Impact of Grid Strength and Impedance Characteristics on the Maximum Power Transfer Capability of Grid-Connected Inverters

: Continuously expanding deployments of distributed power generation systems are trans-forming conventional centralized power grids into mixed distributed electrical networks. The higher penetration and longer distance from the renewable energy source to the main power grid result in lower grid strength, which stimulates the power limitation problem. Aimed at this problem, case studies of inductive and resistive grid impedance with different grid strengths have been carried out to evaluate the maximum power transfer capability of grid-connected inverters. It is revealed that power grids with a higher short circuit ratio (SCR) or lower resistance-inductance ratio (R/X) provide higher power transfer capability. Moreover, under the resistive grid conditions, a higher voltage at the point of common coupling (PCC) is beneﬁcial to increase the power transfer capability. Based on mathematical analysis, the maximum power curves in the inductive and resistive grids can be found. Moreover, a performance index is proposed in this paper to quantify the performance of the system with different parameter values. Finally, the effectiveness of the analysis is veriﬁed by simulation.


Introduction
In the past decades, due to a foreseen exhaustion of conventional fossil-based energies and their climate impact, traditional centralized power generation using fossil fuels has been increasingly considered unsustainable in the long term. Consequently, many global efforts have been devoted to developing more renewable energy sources, such as wind and solar photovoltaics (PVs) [1]. Thus, the proportion of distributed power generation systems has increased quite fast, especially in Europe [2]. Moreover, due to the fact that the best locations to get wind and solar energy are often far from the main grid, the long-distance transmission-line impedance makes the grid weaker than before, which brings a significant challenge to effective power transmission [3]. Although a high voltage direct current (HVDC) system can replace the alternating current (AC) system for long-distance power transmission [4,5], the grid-side converter of the HVDC system still faces the challenge of connecting to a weak grid in some cases [6,7]. Therefore, it is necessary to study the maximum power transfer capability of grid-connected inverters.
The strength of the grid is defined by the short circuit ratio (SCR), which is the ratio of the short circuit power at the point of common coupling (PCC) and the rated power of the inverter. When the SCR is below 6-10, the grid is weak. In the case that the SCR is above 20, the grid is strong [8]. According to the definition of the SCR, the increase of the rated inverter power or the transmission impedance will reduce the SCR and make the grid weaker. The PCC voltage varies easily when a large amount of active power is injected into a weak grid. Thus, the reactive power control has to be used to keep the PCC voltage Figure 1 shows the typical configuration of grid-connected wind and PV generation systems. Generally, a wind generation unit contains an AC/DC converter for maximum power point tracking (MPPT) control and a DC/AC inverter for grid connection. Similarly, a PV generation unit mainly contains a DC/DC converter for MPPT control and a DC/AC inverter for grid connection. In the entire wind or PV power plant, all these generation units are connected at the PCC. The total output power of inverters at the PCC is injected into the high-voltage transmission network connected to the main grid. To simplify the analysis, the transmission network is represented by a transmission line, as shown in Figure 1. In the following sections, this paper will mainly focus on the impact of transmission-line impedance on the maximum power transfer capability of the system. following sections, this paper will mainly focus on the impact of transmission-line impedance on the maximum power transfer capability of the system. Grid-connected inverters usually operate in current-controlled mode, so the whole generation power plant can be treated as an ideal current source. Moreover, the grid can be represented by its Thevenin's equivalent circuit; the equivalent circuit of the grid-connected generation system is shown in Figure 2. Assuming the PCC voltage vector is aligned to the d-axis, Vpcc∠0 is the vector of the PCC voltage, Iinv∠φi is the vector of the inverter current, Vg∠φv is the vector of the grid voltage, and Zg is the grid impedance. It is noted that series-connected resistor Rg and inductor Xg are used to represent grid impedance Zg.
PCC Z g =R g +jX g Wind or PV power plant The stiffness of the grid at the PCC can be described by the SCR, which is expressed as: where SSC is the short-circuit apparent power at the PCC, SN is the total rated apparent power of inverters, Vg is the amplitude of grid voltage, Vpcc(rated) and Iinv(rated) are the amplitude of the rated PCC voltage and rated inverter current. Generally, the rated PCC voltage should be equal to the grid voltage, namely, "Vpcc(rated) = Vg", and the rated current Iinv(rated) is the same as the maximum inverter current Iinv(max), which is determined by the endurable current of the power semiconductor. Thus, the grid impedance can be represented by the SCR according to Equation (1), which is expressed as: According to the equivalent circuit in Figure 2, the relationship between the grid voltage and the inverter current can be derived as: Grid-connected inverters usually operate in current-controlled mode, so the whole generation power plant can be treated as an ideal current source. Moreover, the grid can be represented by its Thevenin's equivalent circuit; the equivalent circuit of the grid-connected generation system is shown in Figure 2. Assuming the PCC voltage vector is aligned to the d-axis, V pcc ∠0 is the vector of the PCC voltage, I inv ∠ϕ i is the vector of the inverter current, V g ∠ϕ v is the vector of the grid voltage, and Z g is the grid impedance. It is noted that series-connected resistor R g and inductor X g are used to represent grid impedance Z g .
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 15 following sections, this paper will mainly focus on the impact of transmission-line impedance on the maximum power transfer capability of the system. Grid-connected inverters usually operate in current-controlled mode, so the whole generation power plant can be treated as an ideal current source. Moreover, the grid can be represented by its Thevenin's equivalent circuit; the equivalent circuit of the grid-connected generation system is shown in Figure 2. Assuming the PCC voltage vector is aligned to the d-axis, Vpcc∠0 is the vector of the PCC voltage, Iinv∠φi is the vector of the inverter current, Vg∠φv is the vector of the grid voltage, and Zg is the grid impedance. It is noted that series-connected resistor Rg and inductor Xg are used to represent grid impedance Zg. The stiffness of the grid at the PCC can be described by the SCR, which is expressed as: where SSC is the short-circuit apparent power at the PCC, SN is the total rated apparent power of inverters, Vg is the amplitude of grid voltage, Vpcc(rated) and Iinv(rated) are the amplitude of the rated PCC voltage and rated inverter current. Generally, the rated PCC voltage should be equal to the grid voltage, namely, "Vpcc(rated) = Vg", and the rated current Iinv(rated) is the same as the maximum inverter current Iinv(max), which is determined by the endurable current of the power semiconductor. Thus, the grid impedance can be represented by the SCR according to Equation (1), which is expressed as: According to the equivalent circuit in Figure 2, the relationship between the grid voltage and the inverter current can be derived as: The stiffness of the grid at the PCC can be described by the SCR, which is expressed as: where S SC is the short-circuit apparent power at the PCC, S N is the total rated apparent power of inverters, V g is the amplitude of grid voltage, V pcc(rated) and I inv(rated) are the amplitude of the rated PCC voltage and rated inverter current. Generally, the rated PCC voltage should be equal to the grid voltage, namely, "V pcc(rated) = V g ", and the rated current I inv(rated) is the same as the maximum inverter current I inv(max) , which is determined by the endurable current of the power semiconductor. Thus, the grid impedance can be represented by the SCR according to Equation (1), which is expressed as: According to the equivalent circuit in Figure 2, the relationship between the grid voltage and the inverter current can be derived as: To simplify the vector expression in Equation (3), the current vector I inv ∠ϕ i can be expressed by the d-axis component "i d = I inv ·cos(ϕ i )" and the q-axis component "i q = I inv ·sin(ϕ i )". Thus, the relationship between the two current components and the PCC voltage can be derived as:

