Page 220 - Mathematical Models and Algorithms for Power System Optimization

Discrete Optimization for Reactive Power Planning 211

(2) Crossover and mutation principles. Basic GA operations are crossover and mutation,
thereby generating new solutions. The essential basis to accept the crossover and mutation
depends on whether the fitness function F is improved or not.
In this section, some improvements have been made to the original GA algorithm by
crossover and mutation operations within a meaningful range, so as not to induce many
meaningless operations. This section follows such a practice; the expert rules give the GA
operation scope where the variables are changing. The tap ratio and capacitor bank shall be
operated based on the expert rules in this section, and the expert rules of crossover and
mutation operations are detailed as follows:
1. Changing range of integer variables to be within (1–3) of initial feasible solutions.
2. No need for crossover among different voltage levels for capacitors due to the locality
of voltages.
3. In case of no directly interconnected lines, capacitors at two nodes are not required
to cross.
4. Crossover is not necessary for the transformer ratios at different voltage levels.
5. In case of no directly interconnected lines, two transformers are not required to cross.
6. If not in the same substation, the transformer ratio and capacitor bank are not required
to cross.
7. The integer variables generated randomly from variations shall be within a
meaningful range, and only one or finite variables are allowed to mutate at a time.
8. Crossover shall be among integer variables of interconnected nodes, and only one or
finite pairs of variables are allowed to cross at a time.

In this section, the VAR units and tap ratios are integer variables, also equivalent to control
variables, and changing of integer variables can alter the system’s power flow distribution. The
method of random number can produce multiple integer solutions. The solutions (if any) of the
optimization calculation and power flow generally could converge after several to dozens of
iterations, namely equivalent to several to dozens of GA generations, the optimization
calculation and power flow calculation may generate the only feasible solution from each
iteration, whereas each GA breeding may produce several generations (feasible solutions). The
larger the calculation scale, the more the combinations of feasible solutions and vice versa.
Therefore, the upper limits of GA breeding generations and maximum generations from each
breeding should be identified according to the calculation scale.

6.6.4.2 Calculation procedure

The main procedure of the GA-based discrete VAR optimization is shown in Fig. 6.11. Because
the existing mathematic model in the power system can be used to obtain the numerical
solutions for GA-based method, the power flow calculation and MILP method in Section 6.3
have been selected. The initial values of MILP method are used at the initial exploration points,
so as to significantly reduce the calculation time of GA-based method.