SLIME MOULD ALGORITHM FOR PRACTICAL OPTIMAL POWER FLOW SOLUTIONS INCORPORATING STOCHASTIC WIND POWER AND STATIC VAR COMPENSATOR DEVICE Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

UDC 621.3

doi: 10.20998/2074-272X.2020.6.07

R. Kouadri, L. Slimani, T. Bouktir

SLIME MOULD ALGORITHM FOR PRACTICAL OPTIMAL POWER FLOW SOLUTIONS INCORPORATING STOCHASTIC WIND POWER AND STATIC VAR COMPENSATOR DEVICE

Purpose. This paper proposes the application procedure of a new metaheuristic technique in a practical electrical power system to solve optimal power flow problems, this technique namely the slime mould algorithm (SMA) which is inspired by the swarming behavior and morphology of slime mould in nature. This study aims to test and verify the effectiveness of the proposed algorithm to get good solutions for optimal power flow problems by incorporating stochastic wind power generation and static VAR compensators devices. In this context, different cases are considered in order to minimize the total generation cost, reduction of active power losses as well as improving voltage profile. Methodology. The objective function of our problem is considered to be the minimum the total costs of conventional power generation and stochastic wind power generation with satisfying the power system constraints. The stochastic wind power function considers the penalty cost due to the underestimation and the reserve cost due to the overestimation of available wind power. In this work, the function of Weibull probability density is used to model and characterize the distributions of wind speed. Practical value. The proposed algorithm was examined on the IEEE-30 bus system and a large Algerian electrical test system with 114 buses. In the cases with the objective is to minimize the conventional power generation, the achieved results in both of the testing power systems showed that the slime mould algorithm performs better than other existing optimization techniques. Additionally, the achieved results with incorporating the wind power and static VAR compensator devices illustrate the effectiveness and performances of the proposed algorithm compared to the ant lion optimizer algorithm in terms of convergence to the global optimal solution. References 38, tables 6, figures 9. Key words: optimal power flow, slime mould algorithm, stochastic wind power generation, static VAR compensators.

Мета. У cmammi пропонуеться процедура застосування нового метаевр^тгческого методу в реальнш електроенергетичнш системi для розв 'язання задач оптимального потоку енергп, а саме алгоритму слизовоТ цвШ, який заснований на поведтщ рою i морфологп слизовоТ ufiim в природь Дане достдження спрямоване на тестування i перевiрку ефективностг запропонованого алгоритму для отримання хороших рШень для проблем оптимального потоку потужностi шляхом включення пристроТв стохастичною втровоТ генерацп i статичних компенсaторiв VAR. У зв'язку з цим, розглядаютьсярiзнi випадки, щоб мiнiмiзувaти загальну варткть генерацп, знизити втрати активноТ поmужносmi i потпшити профть напруги Методологiя В якостi цтьово'Т функцп завдання розглядаеться мтшальна сукупна вартгсть традицшно'Т генерацп електроенергп i стохастичноТ вimровоТ генерацп при задоволенш обмежень енергосистеми Стохастична функщя енергп втру враховуе величини штраф1в через недооцшку i резервш витрати через завищену оцшку доступноТ втровоТ енергИ. У данш роботг функщя щiльносmi ймовiрносmi Вейбулла використовуеться для моделювання i характеристики розподiлiв швидкостг втру. Практична цшшсть. Запропонований алгоритм був перевiрений на сисmемi шин IEEE-30 i великий алжирськоТ тестовш енергосисmемi зi 114 шинами. У випадках, коли мета полягае в тому, щоб звести до мтшуму традицшне вироблення електроенергп, досягнуmi результати в обох тестових енергосистемах показали, що алгоритм слизовоТ цвл функщонуе краще, нЖ iншi iснуючi методи опmимiзaцiТ Крм того, досягнуmi результати з використанням втровоТ енергп i статичного компенсатора VAR тюструють ефективтсть i продуктивтсть запропонованого алгоритму в порiвняннi з алгоритмом опmимiзamорa мурашиних левiв з точки зору збЖностг до глобального оптимальногоршення. Бiбл. 38, табл. 6, рис. 9.

Ключовi слова: оптимальний потж енерги, алгоритм слизово1 цвШ, стохастична генеращя енерт виру, статичш VAR компенсатори.

Introduction. In the last decade, energy developing thanks to the technological advances made in consumption has been increased significantly especially the field of wind generators to reduce the cost of system in developing countries. Renewable energy can be known installations. In addition, the application of flexible AC as green energy or clean energy is one of the best transmission systems (FACTS) controllers such as static solutions to the increasing demand problem, and it is VAR compensators (SVC) devices that considered one of inexhaustible energy that comes from natural resources or the most controllers used in the case of the high demand processes that are constantly replenished [1], even if their for energy to maintain the magnitude of bus voltage at the availability depends on weather and weather conditions, desired level, improve voltage security and minimize the and whose exploitation causes the least possible total power losses.

