J. Cent. South Univ. Technol. (2009) 16: 0683-0689

DOI: 10.1007/s11771-009-0113-4

Multi-objective coordination optimal model for

new power intelligence center based on hybrid algorithm

LIU Ji-cheng(刘吉成), NIU Dong-xiao(牛东晓), QI Jian-xun(乞建勋)

(School of Business Administration, North China Electric Power University, Beijing 102206, China)

Abstract: In order to resolve the coordination and optimization of the power network planning effectively, on the basis of introducing the concept of power intelligence center (PIC), the key factor power flow, line investment and load that impact generation sector, transmission sector and dispatching center in PIC were analyzed and a multi-objective coordination optimal model for new power intelligence center (NPIC) was established. To ensure the reliability and coordination of power grid and reduce investment cost, two aspects were optimized. The evolutionary algorithm was introduced to solve optimal power flow problem and the fitness function was improved to ensure the minimum cost of power generation. The gray particle swarm optimization (GPSO) algorithm was used to forecast load accurately, which can ensure the network with high reliability. On this basis, the multi-objective coordination optimal model which was more practical and in line with the need of the electricity market was proposed, then the coordination model was effectively solved through the improved[A1]  particle swarm optimization algorithm, and the corresponding algorithm was obtained. The optimization of IEEE30 node system shows that the evolutionary algorithm can effectively solve the problem of optimal power flow. The average load forecasting of GPSO is 26.97 MW, which has an error of 0.34 MW compared with the actual load. The algorithm has higher forecasting accuracy. The multi-objective coordination optimal model for NPIC can effectively process the coordination and optimization problem of power network.

Key words: power intelligence center (PIC); coordination optimal model; power network planning; hybrid algorithm

1 Introduction

The coordination and optimization problem of power network planning is a key point in power system planning. However, solving power planning problem from the overall situation of generation, transmission and distribution is the difficulty of planning. So far, there have not many researches on the power network planning from the overall perspective. Most of them are focused on the optimization and forecasting of key factors, such as the simple solution to the issue of optimal power flow and load forecasting. On the one hand, some traditional methods such as nonlinear programming, linear programming and mixed-planning, have been used for solving the simple planning and coordination problems [1]. On the other hand, many intelligent optimization methods have been applied to power network planning, such as the genetic algorithm (GA) [2-3], neural network (NN) [4-5], particle swarm optimization (PSO) [6-7], evolutionary algorithm (EA) [8-9], fuzzy theory [10-11], and immune algorithm [12-14]. These optimization methods have resolved the planning and coordination problems to some extent. However, they often cannot achieve satisfactory results because the problem is a multi-objective coordination optimal one.

LIU et al [15], LIU and NIU [16] proposed the concept of power intelligence center (PIC) and performed some basic researches In this work, on the basis of previous study, through processing three key factors that affect power grid, the corresponding model base was established by the improved intelligence algorithm, and then the solution to multi-objective coordination optimal problem was obtained by calling the model base.

2 New power intelligence center (NPIC) model

In order to make the model simple and convenient for calculation and simulation, the impact of random factors on the NPIC is not considered. The basic idea of NPIC model is established as follows: under the circumstance of PIC, the function departments of power system are divided into generation sector, transmission sector and distribution sector. The key factors corresponding to each department are different, which are respectively power flow, line investment and load. It is necessary to establish a corresponding model base in order to solve the three departments objective optimization problem coordinately from the whole. Initially, a variety of implementation methods are listed, such as genetic algorithm, neural network, and PSO. According to the characteristics of different algorithms, the problems corresponding to the key factors can be solved. Reselect if the method is improper. Once the optimization result is better, it is identified as the method of model base and stored in background model base of the NPIC for the following process. In this way, the multi-objective coordination optimal problem can be solved better by means of the dynamic coordination of model base and power network. The NPIC model is shown in Fig.1.

3 NPIC model library based on hybrid optimal algorithm

3.1 Optimization model for solving optimal power flow problem

3.1.1 Optimal power flow model

Regarding the minimum generation cost as the optimized objective function, the mathematical model of optimal power flow is as follows.

(1) Objective function

        (1)

where  F is the overall generation cost, P_Gi is the active power output of generator i, N_G is the collection of all the generators, a_i, b_i and c_i are the cost factors of generator G.

