3.1. Frame of the Equivalence Method
Synchronized phasor measurement technology has been widely used in the power system dynamic monitoring [
21,
22,
23]. Equivalent model parameters are obtained by making use of dynamic information in the tie line during the disturbance based on the parameter estimation. In this paper, the measurements of voltage, frequency, active power and reactive power in the tie line are extracted as inputs and then the equivalent model parameters are identified, which is α = [
Tj,
xd,
x’d,
T’d0,
Kv,
D,
Ps0,
Qs0,
Np,
Nq]. The overall framework is shown in
Figure 2.
The processes are as follows: firstly, initialize a suspicious parameter α; secondly, take the voltage and frequency responses in the tie line as inputs; thirdly, solve the equations of equivalent model and get active and reactive power; then compare measured and calculated values of the active and reactive power and the error e is generated; finally compare the error with the set threshold, namely if the error is greater, update parameter α by the optimization algorithm and repeat the process; otherwise, output α as the equivalent model parameters.
The electrical power outputs are calculated with the following equations:
The error is calculated with the Euclidean distance formula:
where
N is the total number of sampling sites;
Pl is the measured value of active power;
P′l is the calculated value of active power;
Ql is the measured value of reactive power;
Q′l is the calculated value of reactive power.
In order to obtain the electrical power outputs, the δ and
E′ from different times have to be calculated. The two most classic numerical methods are the improved Euler method and Runge-Kutta method [
24]. The 4th order Runge-Kutta method is used widely when solving ordinary differential equations because of its briefness and accuracy [
25,
26,
27]. In this paper, the 4th order Runge-Kutta method is used to solve generator dynamic equations and the state variables of time
k + 1 moment such as ω, δ,
E′ can be calculated by time
k moment. The initial value of state variables can be determined by the steady-state phasor diagram. The power outputs in the tie line are obtained with the certain equivalent model parameters based on Equations (1)–(4). The frame is shown in
Figure 3.
3.2. Dynamic Multi-Swarm Particle Swarm Optimizer Algorithm
The PSO algorithm is widely used in the optimization of power system because of its good global convergence, easy operation, high efficiency and few parameter settings [
28,
29,
30]. In this paper, equivalent model parameters are identified by the DMS-PSO algorithm [
31].
The optimal solution is found by initializing a group of random particles, iterating them repeatedly with the basic method of PSO. In each iteration process, the particles are updated by tracking two “extremes”: one is the optimal solution found by an individual (personal best, that is
pbest), and the other is the optimal solution found by the entire group (group best, that is
gbest). Each particle updates its velocity and position respectively according to the following equations.
where
vn is the velocity vector of the particle on
n-th iteration; w is the weight factor;
c1,2 is the learning factor;
rand is a random number between 0 and 1.
In order to achieve better results, the following three improvements are used based on the basic PSO algorithm.
The first strategy is DMS-PSO in order to slow down convergence speed and to increase diversity to enhance the global search capabilities. The effect is better on complex problems when PSO with small neighborhoods is used according to many reported researches of PSO. Therefore, in DMS-PSO, the population is divided into small sized swarms. Each particle seeks for better solution in the search space in each small sized swarm. After every
R iterations, the population will be regrouped randomly and then the search will be started in the new swarms [
32].
The second strategy is particle non-uniform mutation for the sake of improving self-help capabilities when the particles get into a local optimum. When the stable frequency of the value of
gbest objective function reaches a certain threshold, current vector is randomly generated so that the particles have a chance to escape from the local optimum and search for the global optimum. The mutation equations are:
where
x′nj is the mutated
xnj;
l is the random number of 0 or 1;
xnj is the
j-dimensional component of the
n-th particle;
Uj is the upper limit of
xij;
Lj is the lower limit of
xij;
b is the random number of 0–1.
The third strategy is decreasing the inertia weight linearly for balancing the local search and global search capabilities. It is conducive to a global search in the earlier stage when
w is larger while being conducive to local development in the later stage when
w is smaller. The formula is:
where
wmax is the upper limit of inertia weight which is set to 0.9;
wmin is the lower limit of inertia weight which is set to 0.4;
n is the current iteration number;
nmax is the maximum number of iterations.
The process of solving the dynamic equivalent model parameters based on the above-mentioned method with DMS-PSO is shown in
Figure 4.
Specific steps: (1) Input measurements, namely, V, ωf, Pl and Ql in the tie line; (2) Initialize the parameters of each particle, swarm and iteration number; (3) Work out the outputs based on Equations (1)–(4); (4) Calculate the objective function and confirm the initial value of pbest and gbest; (5) Update iterations and velocity and position of the particle; (6) Update iterations; (7) Work out the outputs based on Equations (1)–(4); (8) Calculate the objective function and then update pbest and gbest; (9) Determine whether nmax/R is integer or not. If the answer is Yes, regroup the population randomly, otherwise go on; (10) Determine whether the terminal condition is satisfied or not. If the answer is Yes, stop calculating and output gbest, otherwise determine whether mutate or not. If the answer is Yes, generate velocity and position randomly and then return to (6), otherwise return to (5).