ecological damage, does not cause toxic waste and does With the growing penetration of RESs in the power

not cause damage to the environment. They are cleaner, system, the study of optimal power flow (OPF) becomes more environmentally friendly than fossil fuels and fissile necessary to solve power system problems or improve the energies, environmentally friendly, available in large performance of this system. The OPF for the system that quantities around the world. includes RESs such as wind power generators is the

Nowadays, the integration of renewable energy subject of ongoing research models nowadays. It is sources - RESs (i.e., solar, wind, hydropower, etc.) into necessary to confront the stochastic nature of this source the electrical grid is experiencing a rapid increase. Among for analysis of the planning and operation of modern the various RESs, wind energy considered is one of the power systems, in order to obtain much more precise most desirable sources in recent years that keeps -

© R. Kouadri, L. Slimani, T. Bouktir

results [2]. In general, the problem with wind power is the stochastic nature of wind speed. Therefore the model which considers the probability of the available wind power can represent the cost of overestimating and underestimating this power at a certain period.

Recently, OPF with stochastic wind power has extensively been studied by more researchers. In [3] authors proposed a Gbest-guided artificial bee colony algorithm (GABC) to solve the OPF problem in the IEEE 30 bus system incorporating stochastic wind power. In attempting the same problem in [4] author proposed a modified moth swarm algorithm (MMSA) to solve the OPF problem incorporating stochastic wind power. In this work, three different objective functions are considered, which are the minimize the total operating cost, reduce the transmission power loss, and improve the voltage profile enhancement. In another study [5] authors applied the success history-based adaptation technique of differential evolution algorithm to solve the OPF problem comprises of stochastic wind-solar power with conventional thermal generators under various cases. The OPF incorporation with wind power and static synchronous compensator STATCOM was studied in [6] by using a modified bacteria foraging algorithm (MBFA). The results obtained proved that MBFA efficiency and better than the ACO algorithm for solving OPF problems in power systems. Bird Swarm Algorithm (BSA) for solving an OPF problem with incorporating stochastic wind and solar PV power in the power system is studied in [7]. The proposed approach applied in the modified IEEE 30-bus system with objective function is to minimize the total energy generation cost, which is the cost of thermal-wind-solar. In [8] authors applied a modified hybrid PSOGSA with a chaotic maps approach to improve OPF results by incorporating stochastic wind power and two controllers in the FACTS family such as TCSCs and TCPSs. The proposed method is applied in the power systems to minimize the thermal generators' fuel cost and the wind power generating cost.

Several metaheuristic optimization algorithms were developed and applied for the OPF solution. Some of them are: salp swarm optimizer [9], moth swarm algorithm [10], differential evolution [11], glowworm swarm optimization [12], differential search algorithm [12], moth-flame optimizer [14], stud krill herd algorithm [15], artificial bee colony algorithm [16], symbiotic organisms search algorithm [17], improved colliding bodies optimization algorithm [18], firefly algorithm [19], black-hole-based optimization approach [20], the league championship algorithm [21, 22], multi-verse optimizer [23], harmony search algorithm [24], earthworm optimization algorithm [25]. Among several numbers of the available metaheuristic algorithm, a new flexible and efficient stochastic optimization algorithm has been proposed to solve our problem and satisfy our imposed conditions, this technique namely a slime mould algorithm (SMA). SMA is based upon the oscillation mode in nature and simulates the swarming behavior and morphology of slime mould in foraging.

In this paper, a new flexible and efficient stochastic optimization algorithm called slime mould algorithm (SMA) has been proposed with the aim is solving the

OPF problem in power systems incorporating stochastic wind power and SVC devices.

Modeling of SVC. The static VAR compensator (SVC) device is an important member of the FACTS controllers' family. The importance of SVC is to maintain the bus voltage magnitude at the desired level by providing or absorbing reactive energy. In the power system, SVC is modeled by shunt variable admittance. SVC's admittance only has its imaginary part since the SVC device's power loss is assumed to be negligible and is given as follows:

ySVC = jbSVC . (1)

The bSVC susceptance can be capacitive or inductive to provide or absorb reactive power, respectively. In this study, SVC is installed in the power system as a PV bus with the objective is to regulate the voltage magnitude Vk by injecting reactive power to a bus where it is connected. The current ISVC and reactive power QSVC absorbed or injected by the SVC device is calculated as follow:

ISVC = jbSVCVk ; (2)

QSVC - ~Vk bSVC •

(3)

Optimal power flow problem formulation. The

optimal power flow problem solution aims to give the optimum value of the objective function by adjusting the settings of control variables. Generally, the mathematical expression of the optimization problem with satisfying various equality and inequality constraints may be represented as follows:

min F (x, u); (4)

Subjected to g (x, u) = 0; (5)

h(x, u)< 0; (6)

where F(x, u) denotes the objective function that to be optimized, x and u represents the vectors of the state variables (dependent variables) and control variables (independent variables), respectively.

