Page 164 - Mathematical Models and Algorithms for Power System Optimization

Discrete Optimization for Reactive Power Planning 155

practical needs of a power system, it is necessary to consider both the requirements of a
single-operating mode (single state) and multioperating mode (multistate) for the reactive
power compensation equipment.

With the multistate taken into account, discrete reactive power optimization will become even
larger in scale and more complicated. As mentioned previously, no mathematical algorithm to
perfectly solve discrete reactive power optimizations is available thus far. In this chapter,
diagonal block matrix model is used to express a multistate problem, and a reactive variable is
used to express a coupling variable, so it is possible to solve multistate discrete reactive power
optimizations using a decomposition coordination method.

6.2.4 Way of Selecting of Initial Values

The steady-state operation of a power system is normally expressed as a power flow equation,
which is a transcendental equation. Thus, power system operation is a nonconvex problem, that
is, there may be multiple solutions. A loop-iteration linear programming (SLP) algorithm
proposed in the 1960s may be used to solve large-scale nonlinear problems. However, the
convergence of solutions cannot be tested even under a convexity hypothesis. As most
constraints for power system operation problems are nonlinear, when a discrete variable is
introduced, it would be difficult to meet the continuously adjustable requirement. For this
reason, if an SLP algorithm is used to solve MINLP, the obtained discrete feasible solution will
be related to the initial value configuration.

The initial value for discrete reactive power optimization is normally configured in two ways.
The first way is to take the continuous optimal solution and truncate it as the initial value for a
discrete solution. The second way is to take the power flow solution as the initial value, then
truncate it as the initial value for a discrete solution. The former way is more desirable.
However, the power flow solution is taken as the initial value in this chapter in consideration of
the necessity to calculate continuous optimization when calculating discrete optimization and
the use of more computing resources by the continuous optimization.

In fact, the power flow solution is also normally taken as an initial value for continuous
reactive power optimization, so using the power flow solution as the initial value is acceptable
for an operational problem. However, for programming problems, an optimization calculation
may be conducted only after obtaining a converged power flow solution with the help of
expert rules because it is rather difficult to obtain even a converged power flow solution.

6.2.5 Consideration to Obtain Global Optimization

As there is no way to prove reactive power optimization meets a convexity condition, and when
consideringitsdiscretevariable,reactivepoweroptimizationevenbecomesanonconvexproblem;
thus, its optimal solution is usually related to its initial value. Normally, the nonlinear method can