Operational Research https://doi.org/10.1007/s12351-018-0420-3 ORIGINAL PAPER
Optimal reactive power dispatch using water wave optimization algorithm Yongquan Zhou1,2 · Jinzhong Zhang1 · Xiao Yang1 · Ying Ling1 Received: 16 July 2017 / Revised: 11 April 2018 / Accepted: 16 August 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper presents water wave optimization (WWO) algorithm to solve the optimal reactive power dispatch (ORPD) problem with the continuous and discrete control variables in power system. The ORPD problem is defined as a complex, discrete, constrained nonlinear combinatorial optimization problem. The WWO algorithm is utilized to find the optimized values of control variables such as generator voltages, tap positions of tap changing transformers and the amount of reactive compensation devices to achieve minimized value of active power losses. The WWO algorithm not only effectively avoids the shortcomings of local search and poor calculation accuracy, but also accelerates the convergence rate to find the global optimal solution. The WWO algorithm is implemented on standard IEEE 30-bus power system that is to verify the effectiveness and feasibility of the WWO algorithm to tackle with the ORPD problem. Compared with other algorithms, the WWO algorithm can find the set of the optimal solutions of control variables. The simulation experiment indicates that the WWO algorithm has better overall performance to reduce the real power losses. Keywords Water wave optimization algorithm · Optimal reactive power dispatch · Control variables · Active power losses · Simulation experiment
1 Introduction With the development of the economy, the power load increases rapidly and the ORPD problem is an important part for power system to operate safely, which has aroused wide public attention. The power system achieves the optimal dispatch and * Yongquan Zhou
[email protected] 1
College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China
2
Key Laboratory of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China
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control of reactive power, which can improve the quality of voltage and reduce the power transmission losses so as to reduce operating costs and enhance the level of stable operation. The ORPD problem (Alsac and Stott 1974; Lee et al. 1985; Kenarangui and Seifi 1994; Lai 2005; Varadarajan and Swarupa 2008; Duman et al. 2012) is a nonlinear combination optimization problem with discrete, complex, multi-constrained features. The traditional methods to solve the ORPD problem are interior point method (Momoh et al. 1994), linear programming (Deeb and Shahidehpur 1990), nonlinear programming method (Wu et al. 1994), Gradient method (Lee et al. 1985), quadratic programming method (Grudinin 1998), Newton method (Bjelogrlic et al. 1990), which generally causes problems, such as large error, curse of dimensionality, difficult to deal with discrete variable, so that it is difficult to obtain the ideal result. In recent years, the traditional methods have some shortcomings to solve ORPD problem, the heuristic optimization algorithms have been come forward by some scholars to solve the complicated problem. Such as bat algorithm (BA) (Yang and He 2013), flower pollination algorithm (FPA) (Yang 2012), particle swarm optimization (PSO) (Kennedy and Eberhart 1995), sine cosine algorithm (SCA) (Mirjalili 2016), crow search algorithm (CSA) (Askarzadeh 2016), water wave optimization (WWO) (Zheng 2015). The adapted genetic algorithm with adjusting population size is proposed to solve the ORPD problem (Attia et al. 2012). The quasi-oppositional teaching learning based optimization is applied to tackle with the ORPD problem, which accelerates the convergence speed and improves calculation accuracy (Mandal and Roy 2013). The opposition-based gravitational search algorithm has been put forward to solve the ORPD problem, which indicates that its robustness and effectiveness for solving ORPD problem of power systems (Shaw et al. 2014). The harmony search algorithm is used to solve the ORPD problem to find the optimal control variables and the minimal active power loss (Khazali and Kalantar 2011). The gray wolf optimizer is posed to solve the ORPD problem and that is able to achieve less power loss (Sulaiman et al. 2015). Artificial bee colony algorithm is presented to solve the ORPD problem with discrete and continuous control variables (Mouassa and Bouktir 2016). Ant lion optimizer is proposed to solve the ORPD problem in power system, which not only finds the set of optimal control