Control variables. In the OPF the control variables should be adjusted to satisfy the load flow equations. The set of control variables can be represented by vector u as follows:

PG

■Pgng ' W - Pwsnw

VG ...VG

QC ■ ■■ QC NG NT

where PG is the thermal generator active power; PWS is the wind active power; VG is the generator voltage; QC is the reactive power injected by the shunts compensator; T is the tap setting of transformers; SVC is the static VAR compensator; NG is the number of generators; NW is the number of wind farms; NC is the number of shunts compensators units; NT is the number of regulating transformers; NSVC is the number of SVC devices.

State variables. The set of variables which describe the electrical power state can be represented by vector x as follows:

x -[PGslack, QGi ■••QGNG , QWSi ■■■QwSnw ,VLi ■■■VLnl , Sli ■•• SlM J'(8)

where PGsiack is the active power generation at the slack bus; QG is the reactive power outputs of the generators; QWS is the reactive power outputs of the wind farms; VL is the voltage magnitude at load bus; Si is the apparent power flow; NG is the total number of generators buses;

T

,SVCi.

.SVC

NSVC

(7)

2

U =

Nl is the total number of load buses or PQ buses; N is the total number of transmission lines.

Equality constraints. The equality constraints represent in the power system the load flow equations of the balanced powers and reflect the physics of the power system. The equality constraints can be represented as follows:

N i \

PGi + PWS, — Pdt = V Z Vj p j cos Sj + Zj sin Sj) (9) j=1

N i \

Qg,. + Qws, - Qd, = V ZVj (p j sin Sj + z,j cos Sj ) (10) j=1

Inequality constraints. The inequality constraints reflect the limiting of the power system operation. These inequality constraints can be represented as follows:

pG1111 < PG < p?ax;

G, Gi

pmin < p < pmax PWSt < PWSt < PWSj

emin < Qg, < emax;

QWl < Qws, < Q

(11)

WS,

vmin < VG < V™ ; Gi G Gi

71min ^ rp ^Tmax.

lNT < lNTi < lNT ;

Qmvc, < qsvci < ^vc

\sr |< smax.

Objective function. In this study, the objective function is to minimize the total generation cost (TGC) subject to operating constraints. The objective function is formulated as:

N NW

Ftot =ZFi {Pt)+£C

wr i.Pwr )

i —1

i—1

NW

NW

(12)

Cp.wr (Pwr.av Pwr )+ Cr.wr (Pwr Pwr.aw ). i=1 i=1

In the expression of the objective function formulated in the (12), the first term denotes thermal power generation cost, second, third and last term of the objective function shows the costs of wind power, respectively. Details of all terms are explained below.

Fuel cost of the conventional generator. The cost function of the thermal generators as follows: (N \

f, P)— Z a + biPGi + C,PC

Gl

\ i—1

(13)

where PGi is the active power generated from the available thermal generators; a,, bt and c, are the cost coefficients of i-th generator.

The direct cost function for wind power. The grid operators pay the cost of purchasing wind power from a wind power producer based on the power purchase agreement. This cost is termed as the direct cost and is defined as follows [5]:

Cwr(Pwr) = dr'Pwr, (14)

where dr is the direct cost coefficient for the j-th wind generator and Pwr is the scheduled power output.

Cost function due to the underestimation. The underestimation situation is due when the actual wind power is higher than the estimated value. So, the utility operator needs to pay a penalty cost for not using the surplus amount of available wind power [4, 5]. The penalty cost functions due to the underestimation of available wind power represented by (15), it can be given as [26]:

Cp.wr i.Pwr.av Pwr ) kp i.Pw.av Pwr)

Pr.0

— kp J(p - Pwr)• fwPw),

(15)

where Cp.wr is the cost associated with wind power shortage (underestimation); Pp.wr is the actual available power output; kp is the penalty cost coefficient due to underestimation and fw(Pw) represents the probability density function (PDF).

Cost function due to the overestimation. On contrary to the underestimation situation, the overestimation situation is due when the actual wind power is less than the estimated value. So, a spinning reserve is needed for grid operators [5]. The penalty cost function due to the overestimation of available wind power represented by (16) as follows [27]:

Cr.wr (Pwr — Pwr.av ) = kr (Pwr — Pw.av ) =

wr

— kr \(Pwr - W )• fw (Pw ),

(16)

where Cr.wr the cost associated with wind power surplus (overestimation) and kr is the reserve cost coefficient due to overestimation.

Wind power model. The distribution function was used in this work to model and characterize the distributions of wind speed known as Weibull probability density function (PDF) [28], and can be represented as:

fv (V) —

k ( v

c V c

k-1 I _.

(17)

here v is the wind speed; k and c respectively the shape factor and scale factor (m/s).

