Page 45 - Mathematical Models and Algorithms for Power System Optimization

Daily Economic Dispatch Optimization With Pumped Storage Plant 35

period 1; L 2 —load in time period 2; S I , S A —output limits of the power plant in time period 1, 2;
H I , H A —generated energy limits of the power plant in total time periods; A I , A A —generated
output limits of area in time period 1, 2; W I ,W A —limits of water quantity of the pumped
storage plant in time period 1, 2; W 0 –W 24 —balancing value of water quantity of the pumped
storage plant in total time period; X it —generated output of unit i, continuous variable, the
number of variables¼the number of units the number of time periods, t¼1, 2; X jt —generated
output of unit j, continuous variable, the number of variables¼the number of units the
number of time periods, t¼1, 2; X gt —generated output of the pumped storage plant, continuous
variable, the number of variables¼the number of pumped storage plants the number of
time periods, t¼1, 2; Y pt —pumping output of the pumped storage plant, integer variable, the
number of variables¼the number of units for pumped storage plant the number of time
periods, t¼1, 2; t 2 —virtual variable.

(1) Constraint for load balancing in time period 1; X i1 +X j1 165y p1 ¼L 1
(2) Constraint for load balancing in time period 2; X i2 +X j2 165y p2 ¼L 2
(3) Constraint for power plant output in time period 1; S I X i1 +X j1 S A
(4) Constraint for power plant output in time period 2; S I X i2 +X j2 S A
(5) Constraint for area output in time period 1; A I X i1 +X j1 A A
(6) Constraint for area output constraint in time period 2; A I X i2 +X j2 A A
(7) Constraint for generated energy of the power plant in total time periods;

H I X i1 +X j1 +X i2 +X j2 H A
(8) Constraint for reservoir capacity in time period 1; W I 1.36X g1 165y p1 W A
(9) Constraint for reservoir capacity in time period 2;
W I 1.36X g1 +1.36X g2 165y p1 165y p2 W A
(10) Constraint for water storage of the pumped storage plant in total time periods;
1.36X g1 +1.36X g2 165y p1 165y p2 ¼W 0 W 24

The small-scale system shows that the proposed setting of cost coefficient, variables, and
constraint conditions are correct.

2.6.2 Scale of the Practical System

The practical scale system is comprised of 6 areas, 1 pumped storage plant with 4 pumped
storage units, and 27 other power plants, which is a total of 70 hydro and thermal (nuclear)
plants, and purchased units, with a daily load curve divided into 24 time periods. According to
the optimization model, the scale of the mathematical model consists of 743 constraints, 1680
continuous variables, 24 integer variables, and 7368 nonzero elements, with nonzero element
proportion in the constraint matrix of 0.58%. Therefore, the proposed model is a highly sparse
one. The scale of variables of the practical scale system is shown in Table 2.4.