variables, but also gets real power loss (Mouassa et al. 2017). The colonial competitive differential evolution has strong effectiveness and robustness to solve the optimal economic load dispatch problem (Ghasemi et al. 2016). A novel teaching–learningbased optimization algorithm is applied to solve optimal reactive power dispatch problem, the result demonstrates that the optimization efficiency of the modified algorithm is better than that of other algorithms (Ghasemi et al. 2015). The hybrid algorithm combining modified teaching learning algorithm and double differential evolution algorithm has faster convergence speed and better solutions in comparison with other optimization algorithms (Ghasemi et al. 2014). The WWO algorithm based on shallow wave theory mainly simulates propagation, refraction and breaking operations to solve the optimization problem. The WWO algorithm is applied to solve the ORPD problem to obtain the optimal control variables and minimum power losses. The WWO algorithm not only speeds up the convergence rate and calculation accuracy, but also strikes a balance between the global search and local
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search. The robustness and consistency of the WWO algorithm is beneficial to solve the ORPD problem to find the global solution in power system. The WWO algorithm has been tested in IEEE 30-bus in power system, the optimization result of the WWO algorithm is better than that of other algorithms, which shows the WWO algorithm has a strong global search ability. The article has the following sections: Sect. 2 introduces the mathematical formulation of the ORPD problem; Sect. 3 reviews WWO algorithm; the solution procedure of the ORPD problem designed detailed in Sect. 4; the experimental results and analysis presented in Sect. 5. Finally, Sect. 6 discusses the conclusion work.
2 Mathematical formulation The ORPD problem is a constrained nonlinear combinatorial optimization problem in power system. While satisfying all the specified constraints, we obtain the minimization of the total transmission active power losses by setting the optimal values of control variables like reactive power output of generators, tap ratios of transformers and reactive power output of shunt compensators. The total active power losses of the system are minimized as the objective function, and the voltage quality has been improved while reducing the active power losses of the system. The problem’s control decision variables and their notations are given in Table 1. The objective function of ORPD can be described as follows: ( ) ∑ ∑ min Pkloss = gk × v2i + v2j − 2 × vi × vj × cos 𝜃ij (1) k∈NE
Table 1 Control decision variables and notations
k∈NE
Nomenclature WWO
Water wave optimization
BA
Bat algorithm
FPA
Flower pollination algorithm
PSO
Particle swarm optimization
SCA
Sine cosine algorithm
CSA
Crow search algorithm
Pkloss
Active power loss of branch k
gk
Conductance of branch k
NE
A collection of all branches
𝜃ij
Load angle difference between bus i and j
vi,vj
Voltage amplitudes of the bus i and j
Pgi,Qgi
Active and reactive power of generator bus i
Pdi,Qdi
Active and reactive power of load bus i
gij,Bij
Conductance and susceptance between bus i and j
Ng
Number of generators
NB
Number of bus in test system
NT
Number of regulating transformer
NL
Number of transmission lines
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∑ where Pkloss is active power loss of branch k , k∈NE Pkloss represents total active power losses in the transmission system, gk is the conductance of branch k , NE is a collection of all branches, 𝜃ij is load angle difference between bus i and j , vi and vj express the voltage amplitudes of the bus i and j , respectively. The equality constraints can be generated as follow: ∑ ( ) Pgi − Pdi − vi vj gij cos 𝜃ij + Bij sin 𝜃ij = 0 (2) j=Ni
Qgi − Qdi − vi
∑ ( ) vj gij sin 𝜃ij − Bij cos 𝜃ij = 0
(3)
j=Ni
where Pgi and Qgi represent the active and reactive power of generator bus i , Pdi and Qdi represent the active and reactive power of load bus i , respectively. The gij and Bij represent conductance and susceptance between bus i and j. The inequality constraints can be generated as follow: Reactive power generation constraint for each generator bus:
min ≤ Q ≤ Qmax , i ∈ N Qgi gi g gi
(4)
Voltage magnitude constraint for each bus:
vimin ≤ vi ≤ vmax , i ∈ NB i
(5)
Transformer tap-setting constraint:
Tkmin ≤ Tk ≤ Tkmax , k ∈ NT
(6)
The power limit constraint of transmission lines: S ≤ Smax , l ∈ N l
(7)
L
l