The probability density function for the continuous portion of wind energy conversion systems (WECS) power output random variable becomes as follows:

fw (Pw )

k •l • vcut-in ( P + P-1)v,

vcut-in

k-1

exp

(1 + P^)Vcut-» c

(18)

where l = (vmted - v^-m) / v^-m is the ration of linear range wind speed to cut-in wind speed; vcut-in is the wind speed at which wind turbine starts to generate power; vcut-off is the wind speed at which the wind turbine is disconnected; vrated is the wind speed at which the mechanical power output will be the rated power; P = Pw / Pwr is the ratio of wind power output to rated wind power.

p

wr

k

V

c

e

x

c

c

The probability for the discrete portion of the WECS power output is expressed by (19) and (20), respectively as follows [5, 29]:

fw Sw )={w = 0}= 1 - exp

k i

+ exp

vcut-off

k n

(19)

fw (Pw) {Pw Pwr}

exp

I

vrated I

C 1

exp

( \k i

Vcut-off

(20)

X(+1) JXBP) + vbW • TAP)- xBd)) r < p;

[VC • TP), r > p,

(21)

where X denotes the slime mould location; Xb is the individual emplacement with the highest odor concentration currently found; XA and XB are indicated two randomly selected individuals from the swarm; vb is a parameter distributed in the range of [-a, a]; vc decreases linearly from 1 to 0; t shows the current iteration; W represents the slime mould weight and given below by (24); p is the parameter given as follows:

(22)

p = tanh|S(i)- DF\,

where S(i) shows the fitness of X; i e 1, 2, DF is the optimum fitness obtained in all iterations. The parameter of a is given as follows:

( f . \ \

a = arctanh

t

max t

+1

n;

(23)

The expression of W define the location of slime mould and is given as follows:

W (Smelllndex(i) =

. ,bF - S (i) ,,

1 + r • logl-— +1 I, condition;

bF - wF

1 -r-log| +1

bF - wF

(24)

others,

Slime mould algorithm. A slime mould algorithm (SMA) is a new stochastic optimizer technique nature-inspired proposed in 2020 in [30]. This technique based on the oscillation mode of slime mould in nature and simulates the swarming behavior and morphology of slime mould in foraging. The SMA algorithm features a special mathematical model that uses the adaptive weight to simulates the combination of positive and negative feedback from the bio-oscillator-based propagation wave that was inspired by slime mould to form the optimal pathway to connect food. Some of the most interesting characters in the slime mould are the unique pattern based on the various food sources to create a venous network connecting them at the same time. This scheme gives the high capability of escaping from local optima solutions. The algorithm is aroused by slime mold diffusion and foraging behavior. In SMA, slime mould can approach food, depending on the smell in the air. The slime mold morphology varies, with three different forms of contraction. The following section will explain in detail the mathematical model for simulating the behavior of slime mould during the foraging [30].

Approach food. The following formulas for imitating the contraction mode is proposed to model the behavior of slime mould to approaching food according to the odor in the air as follow:

X * =

where condition denotes that S(i) is ranked first half of the population; r represents the random value distributed in the range of [0, 1]; bF and wF are represented the optimal and worst fitness value obtained in the current iterative process, respectively; Smelllndex represents the sequence of fitness values sorted as:

Smelllndex = Sort (S ). (25)

Wrap food. This portion mathematically simulates the contraction mode in the slime mould venous tissue structure while searching. In this context, the higher the food concentration reached by the vein, the stronger the bio-oscillator-generated wave, the quicker the cytoplasm flows and the thicker the vein. The following mathematical formula represents updating the emplacement of slime mould:

rand • < z;

xb( + Vb •(w • Xjt) - IBfi) r < p; (26)

C • Xjt), r > p,

where lb and ub denote the lower and upper limits of the search range, respectively; rand denotes the random value distributed in the range of in [0, 1].

Grabble food. Slime mould is primarily dependent on the propagation wave to change the cytoplasmic flow in the veins, so they appear to be in a better concentration of food. Slime mould can approach food faster when the concentration and quality of food are high, while if the food concentration is lower, approach it more slowly, thus increasing the efficiency of slime mould in selecting the optimal source of food.

In the SMA process, the value of the parameter vb oscillates randomly in the interval between [-a, a] and progressively approaches zero as the iterations increase.

The value of vc oscillates randomly in the interval between [-1, 1] and finally tends to be zero.

The pseudo-code of the SMA to solve the OPF problem is shown in Algorithm 1.

_Algorithm 1 Pseudo-code SMA algorithm_

Read the system data (bus data, line data, and generator data);

Initialize the parameters of search agents, size of the

population, the maximum number of iterations, the number and

position of the control variables;

Initialize the position of the slime mould Xi using (21);

While iteration < Max iteration,

Calculate the fitness of all slime mould using (26);

Update the best fitness, XB

Calculate the W by using (24);

For each search space

Update the parameters of SMA which are: p, vb and vc; Update the best positions of the slime mould; Calculate the best value of the objective function (12); End For iter = iter + 1; End while

Return best Fitness found so far, XB._

V

cut-in

+

C

C

C

Simulations and results. To demonstrate the performance and efficiency of the SMA algorithm to solve the OPF problem by incorporating stochastic wind power and FACTS devices such as SVC, the present work aims to apply the SMA on IEEE 30-bus and Algerian 114-bus systems with different test cases study. In this context, the minimization of total fuel cost and wind power cost is considered as objective functions. The description of all these test cases can be found in the following section. All the simulations are carried out by using MATLAB 2009b and computed with specification Intel® Core™ i5 [email protected] GHz with 8 GB of RAM. For establishing the robustness of the SMA algorithm, 30 independent trial runs are performed for all the test cases. In this work, the population size is 40 and the number of iterations maximal is 500.

