 31 32 33 34 35 36 37 38 39

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0310 0.0697 0.0531 0.0436 0.1320 0.0500 0.0490 0.0570 0.0570 0.0060

42.0 30.3 35.8 28.6 26.0 34.8 26.4 24.3 34.5 500.0

Appendix A. IEEE 30-bus test system data

Appendix B. New England 39-bus test system data

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