In order to make the result satisfy the constraints, the minimum value of the active power losses is taken as the objective function, the penalty function is used to tackle with inequality constraints. So that we can get the augmented objective function as follow:
FP =
∑
k∈NE
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Pkloss + k1 ×
NG ∑
f (Qgi ) + k2 ×
i=1
⎧0 ⎪ f (x) = ⎨ (x − xmax )2 ⎪ (xmin − x)2 ⎩
NB ∑
f (Vi ) + k3 ×
i=1
if if if
xmin ≤ x ≤ xmax x > xmax x < xmin
NL ∑
f (Slm )
(8)
m=1
(9)
Optimal reactive power dispatch using water wave optimization…
Fig. 1 Different wave shapes in deep and shallow water Table 2 Correspondence between problem space and population space Problem space
Population space
The search space of the problem
Seabed area
Solve each solution of the problem
A water wave with height h and wavelength 𝜆
The evaluation function value of each solution
The closer sea level to the seabed, the higher the fitness value. On the contrary, the lower the fitness value
where k1 , k2 and k3 are penalty function coefficient, and the value is 10,000, xmin and xmax are the value rang of dependent variables for each generator bus.
3 WWO algorithm The WWO algorithm mainly simulates the motion of waves to solve the optimization problem, and propagation, refraction and breaking can effectively balance the global search and local search. For each wave, the fitness value of the water wave is related to wave height and wavelength. As can be seen from Fig. 1, the water wave has a higher fitness value, its wave height is higher and wavelength is longer in the shallow water areas; the water wave has a lower fitness value, its wave height is lower and wavelength is shorter in the deep water areas. The correspondence between problem space and population space is given in Table 2.
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3.1 Propagation When each wave is performing propagation, wave height and wavelength will change accordingly because the seabed is uneven. The new water x′ is obtained by propagation the original wave x . Assuming that the position of the original water wave is x(d) and that of the new water wave is x� (d) , the location update formula as follows:
x� (d) = x(d) + rand(−1, 1) ⋅ 𝜆L(d)
(10)
where rand(−1, 1) is a uniformly distributed random number between − 1 and 1, L(d) is the length of d th for search space, 𝜆 is wavelength of water wave x . If the position of wave x′ is not within the search range, then we randomly give a new position in the search space. Compared with the fitness value of original water wave x , if that of the new wave x′ is higher, wave x is replaced by x′ , wave height is defined as hmax . On the contrary, wave x is retained and wave height is decreased by one, which indicates that the wave energy is depleted. The wavelength 𝜆 is updated as follows:
𝜆=𝜆⋅𝛼
−(f (x)−f
/
min +𝜀)
(fmax −f
min +𝜀)
(11)
where 𝛼 is wavelength attenuation coefficient, fmin and fmax are minimum and maximum fitness values, 𝜀 is a positive integer and very small to avoid the denominator is zero. 3.2 Breaking Increasing the wave energy, it makes the crests steeper and steeper. When the speed of crest exceeds that of wave propagation, the water wave will break into a series of solitary waves. In WWO algorithm, the optimal water wave x∗ is performed by breaking operation to improve the diversity of the population. The location update formula is shown below:
x� (d) = x(d) + N(0, 1) ⋅ 𝛽L(d)
(12)
where 𝛽 is a broken wave coefficient, L(d) is the length of the search space for dth . If the fitness value of all solitary waves obtained by breaking operation not better than that of the water wave x∗ , x∗ is retained; on the contrary, the water wave x∗ will be replaced by an optimal solitary wave in the population. 3.3 Refraction In wave propagation, the water wave energy continues to decrease and eventually becomes zero. When the water wave height is zero, refraction operation is used to avoid the search stagnation, which speeds up the convergence of WWO algorithm
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and improves the accuracy of the solution. For water wave x , its refraction operation formula is shown below: ) ( (x ∗ (d) + x(d)) |x ∗ (d) − x(d)| , x� (d) = N (13) 2 2 where x∗ is the optimal solution that found in the current position, N(𝜇, 𝜎) is used to generate mean 𝜇 and standard deviation 𝜎 Gaussian random number. In fact, this allows wave x to learn from the current optimal wave x∗ . At this time, the wave height of wave x′ is hmax , the update formula of wavelength is as follow:
𝜆� = 𝜆
f (x) f (x� )
(14)
The solution procedure of the WWO algorithm is given in Table 3.
