Optimization of daily operation mode of photovoltic systems of enterprises

At present, one of the most important issues at the level of various manufacturing enterprises is the development of ways for building and efficiently using renewable energy sources, especially solar energy. In this regard, this article proposes a mathematical model of the problem of determining the optimal daily mode of operation of a photovoltaic system in an enterprise with a private photovoltaic system connected to the electrical system. The effectiveness of the mathematical model was analyzed on the example of determining the optimal mode of operation of the photovoltaic system located on the territory of Jizzakh Polytechnic Institute. It is shown that the mathematical model can be used to determine the optimal operating modes of existing photovoltaic systems in enterprises and distribution networks, as well as to determine the optimal parameters in their design.


Introduction
Today, the sharp increase in energy demand in the world and the corresponding decrease of fuel reserves have made the use of renewable energy sources one of the most important issues. In addition, the significant damage to the environment caused by the use of fossil fuels is one of the key factors determining the efficiency of the use of renewable energy sources. Therefore, at present, several works are being carried out in the Republic of Uzbekistan on the construction and operation of solar power systems using renewable energy sources. Effective construction and operation of such devices require the determination of their optimal parameters in advance based on the use of appropriate mathematical models and algorithms.
In recent years, many publications on the development of photovoltaic power generation technologies, their rational use, and their optimal integration into power systems have appeared. They are important for the development of research and practical work in this area. However, the methods and algorithms for determining the optimal parameters of photovoltaic plants taking into account all the influencing and limiting factors, cannot be considered sufficiently developed. One of the problems is determining the optimal load schedule for a power system for consumers with a private photovoltaic system. Today, photovoltaic devices that generate electricity from solar energy are widely introduced in many enterprises worldwide. The capacity of a photovoltaic solar station can be determined depending on the solar radiation in the area, the ambient temperature, and the specific characteristics of the module as follows [1]: N is number of modules; s is solar radiation level; U nl is no-load voltage; I sh is shortcircuit current; k is filling coefficient; k V is voltage coefficient; k A is current coefficient; T mod is modul temperature; T A is ambient temperature; T cell is cell temperature; T cell.n is nominal operating temperature for each cell.
In such photovoltaic systems, the batteries can be used to ensure an uninterrupted power supply [2][3][4]. On the other hand, the use of batteries can lead to a sharp increase in the cost of construction and operation of photovoltaic plants. Therefore, the necessity of using batteries in such devices is determined based on solving the relevant optimization problems [2,3,5,6,7].
Under the current conditions, enterprises will also be connected to the power grid to use the photovoltaic plant wisely, increase the reliability of the power supply and choose economically viable options. Given the fact that the definition of energy received from the power system varies throughout the day, the company should develop its own optimal daily load schedule. The decision of such a problem in general cases requires solving the problem of optimization, taking into account the simple, functional, and integral constraints in the form of equality and inequality.
An analysis of available literature shows that there are no rigid methods and algorithms for solving such problems based on the use of appropriate mathematical models. [4] considered the problem of optimization of an autonomous power supply system based on solar and wind energy. In [5,6] proposed the mathematical models and methods for solving the problem of optimization of the location and parameters of battery-powered solar photovoltaic power plants in distribution networks. However, using them directly to effectively solving the problem discussed in this article poses relevant challenges. In [8] described an algorithm for optimization of consumer load schedules, provided that it is possible to regulate loads at individual intervals over a given period without changing the amount of electricity required for that period. It can only be used to solve a problem in special cases. Therefore, developing a mathematical model and algorithm for solving the problem of optimization of consumer load schedules with own power plants operating on renewable energy sources, taking into account all the influencing and limiting factors, remains an important task [9]. Below a mathematical model of the problem of optimization of the daily load schedule of an enterprise with a photovoltaic plant and its solution algorithm are proposed.

