The output model of the photovoltaic station (Yang et al., 2019) is as follows: p s = η p S A p 1000 , (1) where p _s is the power of the photovoltaic station, kW; η _p is the comprehensive power generation efficiency, considering the power degradation of the polysilicon PV module, such as the cable, inverter, and transformer, about 75–82% (Q CPI 173-2015, 2015); S is the solar radiation intensity, W/m²; and A _p is the installation area of the photovoltaic cell array, m².

The output model of the wind station (Zhang, 2020) is given as follows: p w = { 0 v ≤ v c r p r v − v c r v r − v c r v c r ≤ v ≤ v r p r v r ≤ v ≤ v c o 0 v > v c o , (2) where p _w and p _r are the actual power and the rated power of the wind turbine generator unit, kW, and V _cr , V _r , and V _co are the cut-in speed, rated speed, and cut-out speed of wind, m/s.

The output model of the hydropower station is as follows: p h = 9.81 η h Q H , (3) where p _h is the hydropower output; η _h is the power generation efficiency of the hydraulic turbine generator unit; Q is the volumetric flow rate of the hydropower station, m³/s; and H is the head of the hydropower station, m.

The system is operated in a compensation mode using hydropower and pumped storage power for compensation and the energy storage thereof for electricity quantity regulation. Assuming the difference between the total output of photovoltaic, wind and hydro power, and the load as the net load, the formula for net load calculation is l n i = L i − p s i - p w i - p h i , (4) where l n i and L i are the net load and the target load, respectively, kW.

The net load is also the output required for energy storage. When the net load l n is greater than 0, the energy storage unit supplies power; when the net load l n is less than 0, the unit stores power. The superscript i represents the time period.

The energy storage model is s i + 1 = s i − t i n i . (5)

When storing energy, { n i = η i n l n i | l n i | ≤ C i n i n i = − η i n C i n i | l n i | > C i n i . (6)

When supplying energy, { n i = l n i η o u t l n i ≤ C o u t i n i = C o u t i η o u t l n i > C o u t i , (7) where s i is the energy stored in the unit at the beginning of a period, kWh; n i is the virtual energy storage power, kW; t i is the time duration; η i n and η o u t are the efficiency of energy storage and energy supply, respectively; and C i n i and C o u t i are the maximum energy storage power and the maximum energy supply power of the unit in period i, kW. C i n i and C o u t i are determined based on factors such as allowable minimum output and installed capacity of the hydropower and pumped storage power. The energy storage and energy supply process is first realized by changing the way of hydropower station output. Pumped storage power continues to bear the remaining net load when the output or energy storage capacity is insufficient.

Limited by installed capacity, there will be power abandoned due to insufficient energy storage power. The power of such abandonment is expressed as w 1 i = n i - l n i , (8) where w 1 i is the power of abandonment due to insufficient energy storage power, kW.

When the energy storage capacity is used up, power abandonment will also occur. The power of abandonment is the difference between the actual storage power and the planned storage power. The formula for calculation of the power of abandonment is as follows: w 2 i = ( n ′ i − n i ) / η i n , (9) n ′ i = ( s i − s i + 1 ) / t i , (10) where w 2 i is the power abandoned after the energy storage capacity is fully used, kW; n i and n ′ i are the planned energy storage power and the actual energy storage power, kW.

At the end of the period, n ′ i will be smaller than n i when the energy storage capacity is used up.

The energy storage capacity of a pumped storage station is determined by the elevation difference between the upper and lower reservoirs and volume of the upper reservoir, while that of hydropower station is determined by the reservoir capacity and water regulation requirements.

The total output of the system is the sum of the photovoltaic, wind, and hydro output, plus the output of the energy storage unit, subtracting the abandoned power, expressed as P i = p s i + p w i + p h i + n i − w 1 i − w 2 i , (11) where P i is the total system output in period i, kW.

Under fixed installed capacity of multi-energy source of the base, simulation operation can be performed according to the output process of various sources. Thus, the power generation process of the system can be obtained and the output indexes such as the power supply guarantee rate and the power abandonment rate can be calculated, thereby establishing the relationship between the installed capacity and project benefits.

The power supply guarantee rate is the rate of the number of periods during which the system meets the load to the total number of operation periods when the system matches the design load requirement, represented by γ and expressed as γ = n N , (12) where n is the number of periods during which the total output of the system matches the load, and N is the total number of periods for calculation.