Ideal Maximum Power without Current Limitation
Firstly, an ideal case without current limitation is analyzed as follows. Substituting "V pcc = V g " and Equation (2) into Equation (4), the per-unit values of i d and i q can be derived as: Considering Equation (5) as a quadratic equation of i q(pu) , thus, only when Equation (6) is satisfied, does Equation (5) have solutions.
According to Equation (6), the maximum value of i d(pu) can be calculated, as given in Equation (7). Because the PCC voltage is considered a constant in this section, the per-unit value of the current i d is equal to the per-unit value of the inverter active power P inv . Hence, the ideal maximum power of the inverter can be represented by Equation (7).
According to Equation (7), the relationship between the ideal maximum inverter power P inv(pu)max and the SCR with different R g /X g ratios is shown in Figure 3. It can be seen that the relationship between P inv(pu)max and the SCR is linear. In the purely inductive (R/X = 0) case, SCR = 1 is a critical point because when the SCR is lower than 1, the maximum power P inv(pu)max is lower than 1, namely, the maximum output power of the inverter is limited naturally under weak grid conditions. Hence, this natural characteristic between P inv(pu)max and the SCR can be considered a standard to be compared with the maximum power operating point at the stability boundary of different control strategies. To simplify the vector expression in Equation (3), the current vector Iinv∠φi can be expressed by the d-axis component "id = Iinv•cos(φi)" and the q-axis component "iq = Iinv•sin(φi)". Thus, the relationship between the two current components and the PCC voltage can be derived as:

