Chance-Constrained Optimization-Based Unbalanced Optimal Power Flow for Radial Distribution Networks

Optimal power flow (OPF) is an important tool for active management of distribution networks with renewable energy generation (REG). It is better to treat REG as stochastic variables in the distribution network OPF. In addition, distribution networks are unbalanced in nature. Thus, in this paper, a chance constrained optimization-based multiobjective OPF model is formulated to consider the forecast errors of REG in the short-term operation of radial unbalanced distribution networks. In the model, expected total active power losses of distribution lines, expected overload risk and voltage violation risk with respect to N-1 contingencies are minimized, and inequality constraints in the normal state are satisfied with a predefined probability level. Thus, the profitability and security can be balanced in the presence of stochastic REG. The proposed multiobjective OPF problem is solved by the multiobjective group search optimization and the two-point estimate method. Simulation results show that distribution network economy and postcontingency performance deteriorate with increased penetration level of REG, and the penetration level has a greater impact than the forecast errors of REG.

I. Nomenclature

Total active power losses of distribution lines.

Voltage violation risk.

Overload risk.

Active power of th nonrenewable distributed generator.

Reactive output of th distributed generator.

Active and reactive powers at swing bus, respectively.

Reactive power compensation at bus .

The th contingency.

Predefined probability level.

Expectation of a random variable.

Standard deviation of a random variable.

Ratio between expectation and standard deviation.

Occurring probability.

Severity of a contingency.

Voltage magnitude of bus

Apparent power of the th distribution line.

Setting of th tap-changing transformer.

Injected current matrixes of zero- and negative-sequence circuits, respectively.

,

Node voltage matrixes of zero- and negative-sequence circuits, respectively.

Admittance matrices of zero- and negative-sequence circuits, respectively.

Number of buses.

Positive-sequence transfer susceptance between bus and bus .

Positive-sequence conductance between bus and bus .

Positive-sequence voltage angle difference between bus and bus .

Positive-sequence injected active and reactive powers at bus , respectively.

Positive-sequence voltage magnitudes at bus and bus , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Lower and upper limits of , respectively.

Position of a member at the th iteration.

A Pareto-optimal solution at the th iteration.

Head angle of a member at the th iteration.

Search direction at iteration , calculated by polar to Cartesian coordinate transformation.

Normally distributed random number with mean 0 and standard deviation 1.

dimensional vector distributed uniformly in the range (0, 1).

dimensional vectors distributed in the range (0, 1).

Producers, scroungers, rangers.

Maximum pursuit angle.

Maximum turning angle.

Maximum pursuit distance.