Methods
In addition to the solar modules that generate electricity, the photovoltaic plant installed in enterprise includes additional equipment such as power control devices (controllers) generated at the photovoltaic station, inverters, a monitoring system monitors the parameters of the station operation mode. This equipment can also include batteries for the rational use of generated electricity. Figure 1 shows a schematic diagram of a consumer with a photovoltaic station connected to an electric power system (EPS). In solving this problem, the electric capacity of the battery W, the consumer load schedule P L (t) for the planned day, the power schedule produced at the photovoltaic station P R (t), the minimum and maximum limit charging powers of the battery (corresponding to the negative discharge capacity) are known data. The power schedule of the consumer photovoltaic station P G (t) is determined by solving the optimization problem P G (t), the consumer load schedule P PS (t) relative to the EPS, and the battery charging/discharging power schedule P B (t) is determined as a result of solving the optimization problem.
The objective function is a function of the costs associated with generating electricity by photovoltaic plant and EPS during the day, which should be minimized: Here t C is a unit cost of energy received from the system; (t) PS P is an active power from power system; (t) G P is an active power generated by photovoltaic plant;  is specific cost of electricity generated by photovoltaic plant, which includes operating costs [10]. Boundary conditions include: -active power balance in enterprise for each hour of the day: -the condition that for each hour of the day, the power of the photovoltaic plant is equal to the sum of power consumed to charge the battery and power output from the inverter, ie -inequality on the minimum and maximum power that can be produced every hour of the day (determined by the power of inverter): -inequality in the energy capacity of the battery: -inequality on the ability to charge and discharge of battery every hour of the day: Here: In this model, the specific cost, which depends on the operating costs of electricity generated by the photovoltaic plant, is assumed to be constant regardless of the power generated.
Since the objective function and all boundary conditions are linear in nature, the optimization problem based on the formed mathematical model can be solved using the linear programming method -the simplex method.

Results and Discussions
The efficiency of the proposed mathematical model and algorithm for solving the problem under consideration were researched on the example of optimization of loading schedule of Jizzakh Polytechnic Institute, which has a private photovoltaic power plant using solar energy. Table 1 shows the load schedule of the consumer and the daily capacity of the photovoltaic plant and changes in electricity tariffs during the day. The electric capacity of the battery is W = 24 kWh. Residual energy from the previous day in the battery W bal = 0. Maximum charging/discharging capacity of battery P B.max = 4 kW. The unit cost of electricity generated by a private photovoltaic plant is b = 0.088 $ / kWh when all costs are taken into account.  C t , $/kWh 0.14 0.14 0.14 0.14 0.14 0.12 0.09 0.09 Table 2 and Figure 2 the optimization results using the proposed model and the selected optimization method (Gurobi), i.e., the optimal schedules of active power from the photovoltaic plant P G (t) and the active power from the power system P PS (t), as well as the optimal battery charge P ch (t) or discharge P dch (t) are given. Here, the charging capacity of the battery is positive, and the discharging capacity is negative. The total daily cost of electricity from a photovoltaic plant and power system is $ 308,705. Table 2. Optimal daily load charts in the case where the photovoltaic station has a battery. To compare the optimization results, Table 3 shows the optimization results for the case where the photovoltaic plant does not have a battery. In this case, the daily cost of electricity from the photovoltaic plant and the power system is $ 309.12.  Thus, in the example considered, optimization of the load schedule of EPS and photovoltaic plants for the consumer with photovoltaic plant in the presence of battery reveals that the total daily cost of electricity is 1.3% lower than in the absence of a battery. The relatively low efficiency due to battery presence is determined by factors such as the amount of charge and the small size of the photovoltaic power plant, the lack of change in the definition of electricity during the day.
In the example considered, the power schedule produced by the photovoltaic plants was adopted in deterministic form. In general, such schedules may be probabilistic or partially uncertain. Under such conditions, optimization based on the proposed mathematical model and method can be effectively solved based on the use of the algorithms presented in [11][12][13][14].

Conclusions
A mathematical model and method for solving the problem of optimization of load schedules of consumers with their own photovoltaic plants were proposed.
It has been shown that significant economic efficiency can be achieved by optimization the consumer's own photovoltaic plant and EPS load schedules in the face of changing the tariff of electricity throughout the day. The proposed mathematical model and method of optimizing the load schedules of consumers with a private photovoltaic plant can be used to solve the problem of optimal planning of daily loads of enterprises with such plants. The proposed model and method can also be used to select optimal photovoltaic plants' optimal parameters to be installed in the territory of large power consumers.