Page 92 - Mathematical Models and Algorithms for Power System Optimization

82 Chapter 4

4.1 Introduction

Power flow calculation is used to calculate the steady-state characteristics of a power system
under a given condition, which is widely applied in power system operation and planning
calculations. There are many excellent power flow algorithms; Newton-Raphson (N-R) and PQ
decomposition methods are the ones most widely used in current engineering practices. Both
methods are very effective in conventional power flow calculations in terms of calculation
speed and convergence characteristics. However, in some special circumstances, such as power
flow calculation in ill conditions and at a high R/X ratio, the problem of nonconvergence may
still occur. The improvement of convergence characteristics remains a common concern that
deserves further study.
The nonlinearity of power and voltage expressions makes power flow calculation a nonlinear
problem, which is usually solved by the iterative method. In general, the existing power flow
algorithms are nonlinearoneswith nonobjective and unconstrained.Itisnecessary to seta specified
valueatthebeginningofthecalculation.Forexample,PandVneedtobeassignedforPVbus,Pand
QneedtobeassignedforPQbus,andVandθ needtobeassignedforVθ bus.Inaddition,transformer
tap “T” and number of capacitor banks "C" need to be specified. To obtain a feasible solution, these
settings need to be repeatedly adjusted during iterations until solutions for V and θ are within
appropriate limits. This is almost an impossible task for those without prior calculation experience.
For those problems that are difficult to solve by conventional power flow, this chapter
reformulates the power flow model. Based on the introduction of objective function, the
simulated annealing (SA) algorithm is used to provide a new solution to the problem of ill-
conditioned power flow with difficult convergence in engineering practice. Based on the
introduction of constraint function, mixed-integer programming (MIP) is used to solve the
discrete optimal power flow (OPF) problem.

This section first summarizes the traditional power flow problem-solving approach from the
perspective of equation solving and provides a basic reference system for two power flow
algorithms (unconstrained power flow algorithm with objective and constrained power flow
algorithm with objective) in this chapter.

4.1.1 Way of Processing Variables in Traditional Power Flow Equation

The formulation of traditional AC power flow is as follows:
X

P Gi P Li U i U j G ij cosθ ij + B ij sinθ ij ¼ 0
j2i
X

Q Gi Q Li U i U j G ij sinθ ij B ij cosθ ij ¼ 0
j2i
For a network with N buses, each bus has four operation variables (V ¼voltage magnitude,
θ¼voltage phase angle, P¼active power, and Q¼reactive power). In power flow calculation,