Particle Swarm Optimization Implementation

i

ni

N 1 ^(2.147)

From equation (2.147), it is implied that, as the number of units with POZ increases, the total number of decision spaces increases sharply. To compute all those decision spaces classical methods are not viable. Therefore, new approaches that can overcome or better say bypass these diffi culties are needed.

Fig. 2.20 Typical input-output of a unit with POZ.

Fi ($/h)

Pi (MW) PGikl

PGiku

PGik+1l

PGik+1u

Prohibited zone k

Prohibited zone k+1

B. Particle Swarm Optimization Implementation

The practical ED problem with valve-point and multi-fuel effects is represented as a non-smooth optimization problem with equality and inequality constraints, and this makes the problem of fi nding the global optimum diffi cult. To solve this problem, many salient methods have been proposed and genetic algorithm, evolutionary programming, Taboo search, and particle swarm optimization (PSO) are considered powerful methods to obtain usable solutions to power system optimization problems.

Among those methods, PSO is one of the most modern heuristic algorithms and has gained a lot of attention in various power system applications. PSO can be applied to nonlinear and non-continuous optimization problems with continuous variables. It has been developed through simulation of simplifi ed social models.

PSO is similar to other evolutionary algorithms in that the system is randomly initialized with a population of solutions [52].

The algorithm for obtaining the optimal power dispatch with a non-smooth cost function is shown in Fig. 2.21.

As an example, a hybrid particle swarm quadratic programming based economic dispatch (PSO-QP-ED) with a network and generator constrained algorithm is used to illustrate the non-smooth cost function ED.

In the PSO-QP-ED algorithm, the real power generation sets at the generator bus are used as particles in the PSO. The QP based ED with transmission line limit and transformer loading constraints is performed for every generation to obtain the best solution for each population search. PSO-QP-ED is compared to PSO-ED (without QP) on the IEEE 30 bus system under transmission line and transformer loading limit constraints and with generators’ discontinuous fuel cost functions.

Generally, PSO is characterized as a simple heuristic of a well-balanced mechanism with fl exibility to adapt and enhance both global and local exploration abilities. It is a stochastic search technique with reduced memory requirement, computationally effective and easier to implement than other artifi cial intelligence techniques. PSO also has a greater global searching ability at the beginning of the run while conducting a local search near the end of the run. Therefore, when solving problems with several local optimal solutions, there is a high possibility that PSO will explore more local optimal solutions with the potential of global optimal solution after convergence.

Fig. 2.21 Computational procedure of a non-smooth cost function ED.

Calculate the power flow solution of each population

Calculate the fitness or evaluation value incorporating the total cost with system constraints penalization

Update the dispatch level using Artificial Intelligence Technique

Stop

Does the solution converge?

Randomly initialize a set of real power generators as the population in the search space

No

Yes Start

In a PSO system, particles fl y around in a multidimensional search space.

During fl ight, each particle adjusts its position according to its own experience and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors. The swarm direction of a particle is guided by the set of neighboring particles and its historical experience.

In the hybrid PSO-QP-ED algorithm, the set of particles is represented as follows:

where P_Gi is a matrix representing the set of individual searches. More specifi cally, in this case, it is a set of real power generators. The sub-matrix P^j_Gi is the set of the current position of particle j representing the real power generation of the generator connected to bus i (P^j_Gi). Each particle is used to solve the quadratic programming-optimal power fl ow (QPOPF) and the best previous position of the jth particle is recorded and represented as:

T

The index of the best particle among all the particles in the group is represented by the gbest^j. The velocity rate of particle j is represented as

T

The modifi ed velocity and position of each particle can be calculated using the current velocity and the distance from pbest_i to gbest_i as shown in the following formulas:

M number of particles in a group, NG number of members in a particle, k pointer of iterations (generations), w inertia weight factor,

c₁, c₂ acceleration coeffi cients,

u_i^j(k) velocity of particle j corresponding to P_Gi at iteration k,

( ) j k

PGi current position of particle j corresponding to P_Gi at iteration k, rand₁(•), rand₂(•) uniform random values in the range [0,1].

The parameter u_i^maxdetermines the resolution, or fi tness, which regions are to be searched with between present position and target position. If u_i^max is too high, particles might fl y over good solutions. If u_i^max is too low, particles may get stuck to local solutions. From experience with PSO, u_i^max has often been set at 10–20%

in each dimension of the dynamic range of a variable.

The constants c₁and c₂represent the weight of the stochastic acceleration terms that pull each particle towards the pbest_i and gbest_i positions. Low values may move particles past the target regions before being pulled back. On the other hand, high values may result in abrupt movement towards, or past, target regions. Hence, the acceleration constants c₁and c₂ have often been set to 2.0 which works reasonably well for ED problem. A suitable selection of the inertia weight w provides a balance between global and local explorations, thus on average requiring less iterations to fi nd an optimal solution. As originally developed, w often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weight w is set according to the following equation:

M k

where w_max and w_min are maximum and minimum inertia weight factors, respectively, and M is the maximum number of iterations.

