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New Algorithms Related to Power Flow 83

there are generally three types of buses: PQ, PV, and Vθ. For each bus, there are only two
equations (P balance and Q balance), but there are four variables. Therefore, for each bus, two
of its variables are handled as known conditions, whereas the other two are to be solved. In the N
bus system, there is a balance bus (Vθ), MPV buses with 2M equations, and (N – M – 1) PQ
buses with 2(N – M – 1) equations. Thus, there is a total of 2(N – 1) equations and 2(N – 1)
variables. In this way, the number of variables equals the number of equations, of which a
Jacobian matrix is a nonsingular square matrix.

In N-R method of power flow calculation, the polar coordinate matrix of the linearization of AC
power flow calculation model is as follows:

ΔP HN Δθ
¼
ΔQ ML ΔV=V

4.1.2 Way of Processing Variables in New Power Flow Equation

In traditional power flow calculation, V and θ are independent variables, and P and Q are
derived variables. If P and Q are added to the variables of a traditional power deviation
expression, and let X represent includes variables V, θ, P, and Q, then the calculation model of
traditional AC power flow can be expressed as:

F 1 XðÞ ¼ 0
F 2 XðÞ ¼ 0

Where F 1 (X) and F 2 (X) are the active and reactive power deviations, respectively. This is a
completely different solutions (4n unknows for 2n equations), from traditional power flow (2n
unknows for 2n equations), which is a new way to solve the power flow problem. This chapter
provides two new methods to solve the power flow problem. One is to use the SA method by
developing a suitable expression of the objective function. The other is to use the OPF method
by developing a suitable expression of the objective function and constraints.

4.1.3 Overview of Unconstrained Power Flow with Objective Function
(based on SA Method)

The power flow calculation of power systems is used to mathematically solve a set of
multivariate nonlinear equations. The basic principles upon which the existing algorithms
applied are inseparable from the iterative process. In these algorithms, the modified step length
of each iteration process is derived from the previous step. The iterative process approaching to
the final solution is generally monotonous, that is, when the iterative process is a downhill form
of monotonous drop, the solution converges; when the iterative process is an uphill form of
monotonous rise or zigzag, it is difficult to obtain a convergent solution.