The power abandonment rate is the rate of the sum of the power abandoned due to the limitation of installed capacity and energy storage capacity to the natural power output, represented as η , expressed as η = ∑ i [ ( w 1 i + w 2 i ) × t i ] ∑ i [ ( p s i + p w i + p h i ) × t i ] . (13)

Based on the base system model, an optimization model is constructed to optimize the configuration of the installed capacity of various energy sources. The model consists of optimization variables, objective function and constraints, where the objective function is calculated from the optimization variables.

Swarm Intelligence shows outstanding performance in terms of solving problems about complex optimization of various combinations and function optimization (Vitousek et al., 1997). Particle swarm optimization (PSO) is considered an efficient and simple global optimization algorithm for its simple structure, few adjustment parameters, and easy implementation. It has achieved good results in optimization featuring multi-peak, multi-objective, and constraint. ASA, proposed in recent years (WANG et al., 2017) inspired by sheep flock behavior, simulates the behavior of the leading sheep and interaction among the remaining sheep and designs corresponding global exploration and local development operators in the swarm intelligence algorithm. ASA has brought improvement in both effect and efficiency of intelligent optimization.

In order to further improve the optimization quality of ASA, especially to improve its defect of falling into local optimum, the ASA was improved herein. The operators escaping local optimum similar to shepherd dog supervision mechanism were introduced to improve the quality of each iteration (Qu et al., 2018). The improved artificial sheep algorithm (IASA) was thus proposed. A specific optimization process based on IASA is shown in Figure 2, where the part marked by dotted lines was determined and operators escaped local optimum.

Optimization calculations by traditional PSO, original ASA, and IASA were conducted with 10 typical benchmark test functions (Wang et al., 2018) to deal with a minimum value problem, and 30 repetitions were performed. The results are compared in Table 1 with the average convergence value and the standard deviation of the convergence results. The optimization results showed that for functions 3, 4, and 8, equivalent optimal solutions have been obtained, so the performance of the three algorithms was on the same level. For functions 7 and 10, the performance of the original ASA and IASA were equivalent and much better than that of traditional PSO. For the other five functions, the IASA had outstanding performance compared with the other two algorithms.

The stability of the algorithm was also analyzed using the standard deviation of the optimization results. Compared with PSO and ASA, the standard deviation (SD) of IASA was slightly higher for functions 1 and 7, but for all other functions, the standard deviation was more favorable than the other two algorithms. Calculations with IASA show better stability during the optimization process.

Taking two test functions as examples, both Figure 3 and Tables 2, 3 show the optimization convergence process with these three algorithms. From the trend of the curve and the objective function value evolution, IASA demonstrates a better optimization efficiency and faster calculation convergence than the other two algorithms.

From the above analysis, it can be concluded that, compared with the classic PSO and the original ASA algorithms, the main advantage of the IASA algorithm is that a faster and more stable convergence can be obtained. This innovative algorithm has been improved as per the shortcomings of PSO and ASA such as early convergence and local optimum problems.

The Yellow River area was selected as a typical case to study the installed capacity allocation optimization. The base includes four types of power sources: hydro, pumped storage, photovoltaic, and wind power. The annual average unit output process of photovoltaic and wind stations was determined based on NASA meteorological data and the installed capacity and output of the hydropower station were determined based on the river basin planning. In this case, the installed capacity of the hydropower station was 2,000,000 kW and not considered an optimization variable, with its output process varying from month to month. Figure 4 shows the process for output per kW of the installed capacity of the photovoltaic and wind stations and the output of the hydropower station within a year.

Figure 5 shows the daily load process of the DC transmission channel adopted. The process is divided into peak periods from July to August and the off-peak period from September to next June. The late peak of the peak period is 1 h longer than the off-peak period, and loads of such two periods alternate within the year, with an annual average load of 2,100,000 kW.

Technical and economic indexes of various power sources of the base are shown in Tables 2, 3.

