Introduction To Optimal Power Flow

(1) Introduction

In the basic load flow problem, the so-called control variables are specified, permitting the feasibility of solution. The control variables consist of real power generation at all generation buses. The specification of these control variables is based on several considerations. The first major consideration is that of satisfying all power demand within the acceptable tolerance for voltage levels, without violating the limits on generation levels, transmitted currents, and powers (equality and inequality constraints).

In this case, one selects those control variable values which will minimize (or maximize) a desired performance index. One possible performance index can be the total losses in the transmission network. Another one can be the cost of generation needed to meet the demand. A third performance index may reflect a combination of operating cost, security considerations, and possibly pollution levels. In economic dispatching approach (optimization of fossil fueled power systems has been on economic operation only) inequality constraints on power flows and voltages are normally ignored, while accounting for generation real power limits and transmission line losses.

As will be seen below, the problem in this case simplifies to that of static optimization, since the cost of generation is instantaneously related to the heat rate of input energy. A more complicated problem is the optimization of system performance over period of time. Here, several factors have to be taken into consideration. Some of these are:

-    The hourly commitment of units, i.e., the decision whether a unit is on or off at a given hour. This is normally referred to as the unit commitment problem.

-    The hourly productions of hydroelectric plants based on the flexibility obtained by the manipulation of water reservoir levels to improve performance. This is normally preferred to as the hydrothermal coordination problem.

-    The hourly production of-generation, and/or dispersed plants, like solar photo voltaic or wind generation pants. This is referred to as the dispersed generation problem.

-    The scheduling of units maintenance without violating the needs for adequate reserve capacities while minimizing the cost of production. This is referred to as the maintenance-scheduling problem.

For each of the above cases, one optimizes over a practical time range. For example, in the unit commitment problem, the practical time range is in the range of 48 to 72 hours. In hydrothermal coordination, the time range will  vary from a day to a week, a month and finally, a year in maintenance scheduling. Typical time intervals can range from one to three years.

(2) Economic Dispatching

Economic dispatching (ED) refers to classical approaches to the economic operation of power systems. A key element here is the proper modeling of power plant efficiency. Normally, one model the heat rate input to the boiler as a function of output real power. The measured heat rate curve can be quite complex depending on the value positions of the steam turbines. The heat rate characteristics is also dependent on thermodynamic parameters like ambient dry and wet bulb temperatures, operating pressure, water pumping etc. this applies not only to the characteristics curve itself but also to upper and lower limits on generation. Power utilities normally utilize single heat rate curve with upper and lower limits on generation.

The electric power is generated as a result of mechanic rotational energy produced by either steam or combustion turbines. Steam produced in the boiler is the medium of heat transfer to the turbines.

In fossil (fuel-fired steam units, fuel is burnt and energy is released in the form of heat in the boiler, producing high temperature and pressure steam. The steam is led to the turbines where the thermal energy is transformed into mechanical form. The turbine drives the electric generator (alternator). The exhaust of the turbine is cooled in the condenser, and the resulting water is pumped back to the boiler. Our interest in economic operation studies is the input (the fuel cost) in MJ/h or MBTU/h or Kcal/h, output (the active power generation of the unit) in MW, Fig.(1-1) shows a typical thermal input-output curve (c-p). The discontinuities in the cost curves occur as a result of the sharp increases in throttle losses, these points at which a new steam admission value is opened and are called value points.

Fig. (1-2) shows the operating cost Ci of fossil-fuel generating unit, versus the real power output Pi (PGi). The curve is approximated usually by a quadratic polynomial of the form: 

PGi is the MW output of the generator i and  are constant 

coefficients of the fuel cost function.

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Ahmad Atiia

Hi. I’m Designer of Engineering Topics Blog. I’m Electrical Engineer And Blogger Specializing In Electrical Engineering Topics. I’m Creative.I’m Working Now As Maintenance Head Section In An Industrial Company.

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