4 The solution procedure of ORPD problem The effectiveness and feasibility of the WWO algorithm is stable, which accelerates convergence rate to find the global optimal solution. The WWO algorithm is proposed to solve the ORPD problem that can obtain minimized value of power losses in power system. The correspondence between the ORPD problem space and WWO algorithm space is shown in Table 4. Table 3 The solution procedure of the WWO algorithm
Algorithm WWO 1
2 3 4 5 6
Randomly initialize a water wave population P with n waves (solutions), initialize wavelength reduction coefficient 𝛼 , breaking coefficient 𝛽 , and wavelength reduction coefficient 𝜆.
while stop the termination criterion is not satisfied do for each water wave x ∈ P do
Propagate x to a new x′ by using formula (10); if f (x� ) < f (x) then
if f (x� ) < f (x∗ ) then
7
Break x′ by using Eq. (12);
8
Update x∗ with x′;
9
Replace x with x′;
10
else
11 12 13 14 15
Decrease x.h by one;
if x.h = 0 then
Refract x to a new x′ by using Eq. (13) and (14);
Update the wavelengths by using Eq. (11);
return x∗.
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Table 4 Correspondence between ORPD and WWO The ORPD problem space
The WWO algorithm space
A collection contains all the optimization schemes (x1 , x2 , … , xk ) to solve ORPD problem
A water wave population P with (n1 , n2 , … , nk ) waves
An optimal optimization scheme for solving ORPD problem
An optimal water wave
The objective evaluation function of the ORPD problem
The fitness function of the WWO algorithm
In the coding of water waves, the code length is the total number of the control variables and the variables are arranged in a certain order. In this paper, the sequence is the generator bus voltages, tap ratios of transformers and reactive power output of shunt compensators, which constitutes a complete individual structure of the water wave. The values of the control variables are within the valid range. The number of generators is n1 , the number of reactive compensation point is n2 , and the number of transformer branches is n3 , so the code length of a water wave is n = n1 + n2 + n3. The encoding formats are different for different control variables. The generator bus voltages are continuous variables, which is a real number encoding, such as formula (15). Tap ratios of transformers and reactive power output of shunt compensators are discrete variables, and the randomly generated numbers are rounded, such as formula (16).
x1 = xmin + rand(1, 1) ∗ (xmax − xmin )
(15)
x2 = xmin + round(rand(1, 1) ∗ K) ∗ r
(16)
where xmax and xmin are maximum and minimum values of the control variable range, K is total number of gears, r is unit change of discrete variables, rand(1, 1) is random function, round() is a rounding function. The flowchart of WWO algorithm to solve ORPD problem is depicted in Fig. 2 and the solution procedure of the ORPD problem is given in Table 5, VG1 , VG2 , VG5 , VG8 , VG11 , VG13 represent reactive power output of generators (generator bus voltages), T6−9 , T6−10 , T4−12 , T28−27 represent tap ratios of transformers, and QC3 , QC10 , QC24 represent reactive power output of shunt compensators.
5 Experiment results and analysis The section mainly contains two important contents: Content 5.1 simply introduces the experimental setup; Content 5.2 minutely expounds the WWO algorithm to solve the ORPD problem.
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Optimal reactive power dispatch using water wave optimization… Start
Read system data, bus data, line data, and unit data
Initialization of population set parameters
Map control variables from water waves into laod flow data
Evaluation Obtain power losses from power flow calculation (MATPPOWER), then find the fittest water wave
Whether the condition are meet or not ? Yes No Each water wave x performs propagation operation by Eq.(10) to produce a new water wave x
No
f(x )