(2) Equality constraints

         (2)

        (3)

where  Q_Gi is the reactive power output of generator i; P_Di and Q_Di are the active load and reactive load of node i; V_i and θ_i areis the voltage amplitude and phase angle of node i, θ_ij=θ_i-θ_j; G_ij and B_ij are respectively the real part and imaginary part of the element in row i and column

Fig.1 Schematic diagram of NPIC model

j in the node admittance matrix.

(3) Inequality constraints

The cap and floor constrain of P_Gi is

   (i=1, 2, ???, N_G)                  (4)

The cap and floor constrain of Q_Gi is

  (i=1, 2, ???, N_G)                   (5)

The cap and floor constrain of V_Gi is

  (i=1, 2, ???, N_G)             (6)

The cap and floor constrain of T_i is:

  (i=1, 2, ???, N_T)                       (7)

where T_i is the transformation ratio of on-load  transformer of node i.

The cap and floor constrain of Q_Ci is:

  (i=1, 2, ???, N_C)             (8)

where  Q_C is the alternative reactive power of node i.

The cap and floor constrain of V_Li is

  (i=1, 2, ???, N_L)            (9)

where  V_L is the load node voltage of node i.

The cap and floor constrain of S_Li is:

  (i=1, 2, ???, N_L)             (10)

where  S_Li is the line transmission power of node i.

3.1.2 Optimal power flow model based on evolutionary algorithm

Evolutionary algorithm (EA) is similar to genetic algorithm, including initialization, variation, crossover and evaluation processes. The main operation processes to solve the optimal power flow problem using evolutionary algorithm are as follows.

(1) Initialization

In initialization data, each decision parameter in initialization population vector is transformed into a randomly selected parameter value through the following formula:

(i=1, 2, ???, N_p; j=1, 2, ???, D)                    (11)

where   μ_j presents the random number within [0,1], which is used to produce a new individual j;  and  are respectively the upper and lower limits of decision parameter j.

(2) Variation

The variation operation is the combination of a randomly selected vector X_a and the deviation of the other two different random vectors X_b and X_c. The producing formula of a random variation vector  is as follows:

  (i=1, 2, ???, N_p)  (12)

where  a, b and c are randomly selected indicators, a, b, c∈{1, ???, N_p}, and a≠b≠c≠i; scaling parameter Z is the control variable of the algorithm, its range is [0, 2], and it is often used to adjust the amplitude of the variation operation in order to strengthen the convergence of the algorithm.

(3) Crossover

The vector generated from crossover is obtained from the following formula:

            (13)

where  i=1, 2, ???, N_p, and j=1, 2, ???, D; ρ_j represents a random number of uniformed distribution and its range is [0, 1], which is used to produce a new individual j; the crossover vector C_R is used to control differences of population and control algorithm jump local optimum, and its range is [0,1]; q is a randomly selected index and its range is q∈{1, 2, ???, D}.

(4) Selection

The formula of selection operation is as follows:

        (14)

where  i=1, 2, ???, N_p. 

Formula (14) shows that when the individual after crossover is substituted into the fitness function, and the value is smaller than the designed standard value, the conditions are met, and the power flow is the optimal one. Otherwise, the variation operation is returned and the repeated evolution is made until the termination conditions are satisfied. And the fitness function is calculated according to Formula (1).

3.2 Load forecasting model

GM(1,1) is the most commonly used as grey system model and its algorithm can be referred in Ref.[17]. The simple grey forecasting method can bring about larger errors and is easy to generate noise and heavy- tailed phenomenon. In order to increase the forecasting accuracy and computational efficiency, the PSO algorithm is used to improve the simple grey forecasting method and establish the grey forecasting method.

Let t be any number, t∈{0, N-1}. The original load sequence is x⁽⁰⁾(t), the load sequence after one-accumulation is x⁽¹⁾(t), and the load sequence after forecasting is .

We make sum of square error of x⁽⁰⁾(t) and forecasting value  minimum, that is

                      (15)

Through the continuous optimization of PSO, the forecasting accuracy of the gray neural network can be improved.

PSO expresses every possible solution as a particle of the group. Every particle has its own position vector, velocity vector and a fitness value decided by the objective function. All the particles fly at a certain velocity in the search space, and then find the global optimum by following the optimal value of the current search.

