Optimization of the power consumption mode of pumping stations of “Suv Okova” by reactive power

The optimal modes of the existing compensating devices under operating conditions were determined. Minimum power and energy losses were taken as optimality criteria. The unconstrained optimization problem was considered, i.e., finding the absolute minimum. When solving the problem to find the optimal mode of operation of the “SUV OKOVA” network, a relative minimum was found since the system of constraints has a nonlinear form. The problem of conditional optimization in terms of reactive power is solved, for which the gradient method of quadratic programming is used.


Introduction
In recent years, the rapid development of the power industry has posed new challenges for optimizing reactive power in an active distribution network. Traditional stochastic optimization and robust optimization techniques can lead to overly conservative or overly risky decisions when dealing with renewable energy uncertainties. In [1,2], a distributionally robust reactive power optimization model for an active distribution network was proposed; it considers network losses as an aim of optimization and considers reactive power support and switch reconfiguration. A data-driven two-step reactive power optimization model with stable distribution is developed. The model of hybrid optimization of several renewable energy sources used to evaluate the primary data, the development of a load profile, and the determination of the optimal option for the system with the lowest cost was discussed in [3,4]. Methods for determining the size of distributed generation units (DG) and the assessment of optimal locations in electrical distribution networks were proposed in [5][6][7]. To connect DG, some basic optimization principles were introduced, and examples of optimization of the size and location of DG units were presented.
A formalized mathematical description of the optimization problem was considered in [8][9][10]. Constraints in the mathematical description represent various technical, economic, environmental conditions taken into account when solving the optimization problem [9][10][11][12]. Mathematical programming is, as a rule, a repetitive computational procedure leading to the desired optimal solution [9][10][11][12]; for finding an extremum point, it is impossible or very difficult to use analytical methods of solution, in that case, a nonlinear programming method is used to solve optimization problems. As a rule, when solving problems by nonlinear programming methods, numerical methods are used [11][12][13][14][15][16][17][18][19]].

Reactive power balance equation
The total power of the compensating devices (СD) required in the electric power system (EPS) is determined from the reactive power balance equation. This power must be placed in the nodes of the electrical network with minimal costs.
Where Q G  is the total reactive power generated in the EPS, including the reactive power transmitted from the neighboring EPS; When a compensating device ( 0 k Q  ) is installed for the consumer, these losses decrease to the following value: Thus, the compensation of reactive power allows us to reduce the losses of active power in the power supply circuit and, consequently, to improve the technical and economic indicators of this circuit.
Impact of CD on network costs. Let us estimate the impact of СD on network costs. The expression for the total costs of transferring power to the load during the installation of СВ has the form: where З К is the cost for СD; с о ΔР are the costs of covering active power losses in the network; с о is the cost of a unit of lost active power; з к are the unit costs for СD.
To determine the minimum of function 3, we equate its derivative of the variable Q K to zero: The economically feasible reactive power is determined from (4); its transfer from the source to the consumer meets the minimum costs З.
The value of Q Э does not depend on the active power P but depends only on the ratio of the cost indicators з к and с о and the parameters of the network U and R through which the power is transmitted.
Location of CD in power systems. The issue of the location of compensating devices in the electrical network of a real EPS is a complex optimization problem. The difficulty lies in the fact that electric power systems are large systems of interconnected subsystems. It is impossible to consider each individual subsystem since the properties of large systems are determined by the nature of the interconnections of individual subsystems.
In the analysis of large systems, a systematic approach is used [18,19], according to which the analysis of a large system is performed by dividing it into subsystems that are not directly related to each other but affect each other through a higher-level system. Compensating devices can be installed on the buses of higher (jQ кв ) and lower (jQ кс ) voltage of transformers T, and on the buses of 0.4 kV of distribution transformers T1 and in the 0.4 kV network itself (jQ кн ). The value of the СD power is to be determined.
In general, the problem of optimizing the CD location is formulated as follows: to determine the reactive powers in the nodes 6 ... 35 kV of the synchronous machines G, the power of СD in the networks of all voltages Q кв , Q кс , Q кн , and the values of the reactive powers Q эi (i=1, 2, …n), transmitted in the consumers' networks, which ensure the minimum total costs.
Calculations of reactive power compensation for networks of all types are carried out both when designing the development of electrical networks and their operation conditions. In the process of designing, the СD power is determined, and the problem of their distribution in the electrical network is solved. Under operating conditions, the optimal modes of the existing СD are determined within 24 hours. Optimality criteria, in this case, are the minimum power and energy losses and the compliance of voltage deviations with permissible values.
When designing a power supply scheme, the monetary costs for this scheme are minimized as a rule. Reducing power losses due to the installation of СD reduces the costs of the circuit, for the following reasons: • every lost kW of power was generated at power plants and, therefore, money was spent on it; • generation of lost reactive power at power plants is much more expensive (by 3 times!) than consumption.
However, compensating devices are also expensive.

