Page 29 - Mathematical Models and Algorithms for Power System Optimization

Daily Economic Dispatch Optimization With Pumped Storage Plant 19

2.2.3 Way of Processing Integer Variables for Pumped Storage Plant

Setting the power consumption under the pumping condition P pp is a discrete integer variable:
the power consumption for pump water P pp should be an integer variable based on practical
needs, which should be selected within nine options such as 0, 165, 330, 495, 660, 825, 990,
1155, or 1320 (MW). If such nine options are taken as 0–1 variables, then there should be nine
0–1 variables in each time period. That is, 24D time periods require 9 24D variables, 9 24D
(D ¼l, 2, 3), which means if P pp should be represented by 0–1 variable, the number is up to
216–648. Such a huge number of integer variables, together with thousands of continuous
variables, makes optimization a very complex mixed-integer problem. There should be some
other ways to deal with the integer variables as follows:

P pp tðÞ ¼ 0,165,330,495,660,825,990,1155,1320MW:

(1) Let Y p represent an integer variable with the range of 0–9 and with the unit step of 1. When
Y p ¼1, then P pp is 165, and when Y p ¼2, then P pp is 330. Therefore, the number 165 can be
used as the coefficient of variable Y p (t), and the following equation can be used to
represent P pp (t):

P pp t ðÞ ¼ 165Y p t ðÞ,Y p t ðÞ ¼ 0 9:

(2) Setting the operating period of pumped storage plant: as the pumping power and
generation power may be alternatively changed in a very short time, this model does not
consider its continuous operation time limit in the calculation process. Based on the
changing tendency of the load curve, the operating period of a pumped storage plant can be
divided into three periods in advance based on the limits for Y p (t) and P pg (t): storage
period [the upper limit of Y p (t) is not 0], generation period [neither the upper nor lower
limit of P pg (t) is 0], and stop period [the upper limit of Y p (t) is 0, and the upper and lower
limits of P pg (t) is 0].

2.3 Formulation of the Problem

Based on the considerations for the optimization modeling previously mentioned, as well as the
processing methods for objective function, constraint conditions, continuous variables, and
integer variables, and by setting up notations for various variables, the mathematical model of
daily multiarea economic dispatch with a pumped storage could be easily formulated. To
illustrate the correctness and feasibility of a developed mathematical model, it is necessary to
form the basic structure of a constraint matrix based on the mathematical model and provide an
example of the small-scale mathematical model on which verification calculation can be
performed. In the following sections, the details of notations, mathematical models, and basic
structure of constraint matrixes are given.