Pretending in a D-dimensional objective search space, the position vector of particle i is X_i=(x_il, x_i2, ???, x_iD), where the arbitrary particle position is x_id; the the individual extreme point is P_i=(P_il, P_i2, ???, P_iD), and the global extreme point is P_g=(P_gl, P_g2, ???, P_gD); and the velocity vector is V_i=(v_il, v_i2, ???, v_iD). The velocity changed by particles is decided by the following formula:

          (16)

                             (17)

where   d=1, 2, ???, D, i=1, 2, ???, N; N is the population  size; w is the inertia weight; X_i is the current particle position; Vv_i is the current particle velocity and is limited by a maximum velocity Vv_max; r₁ and r₂ are the random numbers distributed in the range [0,1]; c₁ and c₂ are the limited constants related to the position.

Combining the basic algorithm of PSO and the characteristics of GM (1,1) model, and taking Formula (15) as the fitness value of PSO, a new grey forecasting model based on PSO can be created. The basic algorithm is as follows.

(1) Randomly selecting m particles, the limited constants with changed positions c₁ and c₂, the inertia weight w and the maximum evolutionary generations S_max are given. Let the particles’ position vector and velocity vector be respectively: U_i=(u_il, u_i2, ???, u_im), V_i=(v_il, v_i2, ???, v_im), where i=1, 2, ???, m, u_ij and v_ij are the numbers in the range [0, 1], and n is the number of the original load sequence.

(2) Let the sum of square error of grey neural network’s forecasting value and input value be Q, according to Formula (15), we have

Q=.

From the above analysis we can see that when Q is the minimum, the model forecasting accuracy is the highest. Thus, fitness function f(U_i) of individual U_i can be expressed as follows:

                  (18)

where   f(U_i) changes reversely with the model forecasting accuracy, that is, when f(U_i) is smaller, the model accuracy is higher.

(3) Update the location and speed of particles according to Formulae (16) and (17).

(4) When optimization reaches the maximum evolutionary generation S_max, the optimization is finished; otherwise return to Eqn.(2).

4 Multi-objective coordination optimal model for NPIC

According to the model base established, the multi-objective coordination optimal model for NPIC is established. There are three objective functions: the goal of optimal power flow, which is the power system condition required by Formula (1); the goal of load forecasting, which makes Fformula (18) achieve the optimal sequence of load forecasting; the investment cost of line, a route planning that makes investment cost minimum. Thus, the multi-objective coordination optimal model is as follows:

min       (19)

where the parameters and constraints of the first and the second objective fFunction have been described in the former section; the third objective function f ′ is the investment cost for the line; s_i is the investment cost per unit length for the new line; x_i is the number of loop of line i for selection; n is the total number of lines to be chosen; and l_i is the length of line i.

The multi-objective coordination optimal  model is solved by improving PSO algorithm. The specific algorithm is as follows.

Step 1 Because the three objective functions in fFormula (19) are more independent and they all seek the minimum, the penalty factor is introduced, then the multi-objective function is transformed into a single-objective problem. On the one hand the indicators of the power generator can be controlled in the scope of the corresponding variables’ lower and upper limits, on the other hand the single-objective function can be processed as PSO fitness function. The objective function after transformation is as follows:

             (20)

where  F′is the transformed objective function, λ₁, λ₂ and λ₃ are the penalty factors.

Step 2 (Set the parameters of PSO and the parameters of the three objective functions, and randomly generate particles.

Step 3 Set Formula (20) as the fitness function of PSO, set the corresponding penalty factor, calculate the fitness value, and then determine whether the minimum of F′ is obtained. If F′_min is gained, the algorithm is finished; otherwise, continue to the next step.

Step 4 According to the location and velocity formula, calculate the corresponding fitness value and update the particles position and velocity.

Step 5 When optimization reaches the maximum evolutionary generation or the fitness value achieves the minimum, the algorithm is finished; otherwise return to Step 2.

5 Example

In this example, the multi-objective coordination optimal model is used for calculation of optimal power flow and load forecasting of IEEE306 node system. In order to verify the validity of the algorithm, the simulation results of the algorithm proposed this work, genetic algorithm(GA), Newton-Raphson algorithm (NR) and P-Q analysis method of power flow calculation are compared. The reason why the NR method is compared with P-Q method is that P-Q method is derived from Newton-Raphson method which is expressed in polar coordinates and is made the following simplifications. The first simplification is that the reactance of every component in power network is far greater than the resistance, and the second one is that the constraint condition of state variable is not too big. The test on bus system is based on Matlab7.0 platform, the CPU of computer is Intel Pentium IV (2.6 GHz), and the memory is 1  024 Mbit.