Results and Discussion
Determination of the optimal power of CD. In this regard, the problem arises of determining the optimal power of compensating devices that meets the minimum total costs. This problem belongs to the problem of unconstrained optimization and can be solved, for example, by gradient methods. Consider such a problem for the main power supply circuit (Fig.3). It is necessary to determine the power of the compensating devices Q К1 and Q К2 in nodes 1 and 2 based on the condition of the minimum total costs for the installation of these devices and the coverage of active power losses in the circuit.
Initial data: circuit voltage U; resistance of lines R 1 and R 2 ; reactive loads of nodes 1 and 2 Q 1 and Q 2 ; unit costs for the installation of compensating devices z o ; unit costs to cover active power losses с о . The target function, which is the total costs of installing compensating devices and covering active power losses in the circuit, has the following form where а 1 =R 1 •c o •10 -3 /U 2 =0.0006; а 2 =R 2 •c o •10 -3 /U 2 =0.0004. The introduction of a numerical coefficient 10 -3 is necessary to reduce all components of the target function to one dimension (c.u.).
To solve the problem, we choose the coordinate descent method. Let us define the partial derivatives of the target function Z concerning the variables Q 1 and Q 2 : The initial approximation is taken as: (8) For these values, we calculate the target function and its partial derivatives. We assume that the objective function Z decreases more strongly in the direction of the variable Q k2 than in the direction of the variable Q k1 , i.e. 2 1 .
The descent starts in the direction of the variable Q k2 .
Let us take the step size  =400 kVAr. The first approximation (first step) is Q k1 1 =0, Q k2 1 =400 kVAr. We calculate the value of the target function Z 1 . The second step is Q k1 2 =0, Q k2 2 =400 kVAr. We calculate the value of the target function Z 2 . The descent along the Q k2 coordinate should be continued until Z n <Z n-1 . As soon as Z n becomes greater than the previous value of Z n-1 , the descent along the Q k2 coordinate should be stopped and returned to the values of the variables Q k1 n-1 and Q k2 n-1 , obtained at the n-1 step.
Let us take a new step towards another variable Q k1 . The new value of the target function Z is found. The descent along this variable continues in the same way as in the direction of Q k2 -until Z m <Z m-1 .
The point with the obtained coordinates Q k1 m-1 , Q k2 n-1 is in the vicinity of the minimum of the target function Z. With the set step length  = 400 kVAr, a more exact solution cannot be obtained. To obtain a more exact solution, it is necessary to decrease the step and continue the descent. It is absolutely certain that the less the step, the more exact the result. We cannot achieve this accuracy through hand calculation. To solve this problem, it is advisable to use software designed to solve the problem of nonlinear programming with nonlinear constraints. One of these programming languages is C ++. This was the consideration of the problem of unconstrained optimization, i.e., finding the absolute minimum. When solving the problem, to find the optimal operating mode for the "SUV OKOVA" network, it is necessary to find a relative minimum since the system of constraints has a nonlinear form. Thus, the problem of conditional optimization in terms of reactive power is posed, for which we apply the previously selected gradient method of quadratic programming.