The strength of the power grid system on one bus can be evaluated using SCR. When RES is connected to the grid system at the bus i, the SCR at the bus can be defined in (1) (IEEE guide for planning DC, 1997). The tradition SCR is suitable for measuring the system strength of a single machine grid connected system, but it cannot accurately evaluate the system strength of a multi machine system because it does not consider the mutual influence between each grid connected unit (Xiao et al., 2019). S C R i = S a c , i P d , i = U i 2 P d , i • 1 Z i (1) where S a c , i is the system short-circuit capacity at bus i , P d , i is the power injected at bus i , U i is the voltage at bus i , and Z i is the Thevenin equivalent impedance at bus i .

When the reference voltage in the system is the rated voltage U N of the bus and the reference power is the rated power P d , N , the reference impedance of the system Z b is as follows. Z b = U N 2 P d , N

Substituting Z b into Equation 1, when the system uses the rated value as the reference value, the SCR is inversely proportional to the per unit value of the equivalent impedance, that is S C R i = 1 Z i Z b = 1 Z i , p u

Generally speaking, the larger the S C R i , the stronger the system at bus i , as the voltage at bus i is far from its voltage stability limit (Wu et al., 2017; IEEE guide for planning DC, 1997). The value of CSCR is 3 (Kim et al., 2022; Hadavi et al., 2021), and if SCR is greater than 3, the power grid is considered strong at that location, which means that the power grid can accommodate more renewable energy and energy storage devices at this point while maintaining voltage stability. If the SCR value is between 2 and 3, the power grid is considered weak at that point. If the SCR is less than 2, then the power grid is considered very weak at that location.

The SCR defined in (1) only considers the impact of a single RES output power, but the interaction between multiple RESs in the system can affect the system strength. SDSCR was proposed in (Wu et al., 2017) to consider the effects of interactions between multiple RESs. SDSCR is defined as: S D S C R i = U R , i 2 P R , i + ∑ j ∈ R , j ≠ i Z R R , i j Z R R , i i U R , i * U R , j * P R , j Z R R , i i where P R , i , P R , j are the RES output power of bus i and j , Z R R , i j is the i , j t h element in node impedance matrix Z R R , R is the set of the buses connected with RESs.

SDSCR considers the interaction between RESs, expanded the definition of SCR to make it more accurate in evaluating system strength, and provides reference for optimizing the location and capacity of RESs. However, SDSCR did not consider the interaction between ESDs and RESs, and can only reflect the system strength when the power of the power equipment in the system is given. It cannot dynamically reflect the impact of changes in power equipment capacity on the system strength. Therefore, this paper proposes a novel SCSCR index that considers the impact of power devices with different capacity proportions on the strength of the bus in the system.

The multiple RESs and ESDs are connected to the grid as shown in Figure 1. ESD plays an important role in flexible adjustment of RESs (Wang et al., 2021). When ESD operates in the discharged state, it can output energy to the system and can be considered as a power source. The network equation can be expressed as: U G U R U E = Z G G Z R G Z E G Z G R Z R R Z E R Z G E Z R E Z E E I G I R I E where U G , U R , and U E are voltage vectors on the synchronous generator bus, RES bus, and ESD bus; Z G G , Z G R , Z R R , Z R E , and Z E E are the corresponding blocks of the bus impedance matrix; I G , I R , and I E are current vectors connected to the buss of SG, RES and ESD. The voltage at bus i connected to RES can be expressed as: U R , i = ∑ m ∈ G Z R G , i m I G η m + ∑ k ∈ E Z R E , i k I F η k + ∑ j ∈ R Z R R , i j I R η j (2) where the U R , i is the i t h element in U R ; η m is the current proportion of the m t h device in I G ; η k is the current proportion of the k t h device in I E ; η j is the current proportion of the j t h device in I R ; Z R G , i m is the i , m t h element in Z R G ; Z R E , i k is the i , k t h element in Z R E ; Z R R , i j is the i , j t h element in Z R R ; G is the bus set connected to SG, E is the bus set connected to ESD, and R is the bus set connected to RES.

