Power System Optimization in Terms of Uncertainty of Initial Information

In the article the problem of power system optimization under partial uncertainty of initial information is considered. The possibilities of using the methods of determinization to solve the problem of power system optimization taking into account the regime and technological constraints in the form of equality and inequality are researched. In basis of performed calculation studies using the example of the test scheme of power system, it was found that the optimization algorithm by the method of determinization can be effectively used for this proposes.


Introduction
The problem of optimization of power system development and functioning is a difficult problem of nonlinear mathematical programming with many limiting conditions in the form of equality and inequality imposed on adjustable parameters and on some of their functions. These problems are also characterized, like similar problem for any large system, by the presence of various kinds of initial information. Currently, the initial data used to solve such problems can be divided into four types: deterministic, probabilistic, fuzzy, and interval. The deterministic data gives a complete picture of the state of object, which meets the requirements of this problem. The last three types of initial data are uncertain data. With a probabilistic initial data, the laws of distribution of probabilistic quantities in known ranges of their possible values are usually set. Fuzzy initial data suggests their task by formulated linguistic distribution of possible parameter values. In the case of interval initial information, the intervals of possible values of the initial parameter (or state) are usually known, and no laws are known about their possible values (or states) in this interval.
Currently, the methods and algorithms for solving power system optimization problems in the conditions of deterministic initial data are sufficiently developed [1][2][3][4][5][6][7]. The solution of the problem in conditions of probabilistic initial data is carried out by probabilistic-statistical methods. They provide for the use of the mathematical expectation functions of the objective function and the constraint functions of probabilistic quantities. In this case, the calculation algorithms do not differ much from the algorithms for deterministic initial data. The optimization problems in the conditions of fuzzy initial data are solved by methods of the theory of fuzzy sets [8,9]. In this paper, we consider the problem of Power system optimization in the conditions of interval (partial) uncertainty of the initial parameter. In the existing literature, there are no an advanced algorithms for optimization of the development and functioning of power system, taking into account all the constraints in conditions of partial uncertainty of the initial information. In [10][11][12], the efficient algorithm of power system optimization in terms of partial uncertainty of initial information is described. The algorithm is based on the sequential solution of deterministic problem for the values of initial parameter selected in a given interval and select the best plan for various criteria. In this case, preference is given to the criteria of minimax and minmax risk. At the same time, in these works, the questions of taking into account of constraints in the form of inequality remains opened. The algorithms proposed in [13] for taking into account inequality constraints during optimization under the conditions of probability and partial uncertainty of the initial information do not completely overcome the noted difficulties. In some cases, the determination of a conditionally optimal plan, where all the specified constrains are satisfied for all the possible values of the initial parameter in a given range, requires additional complex calculations. In [14][15][16] an algorithm of optimization under conditions of interval uncertainty of the initial parameter by the method of determinization, taking into account the constraints is described. This paper presents the results of study of possibility of using this algorithm to solve problems of optimization of a development and functioning of power system.

The optimization algorithm
We consider the possibilities of using the algorithm of optimization in terms of uncertainty by determinization method for solving the problem of power system development optimization, which is formulated as follows: minimize the function of estimated costs taking into account -simple constraints imposed on adjustable parameters (2) -functional constraints in the form of equality and inequality imposed on some functions of adjustable parameters where l j g , f are specified values of the constraint functions.
At partial uncertainty of initial parameters, when their minimum and maximum possible values are known, all functions, in particular, the objective function and the constraint functions, are also partially indefinite. Then the problem described above is reduced to minimizing the functions of the estimated costs determined at the minimum and maximum possible values of the initial partially indefinite parameters, as in [15]   Solution of the obtained optimization problem (1a) -(4b) reduces to solving the equivalent two deterministic problems. In the first problem, the objective function is represented in the form (1a), and the constraints consist of (2), (3a) -(4b). In the second problem, the objective function form is (1b), and the constraints are (2)

The results of calculation experiments
To study the effectiveness of the described algorithm, we consider its application for solving the problem of power system mode optimization using the example of the test circuit (Fig. 1), given in [4] for given interval (partially undefined) values of active power of load nodes 2, 3, and 4. Minimum and maximum possible values of loads in these nodes are given in table 1. The active power of node 5 is deterministic and P5 = 500 MW.      In the problem under consideration, the constraint in the form of equality is a condition for the active power balance in power system, therefore, the left-hand sides of the constraints in the form (3a) and (3b) are the same. When translating the components associated with the interval loads of the nodes in constraints in the form of inequality on power flows in the controlled NPLs, the left-hand sides of the constraints (4a) and (4b) also become the same. In this case, in each deterministic problem, one constraint in the form of equality takes part. In the first problem, the righthand side of the constraint is equal to the lower boundary value, and in the second, to the upper boundary value of the total load of power system specified in the interval form.
Since the left-hand sides of the constraints in the form of inequality obtained from (4a) and (4b) are the same, one of these constraints is involved in both deterministic problems, the right-hand side of which is less than the second. In this case, the second constraint with a large right-hand side turns into a passive restriction.
Thus, the first deterministic optimization problem, for the considered example, according to the described algorithm has the following form: The objective function min P , P , P , P , P , It should be noted that the first and second deterministic problems, for the considered example, differ from each other only by the right-hand side of the constraint in the form of equality.
In  For the final solution of the problem, according to the above marked rules, we take any point whose coordinates are determined by the powers of the plants in the received intervals. In this case, the lower and upper boundary values of power plant power intervals at nodes 1, 6, and 7 were the same. And the power of the balancing plant can be taken as the average value in the obtained interval. Thus, we take the optimal power of plants as

Conclusion
The problems of power electric system development and functioning optimization taking into account simple and functional constraints in the form of equality and inequality in conditions of partial (interval) uncertainty of initial parameters, can be effectively solved by the method of determinization.
In power system mode optimization on active power and presenting the expression of power flow over a controlled PTL using linear distribution coefficients, solving the problem described by the method of determinization is much easier by halving the number of functional constraints in the form of equality and inequality compared to the classical optimization algorithm in basis of determinization method.