IEEE 30-bus test system. The first test is dedicated to the standard IEEE 30-bus power system in order to verify the performance and efficiency of the SMA for the small scale power system. This system includes 6 generators unit, 41 transmission lines, 4 transformers located at lines 6-9, 4-12, 9-12, and 27-28. Nine reactive compensators are located at buses 10, 12, 15, 17, 20, 21, 23, 24, and 29. The total load is (2.834 +j-0.735) p.u.

The upper limit and lower limit variables are shown in Table 1. In this section, two different parts are considered, the first part is solving the OPF problem under normal conditions and the second part is solving the OPF problem under the contingency state.

OPF solution under normal condition. In this part, the SMA is applied to solve the OPF problem under the normal condition with active power loading is 283.4 MW. Three different cases are examined via SMA as follows.

Case 1: Minimization of total fuel cost. The objective function used in the first case under normal condition is to minimize the total fuel cost according to the optimal power distribution of the production units and is described by (13). Table 3 tabulates the results obtained by the SMA algorithm for Case 1. It can be seen that the optimal settings of control variables are all within their acceptable limits. Furthermore, we can also see that the fuel cost obtained by SMA is 798.9709 $/h, this value is lower and better compared to those obtained by MSA, GSO, MFO, BHBO, ALO, MSCA which are mentioned in Table 1.

Table 1

Comparison of solutions achieved using SMA and different methods for Case 1

Method Fuel cost ($/h)

Slime mould algorithm 798.9709

Moth swarm algorithm [10] 800.5099

Glowworm Swarm Optimization [12] 799.06

Moth-Flame Optimizer [14] 799.072

Black-hole-based optimization [20] 799.921

Ant lion optimizer [31] 799.0133

Modified Sine-Cosine algorithm [32] 799.31

The convergence characteristics of the proposed method and the ALO algorithm are shown in Fig. 1. It can be seen that the SMA algorithm outperforms the ALO algorithm in terms of convergence rate towards the global optimum solution. So, the results achieved showed the SMA superior and robust compared to the ALO algorithm in order to get the best solution to solve the OPF problem.

820

815

CO

o o c

o +-'

<D d <u

CT "rö o

810

805

800

100

200 300 Iteration

400

500

Fig. 1. Convergence characteristics of the SMA & ALO: Case 1

Case 2: Minimization of total fuel cost and wind power cost. In this test case, SMA is applied to solve the OPF problem by incorporating stochastic wind power. Thus, the objective function is minimizing the total generation cost that includes fuel cost and wind power cost. The cumulative cost, described by (13). In this case, the standard IEEE 30-bus system is considered by including two wind farms located at bus numbers 10 and 24. Moreover, the two wind farms (WFs) consist of 30 units of wind turbine generation (WTG) with a nominal power rating of each WTG is 2 MW. Thus, each WF having a total capacity of 30 MW.

Table 2 details the specification of wind turbine characteristics used in all optimization cases in this study concern with incorporating wind power for the IEEE 30-bus system [33].

Table 2

The characteristics of this wind turbine

Parameters Value

k 2

c 3

dr 1.3

P * wr 2000 kW

vcut-in 4 m/s

vrated 12 m/s

vcut off 25 m/s

Kp , (penalty factor) 1 $/MWh

K. (rserve factor) 4 $/MWh

Table 3 presents for case 2 the results obtained by SMA to minimize the total generation costs, which are the total fuel and wind costs. The sizing of the two wind farms can be referred to in the same table. For this case, SMA exhibit bus 10 and 24 as the optimal locations of the wind farm. At active power loading of 283.4 MW, It can be seen that the TGC produced by SMA is reduced from 798.9709 $/h to 725.7113 $/h. Moreover, the active power losses have also increased from 8.5752 MW to 6.2413 MW which is lowered by 27.21 %. Thus, SMA provides the best values to minimize the TGC and reduce the active power losses in the IEEE 30-bus test system by incorporating wind power compared to the case without the implementation of wind farms. In general, the implementation of wind farm installation to the system has significantly reduced the values of the total generation cost and the active power losses.

