C I C = C i c + C i e + C i i + C i o (1) where C _ic is the construction cost; C _ie is the cost of equipment purchase; C _ii is installation engineering fee, and C _io is other expenses.

C O C = C o e + C o h (2) where C _oe is energy consumption and C _oh is labor cost.

C M C = C m r + C m l (3) where C _mr is the cost of daily equipment maintenance and C _ml is the cost of scheduled maintenance.

The actual service life of some batteries may be shorter than the rated service life. Battery change is required to ensure the normal operation of the system. The cost can be expressed as follows: C ex = ( 1 − α ) β n C e × E ESS η (4) where C _ex is ES replacement cost, α represents the annual cost reduction rate of each battery replacement, β is the number of replacement, η is the charge and discharge efficiency.

C F C = ( c f T m + λ T f ) N 1 (5) where c _f is the average repair cost per failure per unit time, N ₁ is the number of failures, T _m is the average repair time, λ is the average penalty cost per failure per unit time, and T _f is the duration of the fault.

C ⋅ C ⋅ K ⋅ d (6) d = 1 − T ∗ K d / K 0 (7) where C _dl is the scrap disposal cost, K ₀ is initial value, d is rate of depreciation, K _d is surplus value at the end of T, and T is life cycle.

C P C = μ ( P C η c ∗ ( 1 − η c ) + P F η c ∗ ( 1 − η f ) ) (8) Where n _c = n _f is the charge and discharge efficiency, P _C = P _F is the amount of charge and discharge each time, and μ is the unit price of charge.

Peak cutting and valley filling mostly refer to ES charging during off-peak load periods and discharging during peak load periods to earn the grid price difference. Additionally, fuel costs and environmental governance costs are reduced by reducing the thermal power unit’s peak load. Therefore, the benefits of peak-cutting and valley-filling are both direct and indirect.

If ES is used exclusively for “peak cutting and valley filling,” ES income will be reduced. Hence, to reflect the ES’s “one standby multipurpose” function, this study proposes the technique of “idle time reuse,” in which ES participates in auxiliary grid service when it is not involved in peak regulation, acting as a backup power source for the grid. At the moment, the income primarily comes from the grid subsidy, which can be expressed: I t y = ∑ t = 1 T ( ∑ t = 1 365 R e P F ) (14) where R _e is the reserve compensation price of the ES, and P _F is spare the capacity.

As the load increases, new equipment is needed to upgrade the distribution network. By constructing ES power stations on the grid side that can release power during peak load, it is possible to reduce the load rate of substations and the capacity demand of the distribution network, thereby replacing the traditional power network expansion scheme. I d e l a y = C i n v × P m × ( 1 − 1 + i t 1 + r ) n n (15) where P _m is the capacity delayed power grid expansion due to ES investment and construction, that is, the rated power of ES, and C _inv is the cost of building additional capacity for the unit, RMB ten thousand/MW, i _t is the rate of inflation.

Summing up the above, dynamic lifetime Net Present Value (NPV) can be represented as: N P V = ∑ k = 1 4 { B k V 0 k − C D C k V 1 k − C I C − ∑ k = 2 k ( C P C + C O C + C M C + C F C + C ex ) k V 0 k } (16) B k = I xf + I ty + I delay (17) V 0 k = ( ( 1 + r ) g ( k ) − 1 ) / ( r ( 1 + r ) g ( k ) ) (18) V 1 k = 1 / ( 1 + r ) f ( k ) (19) where k is the total number of stages in the life cycle, which includes the construction period, trial run period, stable period, and decline period. C _IC occurs primarily in the construction period, C _OC , C _MC , C _FC , C _PC , and C _ex occurs in the trial operation period, stable period, and decline period. C _DC occurs during a recession; B _k is ES revenue. V _0k is the coefficient of the sum of the present values of stage K. V _1k is the discount coefficient of residual value at the end of stage K. r is the social discount rate. g(k) is the duration of phase k. f (k) is the operating life of the storage stage K before operation.

Project investment PP is the time required for a project to recover its initial investment in full. The longer the PP, the higher the investment risk.

If the investment project has the same net cash flow each year, the calculation formula of PP is P P = C F 0 ( C I − C O ) t (20) where PP is the static investment payback period, (CI–CO) _t is the net cash flow in year T, CI is cash inflow, CO is cash outflow, and CF ₀ is the initial investment. If the net cash flow of the investment project is not equal each year, the PP calculation formula is as follows: P P = ( T − I ) + A ( C I − C O ) t (21) where A is the unrecovered investment at the end of t-1, and T is the year when cumulative net cash flow was positive for the first time.

The dynamic PP calculation formula must meet the following requirements: ∑ t = 0 T p ( C I − C O ) t ( 1 + i 0 ) − t = 0 (22) where i ₀ is the discount rate, usually the industry benchmark rate of return.