Before the optimization, the parameters of the algorithm are needed to be set. The parameters of GA are as follows: the population size is the same as that of evolutionary algorithm in this work, crossover probability is 0.9, mutation probability is 0.1, learning probability is 0.5, learning system is 0.6, and penalty factor is 10. In load forecasting, the particle number of the gray particle swarm optimization (GPSO) is 200, the limit constants with changed location c₁ and c₂ are 1.5, the inertia weight factor w is 0.5+0.5σ, where σ is a random number in the range [0, 1], and mutation probability is 0.1, which is the same as that of GA. In the calculation, the construction costs per unit length of each line is assumed to be the same. So the length of such lines can be used to replace the cost to be calculated. The evolutionary algorithm is used for optimal power flow optimization on the IEEE30 bus system. The parameters are as follows: population size N_p is 18, crossover constant C_R is 0.5, the generator’s scaling parameter Z is 0.9, the maximum number G^max is 180. The penalty factors λ₁, λ₂ and λ₃ of multi-objective coordination optimal model are all equal to 1.

Because there are 6 generator units, 4 on-load tap changers and 41 branch circuits in the IEEE30 node system, the fuel cost functions of generators are all quadratic functions. The parameters are shown in Table 1.

IEEE30 nodes are optimized, and the calculation results of four different algorithms are shown in Table 2.

We can see from Table 2 that the largest total generating capacity of the four algorithms belongs to the evolutionary algorithm, and its value is 35.404 MW; and the smallest line investment also belongs to evolutionary algorithm, which is 15.672 k?(RMB). However, the traditional NR algorithm has the smallest total generating capacity of 33.671 MW and it has the largest line investment, which explains that intelligent optimization algorithm has certain advantage in solving optimal power flow problem over the traditional algorithm. It can also be noticed that the optimal results of GA and P-Q are close, while optimal results by the evolutionary algorithm are superior to those of the two methods.

The key factors of multi-objective coordination  optimal model are obtained by short-term load forecasting in 24 h. The used method is the gray particle swarm optimization method. The load forecasting data of IEEE30 node system are shown in Table 3, and the load forecasting results of IEEE30 node system are shown in Fig.2.

The load forecasting curves of Fig.2 and the calculation results of Table 3 show that the average load forecasting of IEEE30 node system is 26.97 MW, which has an error of 0.34 MW compared with the actual load. The accuracy is high and the algorithm reaches convergence.

The above results show that it is feasible for the multi-objective coordination optimal model to optimize three key factors that affect the power network. The model can achieve convergence and make the total objective function minimum. Thereby, it lays a theoretical

Table 1 Parameters of generator’s quadratic cost function

Table 2 Calculation results of optimal power flow of four different algorithms (IEEE30 node system)

Table 3 Load forecasting results of GPSO algorithm (IEEE30 node system)

Fig.2 Load forecasting results of IEEE30 node system for nodes 1, 2 and 5 (a) and for nodes 8, 11 and 13 (b)

foundation for building NPIC in local areas to deal with the key factors of the power network and optimization network.

6 Conclusions

(1) The key factors in the three kinds of power network planning are considered. The intelligent optimization method is proposed to solve the problems arising from the key factors, which lays the foundation for cooperation and decision-making of power supply chain alliance system.

(2) The optimal power flow problem is effectively solved and the minimum cost of line investment can be calculated by using evolutionary algorithm. The high precision load sequence is got by using the GPSO method, which provides a model reserve for building multi-objective coordination optimal model. Thereby, the model base of the NPIC is established, which provides the convenience for solving the same problem.

(3) The examples show that the intelligent optimization method has a greater advantage in resolving the issue of multi-objective coordination optimal than the traditional planning methods and verifies the validity of the model established in this work.

(4) Merely considering the model with one key factor is a special case of multi-objective coordination optimal model. And multi-objective model can be extended to taking into account three or more key factors.

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(Edited by CHEN Wei-ping)

Foundation item: Project (70671039) supported by the National Natural Science Foundation of China

Received date: 2009-02-25; Accepted date: 2009-04-20

Corresponding author: LIU Ji-cheng, Doctoral candidate; Tel: +86-13601030970; E-mail: ljc29@163.com