Table 3

Best control variable settings obtained via SMA for IEEE 30-bus system including WPG and SVC devices

Control Variables Limits Active power loading 283.4 MW Active power loading 410.93 MW

Min Max Case i Case 2 Case 3 Case 4 Case 5 Case 6

pgI(MW) 50 2oo i77.5784 i39.3865 i 39.6782 i99.9977 i95.22o7 i95.2576

PG2(MW) 20 8o 48.677o 39.62i6 39.48o3 78.82i8 57.6992 57.8394

pg5(MW) i5 5o 2i.2668 i8.6332 i 8. 5 i 44 42.42ii 32.9495 32.7988

PG2(MW) io 35 2i.23i6 io.oooo io.o292 34.99i5 34.9999 34.9896

PgII(MW) io 3o i2.o89o io.oooo io.oo25 29.9997 2 i. 9266 23. i78i

PgI3(MW) i2 4o i2.oooo i2.oooo i2.oo42 38.2946 2o.3897 i9.i394

PwsI(MW) o 4o - 3o.oooo 3o.oooo - 3o.oooo 3o.oooo

PWS2(MW) o 4o - 3o.oooo 3o.oooo - 3o.oooo 3o.oooo

^Gl(P-U) o.95 i.iooo i.iooo i.iooo i.iooo i.iooo i.iooo

VG2(p.u) o.9 i.o879 i.o894 i.o873 i.o843 i.o8o4 i.o8i8

Vg5(P-U) o.9 i.o6i8 i.o644 i.o597 i.o286 i.o264 i.o263

Vg8(P.U) o.9 i.o7oi i.o76o i.o7i9 i.o6i6 i.o669 i.o694

VGll(p.u) o.9 i.iooo i.o539 i.o233 i.iooo i.iooo i.o964

VG13(P.U) o.9 i.iooo i.oi83 i.oi5o i.iooo i.o5i6 i.o37i

7,11(P.U) o.9 i.o259 i.o9o3 i.o989 i.oi89 i.o896 i.iooo

Tl2(p.u) o.9 o.9oio i.o286 i.o887 i.o2ii i.o99i i.o993

T^CP.U) o.9 o.98o3 i.o98o i.o786 i.o5ii i.o997 i.o974

?36(P.U) o.9 o.9568 i.o594 i.o429 o.96o9 i.o272 i.o455

ßcio(Mvar) o 5 4.38o6 o.oi39 i.7i5o 4.88i3 4.i783 3.8886

gci2(Mvar) o 5 4.779o 2.858i o i.9i64 4.89oi o.856o

gci5(Mvar) o 5 4.8272 o 4.7o98 3.iio9 3.i556 i.6o88

gcn(Mvar) o 5 4.9942 2.272i i.463i 4.9727 4.96i7 5.oooo

gc2o(Mvar) o 5 2.565i 2.7844 i.oi3i i.39i5 i.i554 4.i684

gc2i(Mvar) o 5 2.8396 5.oooo 4.8532 4.9937 o.oo66 4.9944

gc23(Mvar) o 5 3.46o9 4.8785 o.5928 2.98o8 2.7736 4.7325

gc24(Mvar) o 5 4.9957 o.2i67 i.8i72 4.63o7 i .3769 o.o423

gc29(Mvar) o 5 i.i562 o.9389 o.49oo i. i 98i i.29oo 4.8493

gwsi(Mvar) -i5 4o - -3.93i9 39.48o3 - 4.7442 57.8394

gws2(Mvar) -i5 4o - 3.3754 o.87i9 - io.324o 32.7988

gsvc3o(Mvar) -25 25 - - 5.6479 - - 6.67i6

Total generation cost ($/h) 798.97o9 725.7ii3 725.8855 i 339.4776 ii98.i826 ii98.2o92

Power losses (MW) 8.5752 6.24i3 6.3o87 i3.5964 i2.2555 i2.2729

Voltage deviation (p.u.) i.4494 o.6285 o.5i95 o.74i3 o.6o66 o.5465

Reserved real power - 53.5o74 53.5o74 - 53.5o74 53.5o74

The convergence curves of the SMA and ALO for case 2 are shown in Fig. 2, which allows us to note, in the first place, that the SMA converges towards the global optimum value at iteration 120 compared to the ALO, that the convergence towards the optimal solution is reached at iteration 270.

Case 3: Minimization of fuel cost and wind power cost by considering the SVC device. In this case study, SMA is applied for solving the OPF problem by incorporating wind power and SVC devices. The optimal location of the SVC device for the IEEE 30-bus system found by SMA is bus N°30. The objective function used is to minimize the TGC as described by (13). From this case, It can be seen that the voltage deviation is reduced from 1.4494 p.u (case 1) and 0.6285 (case 2) to 0.5428 p.u. The voltage profile obtained by the SMA algorithm for cases 2 and 3 is shown in Fig. 3. It is seen that the effect of the SVC device to improve the profile voltage, especially in

the busses far from generators units such as bus N°25 until bus N°30.

Iteration

Fig. 2. Convergence characteristics of the SMA & ALO: Case 2

OPF solution under the contingency state. In this part, the SMA is applied to solve the OPF problem under

the contingency state, which is increased loading at 45 %. Thus, the active power loading is 410.93 MW. Three different cases are considered for this part.

10

15

20

25

30

1500

1450

2?

to 1400

o

o

n

oi 1350

tar

re

n e 1300

ro

to 1- 1250

1200

SMA ALO

Fig. 3. Profile Voltage magnitudes for case 2 and case 3

Case 4: Minimization of total fuel cost. In this case, the objective function is to optimize the total fuel cost in the IEEE 30-bus system with increased loading at 45 % and is described by (16) addition to the penalty of line power. From the results given by the SMA algorithm for the case N°5, It can be seen that most generators work near their maximum limits, due to the increased load compared to the results given in case 1 without increased load. Moreover, we can also see that the fuel cost, active power losses, and voltage deviation are increased as presented in Table 3. The convergence characteristics of the SMA and ALO for case 4 are shown in Fig. 4.

