Structure of Chromosomes

The chromosome structure of Fig. 10.9 is used.

Fuzzy Fitness (Suitability) Functions

The suitability level of each chromosome (S_chrom) is computed by defining fuzzy membership functions for suitability of total harmonic distortion (μ_STHD), voltage (μ_SV), and cost (μ_cost). A fuzzy expert system (FES) uses the number of buses with high THD_v and voltage values as well as the average of the THD_v and voltage deviation (at these buses) to determine the suitability of THD_v and voltage.

A concern for the development of fuzzy expert systems is the assignment of appropriate membership functions, which could be performed based on intuition, rank ordering, or probabilistic methods. However, the choice of membership degree in the interval [0,1] does not matter, as it is the order of magnitude that is important [1].

Suitability of THD_v (S_THD)

For a given chromosome, define the average unacceptable value of THD_v as

$TH{D}_{avg}^{\mathrm{high}}={\displaystyle {\sum }_{j=1}^{N_{THD}}TH{D}_j/N_{THD}}\text{,}$

where N_THD is the number of buses with high total voltage harmonic distortions (e.g., THD_j > THD_v^max). The following steps are performed to compute S_THD:

• Fuzzification of THD_avg^high and N_THD (based on membership functions of Fig. 10.12a and 10.12b, respectively) to compute the corresponding membership values (μ_THD and μ_NTHD).

• Fuzzy inferencing and defuzzification based on the decision matrix of Table 10.5, the membership function of Fig. 10.13, Mamdani max-prod implication, and center average methods [50]:

$S_{THD}={\displaystyle \sum \overline{y}{\mu }_{N_{THD}}{\mu }_{THD}/}{\displaystyle \sum {\mu }_{N_{THD}}{\mu }_{THD}}\text{.}$

Suitability of Voltage (S_V)

For a given chromosome, define the average unacceptable value of voltage deviation as

$\mathrm{\Delta }{V}_{avg}=\frac{{\displaystyle {\sum }_{j=1}^{N_{\mathrm{\Delta }\left|V\right|}}\mathrm{\Delta }{V}_j}}{N_{\mathrm{\Delta }\left|V\right|}},\left\{\begin{array}{l}\mathrm{\Delta }{V}_j=\left|V\right|-V^{\text{max}}\left(\mathrm{if}\left|V\right|>V^{\text{max}}\right)\hfill \\ \mathrm{\Delta }{V}_j=V^{\text{min}}-\left|V\right|\left(\mathrm{if}\left|V\right|<V^{\text{min}}\right)\hfill \end{array}\right.$

where N_Δ|V| is the number of buses with unacceptable (high) values of voltage. V^max = 1.1 pu and V^min = 0.9 pu are upper and lower bounds of rms voltage, respectively. The following steps are performed to compute S_V:

• Fuzzification of ΔV_avg and N_Δ|V| (based on membership functions of Fig. 10.14a and 10.14b, respectively) to compute the corresponding membership values (μ_Δ|V| and $N_{N_{\mathrm{\Delta }\left|V\right|}}$).

• Fuzzy inferencing and defuzzification (using the decision matrix of Table 10.5, Fig. 10.13, and Eq. 10-18) to determine the suitability of voltage (S_V).

Cost Index (cost)

Compute the cost for all chromosomes (Eq. 10-8) and linearly normalize them into [0, 1] range with the largest cost having a value of one and the smallest one having a value of zero.

Fitness (Suitability) of a Chromosome (S_chrom)

Suitability level for a given chromosome is computed from S_THD, S_V, and cost using the following steps:

• Fuzzification of S_THD, S_V, and cost (based on Fig. 10.13) to compute the corresponding membership values (μ_STHD, μ_SV, and μ_cost).

• Fuzzy inferencing and defuzzification (using the decision matrix of Table 10.6, Fig. 10.14b, and Eq. 10-18) to determine the suitability of the chromosome (S_chrom).

Genetic Operators

Three genetic operators including reproduction (based on the “roulette-wheel” mechanism), crossover (with crossover probability of 0.6 to 1.0), and mutation (with mutation probability of 0.01 to 0.1) are used.

Figure 10.15 illustrates the block diagram of the fuzzy expert system for the determination of the suitability of a chromosome for capacitor placement.

Analysis and Convergence Criterion of the GA-FL Algorithm

The iterations of the GA-FL algorithm are continued until all generated chromosomes become equal or the maximum number of iterations is reached.

At each iteration, a fuzzy expert system (Fig. 10.15) is used to compute the fitness function of each chromosome and to select the parent population. As a result, new generations have more variety of chromosomes and higher probability of capturing the global solution. As demonstrated in Fig. E10.7.1, the relatively narrow search band of the conventional GA does not allow a large variation of chromosomes between iterations and the solution (e.g., benefits) is trapped at a near global point after the 14th iteration. However, the larger search space of the GA-FL approach (allowing large variations of benefits in consecutive iterations) has resulted in a better near global solution at the 35th iteration.

The inclusion of objective function and power quality constraints will automatically eliminate all solutions generating extreme values for voltages and/or currents and prevents fundamental and harmonic parallel resonances.

Analysis and Convergence Criterion

The generations (iterations) of the GA-FL algorithm are continued until all chromosomes of the population become equal or the maximum number of generations is achieved. The initial conditions for Eq. 10-8 (e.g., the initial capacitor values) do not usually reside inside the permissible solution region. In this section, a GA-FL approach is used to minimize the objective function while directing the constraints toward the permissible region.

To select the parent population at each generation, a fuzzy expert system (Fig. 10.15) is used to compute the fitness function of each chromosome considering the uncertainty of decision making based on the objective function and constraints. As a result, new generations have a higher probability of capturing the global solution.

As demonstrated in Fig. E10.8.2, the relatively narrow search band of the conventional GA does not allow a large variation of chromosomes between generations, and thus it is more difficult to escape from local optima. For the 18-bus and 123-bus systems, the solution is trapped at a near global point after the 14th and 22nd iterations, respectively. In the GA-FL approach, there is more variety in the generated parent population as the chromosome evaluation is based on fuzzy logic. The larger search space (allowing large variations of benefits in consecutive iterations) has resulted in a better near global solution at the 35th and 29th iterations for the 18-bus and 123-bus systems, respectively.

The inclusion of objective function and power quality constraints will automatically eliminate all solutions that generate extreme values for voltages and/or currents and prevents fundamental and harmonic parallel resonances.