## 1 Introduction
### Methods for Assessing the Economic Performance of Energy Systems with Renewable Energy Sources  
The creation and operation of renewable energy sources, as well as hybrid energy systems that include them, can be considered as an investment project; to evaluate its economic feasibility one applies standard methods [65]. The criterion to judge the performance of an investment project is the net present value (NPV):  
\[\hat{E}=\sum_{t}(R_{t}-Z_{t})(1+d)^{-t} \tag{1}\]  
Figure 2: Dependency of thermal efficiency of a gasification-based plant on its fuel consumption capacity according to [38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64] (ICE—internal combustion engine; GTU—gas turbine; FC—fuel cell). where \(R_{t}\) and \(Z_{t}\) are the outcomes and costs of the project in the year \(t\) expressed in monetary terms and \(d\) is the annual discount rate. The project outcomes are equal to the proceeds from the sale of the generated energy \(R_{t}=p_{t}Q_{t}\) (\(p_{t}\) is the price, \(Q_{t}\) is the annual sales volume). The key defining feature of solar and wind installations is the operation in an uncontrolled stochastic mode. In this regard, the following factors should be taken into account when evaluating the performance of wind turbines and analyzing the feasibility of their adoption:  
* Dependency of output on probabilistic characteristics;
* Operation in an energy system that includes backup energy sources;
* The ability to accumulate generated energy and its subsequent use during periods of idling operation. For preliminary evaluations and comparisons of energy sources of different types, the cost of energy (in particular, electric power) is used [9; 10; 65]. The cost of energy is by definition equal to such a fixed price (\(p_{t}=p=\text{const}\)) that yields the NPV (1) equal to zero. Then, it follows from (1) that:  
\[p=\sum_{t}Z_{t}(1+d)^{-t}/\sum_{t}Q_{t}(1+d)^{-t} \tag{2}\]  
The cost of energy (2) is the minimum price at which the energy supply project is still feasible (\(\hat{E}=0\)); it is equal to per-unit costs (the ratio of total discounted costs to total discounted energy output). The best energy source is deemed to be the one that ensures the lowest cost of energy. In the simplest case, the Formula (2) can be reduced to the form [65]:  
\[p=\frac{k}{CF\cdot H}(CRF+\mu)+\frac{p_{F}}{8.15\cdot 10^{3}\eta} \tag{3}\]  
Here, \(k\) is specific investment, \(CF\) is the capacity factor, \(H=8760\) h/yr, \(CRF\) is the capital recovery factor, \(\mu\) is annual fixed costs (as a percentage of capital expenditures), \(p_{F}\) is the fuel price (tons of coal equivalent), and \(\eta\) is the efficiency. The last summand for wind and solar energy sources is zero, as the price of fuel for them is zero (they consume a free resource). If necessary, Formula (3) can be transformed to take into account the terms of construction, operation, and dismantling of the energy source, co-generation of electricity and heat, energy losses and auxiliary energy costs, the escalation of the fuel price over time, and other factors [65]. The energy cost criterion is obtained by assuming that all of the energy produced is delivered to the consumer. Renewable energy sources with a stochastic mode of operation almost always operate in the energy system, which, by backing them up with controlled energy sources and storage batteries, makes it possible to provide a reliable energy supply to consumers. In hybrid energy supply systems, estimates of the cost of energy should be supplemented by taking into account the system effects that arise due to the presence of interconnections between energy sources of different types and energy storage facilities [65]. Such system effects are usually taken into account by applying dedicated mathematical models. Mathematical modeling of energy systems with renewable energy sources has been the subject of a vast body of research [66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78].