NPV is the sum of the present value of the discounted project at the beginning of maturity. It is a method for evaluating investment proposals; if the NPV is positive, the investment scheme is acceptable; if the NPV is negative, the investment scheme is theoretically unacceptable.

The Internal Rate of Return (IRR) is the actual expected rate of return of a project investment. The bigger the IRR, the better the investment. IRR is calculated as follows: ∑ t = 0 n ( C I − C O ) t ( 1 + I R R ) t = C F 0 (23)

It is proposed in this section that the sum of the present value of the benefits of charging and discharging of an ES system in each period equal the present value of the cost of the whole life cycle of charging and discharging in each period be used to determine the economic boundary of ES (Qin, 2020). Namely: { ∑ i = 0 n N P V = 0 ∑ i = 0 n R i ( 1 + r ) i = ∑ i = 0 n C i ( 1 + r ) i (32) where R _i is the income of ES system in the i year; C _i is the i th annual cost of ES system, and n is the life of ES system.

Additionally, the sum of the discounted value of the product of the annual leveling KWH cost of ES P _dj and the current electricity production E _n is equal to the sum of the discounted value of the annual life-cycle cost of the ES, namely: ∑ i = 0 n P d j _ i × E n ( 1 + r ) i = ∑ i = 0 n C i ( 1 + r ) i (33)

The return on the investment period may be computed using Eqs 27, 28 for a certain level of electricity cost and power production E_n.

To enhance the economic border mathematical model of ES, this part develops a crucial value model for resolving the equalization KWH cost P _dj over the entire life cycle. Assuming that the leveling KWH cost P _dj remains constant during the life of the ES system, it may be calculated using Eq. 1. To enhance the economic bounder mathematical model of ES, this section develops a critical value model for resolving the equalization KWH cost P _dj in the whole life cycle. Assuming that the leveling KWH cost P _dj remains unchanged throughout the whole life cycle of ES, it can be calculated using Eq. 28: P d j = ∑ i = 0 n C i ( 1 + r ) i ∑ i = 0 n E n ( 1 + r ) i (34)

Additionally, when taken with the life-cycle cost of ES in Section 2: P d j = ∑ i = 0 n C D C k V 1 k + C I C + ∑ k = 2 k ( C P C + C O C + C M C + C F C + C ex ) k V 0 k ( 1 + r ) i ∑ i = 0 n E N × h i × η i ( 1 + r ) i (35) Where E _N is the rated capacity of ES, h _i and η I are the utilization hours and efficiency of ES in the first year, respectively.

It is assumed that the maximum iteration number of the GA algorithm is 500, the iteration termination error is 10–10, the population size is 200, and the crossover probability is 0.3 and 0.1, respectively. Annual costs and revenue are shown in Table 4. The curve of “peak cutting and valley filling” is shown in Figure 6.

It can be seen from Figure 6 that the peak value of the load curve is reduced from 205.32 to 177.24 MW through the function of ES peak clipping and valley filling, and the peak clipping rate reaches 13.68%, alleviating the peak load pressure of the power grid. When combined with Table 4, the initial investment cost accounts for 98% of the total cost, and the scrap cost includes the residual value provided by the ES battery resource recovery, which is negative, showing that the scrap cost can bring additional revenue. It has a 9.26 year investment recovery duration and a 10 year ES life cycle, indicating that the cost can be recovered within the life span.

Figure 7 shows the output value of ES after adopting the control strategy of “idle time reuse,” where standby is mainly to suppress equivalent load disturbance. The ES operation times and utilization rates of “peak clipping and valley filling” and “disturbance suppression” are shown in Table 5. Table 6 shows the annual cost, annual income, and annual net income under the control strategy of “off-time reuse.”

When the “off-time reuse” control strategy is adopted for ES, the output power of ES is consistent in the period of “peak cutting and valley filling,” as seen in Figure 7. When the system load increases, the ES discharges, and when the load decreases, the ES utilization rate effectively increases by 16.25%.

When Table 4 and Table 6 are compared, the investment and construction costs remains unchanged, while the operation and maintenance costs increase slightly, due to the standby state contributing to load disturbance suppression and increasing the number of operations. However, because the ES contributes to suppressing load disturbance during idle periods, the reserve power gain is greater than the cost. Therefore, annual net income increased by 201,200 yuan when only “peak cutting and valley filling” were used, and the investment recovery life was reduced by 1.17–8.09 years.

To summarize, by using an “off-time reuse” control method for ES, not only can annual net income be increased and the investment recovery period shortened, but also the use of ES can be effectively increased. Then, realize the dual purpose of improving ES use and grid economy, which has good research value.