OPTIMIZATION PROBLEMS OF ELECTRIC POWER SUPPLY Томский политехнический университет

2 When designing and operating electrical systems often engineers have to deal with the multiple-choice tasks, i.e. with problems in which out of some set of admissible solutions for technical reasons it is required to choose one which best suits any criterion. This solution is called optimal, and tasks to be searched for such a solution are called optimization problems. With regard to power supply systems, optimization problems have to be solved selecting power grid voltage, number and capacity of power sources, optimal configuration of power grid, sections of the conductors, determination of rational distribution of reactive power sources, installation sites of power supplies, etc.

3 Томский политехнический университет As criteria of optimality in most practical problems of power supply economic indicators (cost, profit, financing costs, etc.) are used, although in some cases may be used and others: minimum voltage losses (power), reliability of power supply, power quality, etc. Thus, optimality criterion is a quantitative evaluation of the optimized object quality. Based on the selected optimality criterion, objective function is made, which is the dependence of optimality criterion on parameters that affect its value. Optimality criterion or objective function is defined by a specific optimization problem. Accordingly, optimization purpose is to find the extremum of objective function.

4 Томский политехнический университет In general, mathematical model of optimization problem consists of three basic components: objective function, restrictions and boundary conditions. Objective function is a mathematical notation of optimality criterion: Z(x 1, x 2, …., x n ) → extr (3.1) where x 1, x 2, …., x n – unknown variables that must be determined in the process of solving the problem. Restrictions are a variety of technical, economic and other conditions that must be taken into account when solving the problem: f j (x 1, x 2, …., x n ) (≤ or =) b j, where j = 1,2, … m (3.2) Boundary conditions are the variation range of the unknown variables: d i ≤ x i ≤ D i, where i = 1,2, … n (3.3) d i, D i – the lower and upper boundary of the variation range of variable x i, respectively.

5 Томский политехнический университет In order to solve optimization problems special mathematical techniques and methods are used, which are called methods of mathematical programming. In accordance with the relation between variables when expressing objective function, optimization problems are classified into problem of linear programming and nonlinear programming. In addition, by variation unknown variables can be continuous, integral or discrete. Accordingly, optimization problem that contain integral or discrete variables are divided into problems of integral or discrete programming.

6 Томский политехнический университет If in a mathematical model consisting of optimization problem the objective function is linear but set in which extremum of the objective function is sought is defined by the system of linear equations and inequalities, then it’s an LP problem. The purpose of linear programming is to minimize (maximize) linear function: Z = c 1 x 1 + c 2 x 2 + … + c n x n (3.4) n variables x 1, x 2, …, x n, satisfying non-negativity conditions x 1  0, x 2  0, …, x n  0 (3.5) and m linear restrictions a 11 x 1 + a 12 x 2 + … + a 1n x n  (=,  ) b 1, a 21 x 1 + a 22 x 2 + … + a 2n x n  (=,  ) b 2, (3.6) ……………………………………… a m1 x 1 + a m2 x 2 + … + a mn x n  (=,  ) b m.

7 Томский политехнический университет Solution of nonlinear programming problems fundamentally does not differ from solving linear and integer programming. However, search procedure for solving nonlinear programming problems is more critical to the original initial data. To solve the problems of nonlinear programming in Excel two methods are implemented: Newton's method and method of conjugate gradient Fletcher-Reeves. Selection of solution method is made in the dialog box "Solution Settings". As a breaking criterion to find a solution in Excel the following condition is used: (3.7) The value of ε is entered into the box "Solution Settings" in the line "Relative error".

8 Томский политехнический университет Transportation problem (TP) is a special type of linear programming problem and is defined as the task of developing the most economical plan for transportation of products of one species from several departure points to destination. Value of transport costs is directly proportional to the volume of transported goods and is given by the rates for transportation unit. Problems of transportation type are common in practice. They reduce many problems of linear programming - assigning problems, grid, scheduling, etc. Although TP can be solved by any method of solving linear programming problem and its mathematical model and the structure of restrictions have a number of specific features.

9 Томский политехнический университет Standard TP is as follows. There are m points of origin (or point of production) А i …,А m, which concentrate the reserves of homogeneous products equal to a 1,...,а m units. There are n destinations (or points of consumption) В 1,..., В m, whose need for these products is b 1,..., b n units. There are also transport costs С ij, associated with transportation of one unit from the point A i to В j, i 1,2…, m; j = 1,2..., n. It is required to make up such a transportation plan (where and how many units to transport) in order to meet the demand of all consumption points by means of disposal of all units manufactured by all the production points with a minimum total value of all shipments.

10 Томский политехнический университет Let х ij – number of units delivered from point A i to В j. Minimizing total costs of transportation from all production points to all consumption points are expressed by the formula: → min(3.8) Thus, objective function of TP is transportation cost for the implementation of all shipments in general. A mathematical model of TP also contains two groups of restrictions.

11 Томский политехнический университет The first group of restrictions indicates that goods reserves at any departure point should be equal to the total volume of shipments from this point., where i = 1. …., m(3.9) The second group of restrictions indicates that the total shipments to some point of consumption should fully meet the demand for the products at this point., where j = 1. …., n(3.10)

12 Томский политехнический университет Transportation volumes – non-negative number, since shipments of units from consumption points to the production points are excluded: x ij 0, i 1,..., m; j 1,..., n.(3.11) From (2.8) and (2.9) it follows that the sum of reserves in all departure points should be equal to the total needs in all consumption points, that is (3.12) If the condition (3.12) is performed, then TP is called balanced (closed model), otherwise – unbalanced (open model). As restrictions of the model TP (3.9), (3.10) can be performed only at balanced TP, then when designing transport model it is necessary to check the balance condition (3.12).

13 Томский политехнический университет In the case when total reserves exceed total needs, it is required an additional fictitious point of consumption, which will formally consume excessive reserves, that is (3.13) If total demand exceeds total reserves, then an additional fictitious point of departure is needed, that formally compensates the lack of existing products at points of departure: (3.14)