Equation 2 can be transformed into U R , i − U G , m Z R R , i m = I e , i = S e , i U R , i * (3) where U G , m = ∑ m ∈ G Z R G , i m I G η m I e , i = ∑ j ∈ R Z R R , i j Z R R , i i I R η j + ∑ k ∈ E Z R E , i k Z R R , i i I E η k S e , i = U R , i I e , i * = U R , i ∑ j ∈ R Z R R , i j Z R R , i i I R η j + ∑ k ∈ E Z R E , i k Z R R , i i I E η k * = ∑ j ∈ R Z R R , i j * Z R R , i i * U R , i U R , j S R η j + ∑ k ∈ E Z R E , i k * Z R R , i i * U R , i U R , k S E η k (4)

Equation 3 is the same power flow equation for a dual bus system, as shown in Figure 1. S e , i , U G , m and Z R R , i m respectively represent the equivalent input power, voltage source, and Thevenin impedance at bus i . From Equation 4, it can be seen that in Figure 2, S e , i includes the output power of RES in the system, i.e., S R , and the output power of ESD, i.e., S E . Therefore, S e , i can reflect the interaction effect of the output power of RESs and ESDs located on different buses of the system.

According to Wu et al. (2017), the boundary conditions for voltage stability in power systems can be derived as U R , i cos ⁡ θ G m , R i U G , m = Re U R , i U G , m = 1 2 (5) where θ G m , R i = θ G m − θ R i , is the phase difference of voltages U G , m and U R , i . Substitute Equation 3 into Equation 5: U R , i 2 ∑ j ∈ R Z R R , i j Z R R , i i U R , i * U R , j * S R * η j + ∑ k ∈ E Z R E , i k Z R R , i i U R , i * U R , k * S E * η k Z R R , i m = 1 (6)

In the distribution network with multiple new energy sources connected, we defined the SCSCR based on Equation 6 to quantify the strength of the system. SCSCR can reflect the distance to the voltage stability boundary.

When ESD operates in a discharge state, SCSCR is represented as: S C S C R i = U R , i 2 ∑ j ∈ R Z R R , i j Z R R , i i U R , i * U R , j * P R η R , j + ∑ k ∈ E Z R E , i k Z R R , i i U R , i * U R , k * P E η E , k Z R R , i m (7) ∑ j ∈ R ∑ k ∈ E η R , j + η E , k = 1 P R + P E = P where P R and P E represent the total output power of RESs and ESDs, η R , j and η E , k represent the capacity proportions of each RESs and ESDs, and P represents the total output power of RESs and ESDs in the system.

It can be seen from (7) that SCSCR is a generalized form of SCR defined in (1), when only one RES is connected to the power system, the power of other RESs and ESDs in the equation is zero. At this time, the expression of SCSCR is simplified to the same expression as SCR in (1), which means that SCR is actually a special case of SCSCR in the case of a single RES connection. In addition, SCSCR retains the same physical interpretation as SCR, which evaluates system strength based on the distance between the system and the voltage stability limit.

To validate the efficacy of the multi-source collaborative optimization technique proposed for distribution network planning utilizing SCSCR, we employed the IEEE 9 bus model to simulate a RES and ESD integrated distribution network. As depicted in Figure 4, bus 1 serves as an idealized voltage source, replicating the power system with adjustable frequency and voltage capabilities. The branch parameters are presented in per-unit values, the base voltage V B is 230 kV and the base power S B is 100 MW. Following 5 buses as potential buses: 2, 3, 5, 6, and 8 are chosen as the RES potential buses, ESD potential buses are 2, 3, 4, 5, 6, 7, 8, and 9.

This section further validates the effectiveness of the proposed method using the IEEE39 bus system, as shown in Figure 6. The system is divided into two regions for optimization, and bus 30 is used as the interconnection point with the power grid. The interaction between power devices clustered in the two regions is considered, and the optimal locations and capacities of RESs and ESDs in the two regions are obtained. The potential buses for RESs are 8, 31, 32, and39 in region 1 and 33, 34, 35, and 36 in region 2. The potential buses for ESDs are 8, 10, 11, 13, 31, and 33 in region 1 and 16, 19, 21, 22, 23, 33, and 35 in region 2.

When the total capacity of four RESs is fixed at 200 MW, RESs can be installed at different buses with different capacity proportions. The values of MSCSCR are shown in Table 3, and MSCSCR has the highest value in case 1, indicating that the optimal proportion of the four RESs is η R , 8 = 10 % , η R , 39 = 10 % , η R , 33 = 40 % , η R , 35 = 40 % , The optimal installation locations are bus 8, 39, 33, and 35.

Assuming that RES accounts for 80% of the total capacity and two ESDs accounts for 20%, four RESs are installed at bus 8, 39, 33, and 35 with a capacity proportion of η R , 8 = 10 % , η R , 39 = 10 % , η R , 33 = 40 % and η R , 35 = 40 % , respectively. By changing the ESD capacity proportion under each ESD installation situation, it can be seen from Table 4 that when two ESDs are installed at bus 8 and 19 with η E , 8 = 3.13 % and η E , 19 = 16.87 % respectively, MSCSCR is maximized.