The variation of output indexes under different allocations of installed capacity was analyzed for the base. as given in Figure 6when increasing the installed capacity of the photovoltaic station from 5000 to 30000 MW while maintaining that of other stations, the power abandonment rate showed a trend of increasing from 0.28 to 39.20%, and the power supply guarantee rate showed a trend of increasing from 73.21 to 97.74%. When increasing the installed capacity of the wind station from 1000 to 10000 MW while maintaining that of other stations, the power abandonment rate showed a trend of increasing from 8.52 to 17.39%, and the power supply guarantee rate showed an increase from 84.63 to 94.39%. As for the installed capacity of the pumped storage station, when increased from 1000 to 10000 MW while maintaining that of other stations, the power abandonment rate showed a trend of decreasing from 15.66% down to 11.48%, meanwhile, the power supply guarantee rate increased from 85.44% up to 93.00%. It indicates that increasing the installed capacity of photovoltaic or wind stations independently can improve the power supply guarantee rate but increase the power abandonment rate at the same time, while the energy storage unit plays a positive role in reducing power abandonment and ensuring power supply by storing electric energy at low load and compensating for power supply at high load.

The optimization model established was used to analyze the capacity allocation, in which the variables were installed capacities of photovoltaic, wind, and pumped storage power, and the objective function was a minimum total investment of the base.

The power abandonment rate was set with a constraint threshold step size of 0.5%, the rate varying between 13 and 20%, while the power supply guarantee was set with a constraint threshold step size of 0.5%, the rate varying between 90 and 94%. In such variation intervals, optimization tests were carried out repeatedly with the same optimization calculation parameters to obtain the most economical initial investment under different constraints. The initial investment was the least when the power abandonment rate was the maximum and the power supply guarantee rate was the minimum. Such minimum investment was taken as a reference value to get the ratio of the optimal investment under other constraints to the reference value, which indicated that the initial investment would gradually increase along with the gradual increase in power supply guarantee rate or gradual decrease of power abandonment rate.

Table 4 shows the calculation results of the ratio of the optimal investment under different constraints to the reference value. It indicates that when the power supply guarantee rate is fixed at 90%, the optimal power abandonment rate can be controlled at a level of 18%, and the total investment of the base is only increased by about 1% compared to the reference value; when there is a slight increase of power supply guarantee rate, the investment increase significantly. Based on this, a power abandonment rate of 18% and power supply guarantee rate of 90% were selected for the final optimization calculation.

When the power abandonment rate is small or the power supply guarantee rate is large, the initial investment in the base is relatively high. When the power abandonment rate is large or the power supply guarantee rate is small, the initial investment is relatively low.

To analyze the influence of these two constraints on the higher investment, a comparison was made for variation of investment correlated with power supply guarantee rate when the power abandonment rate was the minimum and that correlated with power abandonment rate when power supply guarantee rate was the maximum. Similarly, to analyze the influence of these two constraints on the lower investment, a comparison was made for variation of investment correlated with power supply guarantee rate when power abandonment rate was the maximum and that of investment correlated with power abandonment rate when power supply guarantee rate was the minimum. The comparative analysis curve is shown in Figure 7, which indicates that under the same variation rate of 0.5% for the gradual increase of power supply guarantee rate and gradual decrease of power abandonment rate, the variation of power supply guarantee rate leads to a faster increase of investment, that is, the initial investment is more sensitive to the power supply guarantee rate.

The cumulative histogram in Figure 8 shows the correlation between the installed capacity allocation of three types of energy in the optimal capacity allocation scheme and the power abandonment rate under the case of a fixed power supply guarantee rate of 90% and varying power abandonment rate. When constraint on power abandonment rate was decreased from 20 to 13% in the optimal scheme, the proportion of installed capacity of photovoltaic power tended to decrease, while the proportion of installed capacity of both wind and pumped storage power tended to increase, and installed capacity proportion of the three power sources gradually tended to balance. However, when the power abandonment rate was fixed and the power supply guarantee rate varied, there was no obvious law in the variation of the installed capacity proportion of the three power sources. It qualitatively shows from another perspective that the power supply guarantee rate has greater influence on the initial investment than the power abandonment rate.

The general PSO and the IASA were used for multiple trial calculations under a power abandonment rate of 18% and power supply guarantee rate of 90%, obtaining optimization results as shown in Table 5 and Figure 9. Table 4 compares the installed capacity allocation and corresponding investment obtained by the two optimization methods, indicating that the total investment of the base obtained by the IASA is lower and the installed capacity allocation for various energy sources is more reasonable and elaborate.

Figure 9 gives the change of investment of various energy sources and the total investment of the base corresponding to the installed capacity allocation obtained by the two optimization methods, indicating that the IASA produces lower total investment of the base and better project economic benefits. With the installed capacity of hydropower as the benchmark, the final installed capacity proportion of photovoltaic, wind, pumped storage, and hydropower is 4.6:1.4:1.7:1.