50 100 150 200 250 300 350 400 450 500 Iteration

Fig. 5. Convergence characteristics of the SMA & ALO: Case 5

Case 6: Minimization of total fuel cost and wind power cost by considering the SVC device. In this case, we have study the influence of SVC devices on a power system to improve the voltage profile. The voltage profile for case 5 and case 6 are shown in Fig. 6. Unlike case 5 where profile voltage decreases after overloading, adding the SVC to the power system, in this case, improves the voltage as seen in Fig 6. Through the given results, we note that the effect of SVC is significant in the case of increased load.

1.1

1.05

a> if=

! 1

<D

jS 0.95 o

> 0.9

0.85

0

5

25

30

50 100 150 200 250 300 350 400 450 500 Iteration

Fig. 4. Convergence characteristics of the SMA & ALO: Case 4

Case 5: Minimization of total fuel cost and wind power cost. The minimization of total fuel cost and wind power cost, in this case, is formulated as the objective function, which is described by (13). At higher active power loading of 410.90 MW, SMA provides 1198.1826 $/h for the TGC, this value better than a value obtained in a case without incorporating wind power. On the other hand, the implementation of wind farms has reduced the active power losses and the deviation voltage in the system.

The convergence characteristics of the SMA and ALO for case 5 are shown in Fig. 5. From this figure, it demonstrates that the SMA algorithm can converge to the global optimum at iteration 170, while ALO towards the optimal solution is reached at iteration 230.

10 15 20

Bus number

Fig. 6. Profile voltage magnitudes for case 5 and case 6

Algerian electrical network system. In order to verify the performance and efficiency of the ALO to solve nonlinear problems in larger-scale dimensions, OPF is performed on the Algerian electrical network system. This system includes 15 generators, 175 transmission lines, and 16 located from line 160 to line 175. The technical and economic parameters of generator units of the Algerian electrical network system are presented in [34].

Case 7: Minimization of total fuel cost. In this case, SMA is tested to identify the optimal fuel cost on the large-scale Algerian electrical network system with 114 buses. Table 4 presents the optimal settings of control variables reached by SMA with three different cases taking into consideration the vector of control variables contains the active powers generated and the generator voltages. The best value of fuel cost obtained by SMA for the vector of control variables contains the active powers generated is 18914.105 $/h and better than other methods as well as previously reported methods in Table 5.

The convergence characteristics of the proposed algorithm and ALO algorithm for case 7 are shown in Fig. 7. It can be seen that the SMA algorithm outperforms the ALO algorithm in terms of convergence rate towards the global optimum solution.

0

5

Table 4

Best control variable settings obtained via SMA for ALG 114-bus system including WPG and SVC devices

Control Variables Case 7 Case 8 Case 9 Control Variables Case 7 Case 8 Case 3

PG4(MW) 451.3078 444.8246 446.5335 VG4(p.u) 1.0997 1.1000 1.0999

PG5(MW) 451.1405 446.1754 443.8411 VG5(p.u) 1.1000 1.1000 1.1000

PgH(MW) 99.9998 99.9992 99.9993 VG11(p.u) 1.0954 1.0990 1.0993

Pg15(MW) 193.3981 190.5629 188.6959 VG15(p.u) 1.1000 1.1000 1.0993

Pg17(MW) 446.9078 439.3309 441.6877 VG17(p.u) 1.1000 1.1000 1.1000

Pg19(MW) 194.8571 190.8661 189.4341 VG19(p.u) 1.0599 1.0523 1.0590

Pg22(MW) 191.8038 190.0866 186.7558 VG22(p.u) 1.0620 1.0589 1.0683

Pg52(MW) 188.5324 186.9000 185.9111 VG52(p.u) 1.0661 1.0622 1.0668

Pg80(MW) 190.4592 184.5212 186.0970 ^G8ö(p.u) 1.1000 1.1000 1.0998

Pg83(MW) 187.8661 181.9296 183.6420 VG83(p.u) 1.1000 1.1000 1.1000

Pg98(MW) 188.6026 183.2775 184.3464 VG98(p.u) 1.1000 1.1000 1.1000

Pg100(MW) 600.0000 599.9998 600.0000 VG100(p.u) 1.1000 1.1000 1.1000

Pg101(MW) 200.0000 200.0000 200.0000 VG101(p.u) 1.1000 1.1000 1.1000

Pg109(MW) 100.0000 99.9995 99.9985 VG109(p.u) 1.1000 1.1000 1.0998

Pg111(MW) 99.9976 100.0000 100.0000 VG111(p.u) 1.0701 1.0650 1.0792

Pra(MW) - 15.0000 15.0000 ßsFC68(Mvar) - - 22.000

Prs2(MW) - 30.0000 29.9999 Qsvc89(Mvar) - - 32.800

Case 1 Case 2 Case 3

Fuel cost ($/h) 18914.105 18624.9978 18610.7234

Power losses (MW) 57.8726 56.4733 54.9422

Voltage deviation (p.u.) 4.9714 4.8197 4.5968

Reserved real power - 41.0227 41.0227

Table 5

Comparison of solutions achieved using SMA and different methods for Case 7

Method Fuel cost ($/h)

