Optimal Power Flow (OPF) is the process of determining the dispatch schedule of power generators with minimum cost while satisfying the system constraints like the upper and lower power and reactive power generation limit, upper and lower voltage level limit, and line flow limit. The minimization of the generation cost will result to a lower cost of electricity paid by the consumers.

A sample transmission line with 4 busses odes), 4 generation plants, and 4 loadsThe Optimal Power Flow ProblemThe Optimal Power Flow Problem is not an easy optimization problem because it involves the non-linear characteristics of the transmission systems where the real and reactive power flows, and the nonlinearity of the cost functions.

In actual OPF problem, speed in calculations is a great concern. Note that a practical system is composed of several hundreds to several thousands of busses or nodes. The Luzon grid, for example, is composed of not less than 600 busses; and to solve the power flow of a transmission system with 600 busses, it will require to build an approximately 1200 x 1200 Jacobian Matrix. That is why it is necessary to make some approximations in the formulation process. Most OPF solutions approximate the AC power flow into DC power flow. In DC power flow, the speed of calculations will increase, however, some important parameters like voltage angle and magnitude, and line resistances are approximated.

The Problem Formulation

The objective function, Cost Function, C, is a non-linear function, which is a function of power, and sometimes, length of down time and start-up of generators. Our objective in this problem is to minimize the cost function or the cost of generation while satisfying all accompanied constraints. However, since the cost function is non-linear, we will approximate the cost function using piece-wise linear approximations.

The piece-wise linear approximation of the cost function for Generator iThe formulation of the problem for linear programming is given by the following sets of equations;Minimize

Subject to:

, Real power output constraint

, Reactive power output constraint

, upper and lower voltage limit

, line flow limit

, the total generated power must be equal to the total load plus energy loss at all times

Where;

Sij is the cost of electric energy generated by generator i at output level j in $/MWh

Pij is the generated energy by generator i at level j in MWh

Pimin, Pimax, the minimum and maximum output power of generator i in MW

Pi is output power of generator i in MW

Qimin, Qimax, the minimum and maximum output reactive power of generator i in MW

Qi is output reactive power of generator i in MW

Vkmin, Vkmax, is the lower and upper voltage limit at any bus k in KV

Vk is the voltage at bus k in KV

Slmax is the maximum line flow at line l in MW

Sl is the line flow in line l in MW

Ploss is the total transmission line loss in MWh

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Filed in: Optimization and Economic Dispatch