The evaluation value is normalized into the range between 0 and 1 as:

)

F_max maximum generation cost among all individuals in the initial population, F_min minimum generation cost among all individuals in the initial population, BG number of buses with generation.

To limit the evaluation value of each individual of the population within a feasible range, before estimation, the generated power output must satisfy all constraints. If one individual satisfi es all constraints, it is a feasible individual

and F_cost has a small value. Otherwise, the F_cost value of the individual is penalized with a very large positive constant. The computational procedure is shown in Fig. 2.22.

Fig. 2.22 PSO-QP-ED Computational Procedure.

Start

j = 1

Randomly searching initial point for power generation of generator i of the population j (P_Gi^{j (k)} for i = 1,…,NG)

Solve the OPF wiyh the power generation obtained by of the population j (P_G^j(k) =[P_G1^{j (k)} P_G2^{j (k)} … P_{G NG}^{j (k)} ]^T ) using QP

All P_Gi^{j (k)} for i = 1,…,NG are with in their limit?

Compute the EV^{j (k)} Does the OPF converge ?

Record the EV^{j (k)}

Selects P_G^jthat gives the best EV^{j (k)} as Pbest^(k) j = j + 1

j > Specified number of populations?

k > Specified number of generations?

k = 1, EVgbest = 0

k = k + 1 EV^{j (k)}> EVgbest?

EVgbest = EV^{j (k)} , gbest = pbest^(k)

Update u^{j (k+1)} = [u₁^{j (k+1)} u1 j (k+1)

… uNG j (k+1)]^T and P_G^{j (k+1)} =[P_G1^{j (k+1)} PG2j (k+1) … PG NGj (k+1) ]^T

P_Gi = gbest F_cost^j= 10e¹²

No

Yes

Yes

Yes

Yes

Yes No

No No

No

Example 2.8

The IEEE 30 bus system in example 2.6 is used as test data. The generator fuel cost functions and prohibited operating zones are given in Table 2.16. The parameters of the proposed PSO-QP-ED and the PSO-ED are as follows:

Population size = 200 Generation (M) = 10 w_min = 0.4, w_max = 0.9

max max 0.5 _Gi

i P

u ˜ ^, u_i^min 0.5˜P_Gi^min

1 2

c , and c₂ 2

Table 2.17 addresses the summarized results of PSO-ED and the proposed PSO-QP-ED from 50 trials. The convergence properties of the best solutions of PSO-QP-ED and PSO-ED are shown in Fig. 2.23. Figure 2.24 shows the total system operating cost from 50 trials of POS-ED and PSO-QP-ED. The results show that the total generator fuel costs of the proposed PSO-QP-ED are lower than that of the PSO-ED. The computational time of PSO-QP-ED is longer than that of PSO-ED because PSO-QP-ED solves QPOPF for each individual search in the search space. The computational time of the proposed method could be decreased by parallel computation. However, the PSO-QP-ED offers a higher probability of obtaining the global minimum of total generator fuel cost.

Table 2.16 Generator operating cost functions and constraints.

Gen bus F(P_Gi) = a_i + b_iP_Gi + c_iP_Gi² m in

PGi P_Gi^max Generator Prohibited Operating Zone

a_i b_i c_i MW MW From

MW To MW

From MW

To MW

1 0 2 0.00375 50 200 100 120 150 160

2 0 1.75 0.0175 20 80 25 30 40 60

5 0 1 0.0625 15 50 20 25 40 45

8 0 3.25 0.00834 10 35 15 20 25 30

11 0 3 0.025 10 30 15 18 22 25

13 0 3 0.025 12 40 20 25 30 35

Table 2.17 Results of the IEEE 30-bus test system.

PSOED PSO-QP-ED

Total system operating cost ($/h)

Min Aver. Max Min Aver. Max

806.10 818.13 834.05 805.14 809.46 816.47 Computation time of the best

trial solution (sec) 52.67 69.39

Fig. 2.23 Convergence properties of the IEEE 30 bus test system.

Fig. 2.24 The results from 50 trials with the IEEE 30 bus test system.

PSO-ED

PSO-QP-ED

Iteration

Total Generator Fuel Cost ($/h)

Color image of this figure appears in the color plate section at the end of the book.

Color image of this figure appears in the color plate section at the end of the book.

A hybrid particle swarm optimization and quadratic programming for economic dispatch (PSO-QP-ED) is effectively minimizing the total generator fuel cost while satisfying transmission line limits and transformer loading constraints with generator prohibited operating zone constraints. PSO-QP-ED achieves lower minimum total generator fuel cost in the constrained ED than PSO-ED.

In document artificial-intelligence-in-power-system-optimization.pdf (Page 70-77)