Slime mould algorithm 18914.105

Differential evolution [34] 19203.340

Grey wolf optimizer [35] 19171.958

Hybrid GA-DE-PS [36] 19199.444

M-objective ant lion algorithm [37] 19355.859

-C

tn o o c o

0 c 0 m

o

2.3 2.25 2.2 2.15 2.1 2.05 2

1.95 1.9

x 10

SMA ALO

located at busses 99 (Setif) and 107 (Djelfa). Moreover, the two wind farms (WF) consist of 40 units of wind turbine generation (WTG) are connected to the system at busses 10 and 24 with a nominal power rating of each WTG is 1.5 MW. Weibull settings for the sites that have been chosen are taken from [38]. The choice of the turbine has been set for General Electric GE 1,5-77 machines. The characteristics of this wind turbine are shown in Table 6.

Table 6

The characteristics of this wind turbine

Parameters Wind turbine1 Wind turbine

k 1.425 2.008

c 4.083 5.178

dr 1.75 2

P wr 15 MW 30 MW

vcut-in 3.5 m/s 3.5 m/s

vrated 12 m/s 12 m/s

vcut-off 25 m/s 25 m/s

Kpj (penalty factor) 1.5 $/MWh 1.5 $/MWh

Krj (rserve factor) 3 $/MWh 3 $/MWh

50 100 150 200 250 300 350 400 450 500 Iteration

Fig. 7. Convergence characteristics of the SMA & ALO: Case 7

Case 8: Minimization of total fuel cost and wind power cost. In this case, SMA is applied to solve the OPF problem on the large-scale power system by incorporating stochastic wind power. The Algerian power system ALG 114-bus is considered by including two wind generators

Table 4 summarizes the best results reached by SMA to minimize total generation cost, reduce active power losses and improve the voltage profile by incorporating two wind farms. Based on the results achieved by the SMA in case 7 compared to case 8, the incorporation of wind farms into the system in the ALG 114 system gave more significant profit in TGC and reducing active power losses. The convergence characteristics of the SMA for case 8 are shown in Fig. 8. The convergence of the SMA is reached in the first 170 iterations, while the convergence of the ALO towards the optimal solution is reached at iteration 230.

x 10

SMA ALO

50 100 150 200 250 300 350 400 450 500 Iteration

Fig. 8. Convergence characteristics of the SMA & ALO: Case 8

Case 9: Minimization of total fuel cost and wind power cost by considering the SVC device. In order to illustrate the effectiveness of the SMA in presence of SVC devices on the power system, the ALG 114-bus is considered by including two SVC devices at busses N°68 (Sedjerara) and bus N°89 (Souk Ahras). These locations of SVC devices are considered the optimal placement in the Algerian 114-bus system found by the SMA algorithm. After the results of the simulation, the installation of the SVC improved considerably the total generation cost, the active power loss. Figure 9 represents that the effect of SVC devices is significant in the Algerian 114-bus system to maintain the voltages within the acceptable limits.

1.1

S 1.05

<D ra

o 095

>

0.9

0.85 L

0 20 40 60 80 100 Bus number

Fig. 9. Profile voltage magnitudes for case 8 and case 9

Conclusion. This paper proposed a recent metaheuristic technique called a slime mould algorithm to solve the optimal power flow problem incorporating stochastic wind power and static VAR compensator devices. In this study, nine cases have been considered and examined via the proposed algorithm on the IEEE 30-bus system and practical Algerian power system ALG 114-bus. The objective function solved is a minimization of the total generation cost that includes fuel cost and wind power cost. Also, the nature of the wind output function used is based on the Weibull probability distribution model. For the case without considering wind power and static VAR compensator devices, it is worth mentioning that the proposed algorithm is capable of achieving and getting the best global optimal solution for

both of the testing systems compared to the other methods in the literature mentioned in this paper. With considering wind power and SVC devices, the numerical results obtained show a better performance of the proposed algorithm to solve the optimal power flow problem compared to the ant lion optimizer algorithm. Additionally, incorporating the wind power and static VAR compensator device has a high influence on the power system through minimize the total generation cost, reduce the active power loss as well as improve the voltage profile. Thus, the results obtained prove the merits and efficiency of the proposed algorithm to solve the stochastic optimal power flow problem.

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Received 11.08.2020 Accepted 02.11.2020 Published 24.12.2020

Ramzi Kouadri1, Ph.D Student, Linda Slimani1, Professor, Tarek Bouktir1, Professor, 1 Department of Electrical Engineering, University of Ferhat Abbas Setif 1, 19000, Setif, Algeria. e-mail: [email protected], [email protected], [email protected]