Ideal Maximum Power without Current Limitation
Firstly, an ideal case without current limitation is analyzed as follows. Substituting "Vpcc = Vg" and Equation (2) into Equation (4), the per-unit values of id and iq can be derived as: Considering Equation (5) According to Equation (6), the maximum value of id(pu) can be calculated, as given in Equation (7). Because the PCC voltage is considered a constant in this section, the per-unit value of the current id is equal to the per-unit value of the inverter active power Pinv. Hence, the ideal maximum power of the inverter can be represented by Equation (7).
According to Equation (7), the relationship between the ideal maximum inverter power Pinv(pu)max and the SCR with different Rg/Xg ratios is shown in Figure 3. It can be seen that the relationship between Pinv(pu)max and the SCR is linear. In the purely inductive (R/X = 0) case, SCR = 1 is a critical point because when the SCR is lower than 1, the maximum power Pinv(pu)max is lower than 1, namely, the maximum output power of the inverter is limited naturally under weak grid conditions. Hence, this natural characteristic between Pinv(pu)max and the SCR can be considered a standard to be compared with the maximum power operating point at the stability boundary of different control strategies. Figure 3. Relationship between ideal maximum inverter power Pinv(pu)max and SCRs with different Rg/Xg. Figure 3. Relationship between ideal maximum inverter power P inv(pu)max and SCRs with different R g /X g .

Realistic Maximum Power with Current Limitation
In real applications, the maximum inverter current I inv(max) is determined by the endurable current of the power semiconductor. Thus, considering the current limitation of the inverter, the d-axis inverter current should be limited by the expression in Equation (8).
To analyze the maximum transferred power of a grid-connected inverter, the d-axis inverter current should be equal to the limiting value in Equation (8). It is worth mentioning that since the active power is transferred from the wind or PV power plant to the grid under normal circumstances, the active power P inv should be positive. Thus, the current i d should also be positive; hence, a negative i d is not considered in this paper.
It is worth noting that this paper investigates the system without reactive power compensation devices; thus, all the reactive power is provided by the inverter itself. Due to the flexible control ability of inverters, the suitable reactive power from the inverter can be controlled to make sure the PCC voltage operates near its rated value [23,24]. To be easily understandable, the analysis starts with the case where the PCC voltage equals its rated value.
By substituting "V pcc = V g " and Equation (8) into Equation (4), a standard quadratic equation for the q-axis inverter current i q can be derived as: Based on Equation (9), positive and negative solutions of the q-axis inverter current can be obtained. As the control law in the voltage PI controller is only suitable for the positive solution, the negative solution is unattainable. Hence, it is disregarded. The positive solution is given by Equation (10).
If the solution listed in Equation (10) exists, Equation (11) must be satisfied.
Substituting Equation (2) into Equation (11), the range of the SCR is derived as: The expression in Equation (12) shows that the PCC voltage cannot be maintained at the rated value if the SCR is lower than 0.5.
Moreover, when Equations (2) and (10) are substituted into Equation (8), the expression of the d-axis inverter current i d is derived as: In order to visualize the aforementioned analytical analysis, the voltage and current vector diagrams of the grid-connected system at the d-q plane are shown in Figure 4. The purely inductive, purely resistive, and hybrid impedance cases are shown in Figure 4a-c, respectively. The vector amplitudes I inv and V g are invariable, while the phase angles ϕ i and ϕ v are adjusted to match different values of Z g . It can be seen from Equation (13) that the d-axis current i d is always positive because only the inverter mode is considered in this paper. Moreover, according to Equation (10), the q-axis current i q is negative in the purely inductive case, which is shown in Figure 4a. However, the q-axis current i q is positive in the purely resistive case, which is shown in Figure 4b. Then, in the hybrid case, the current vector should be in the middle area between these two extreme cases, which is shown in Figure 4c. and φv are adjusted to match different values of Zg. It can be seen from Equation (13) that the d-axis current id is always positive because only the inverter mode is considered in this paper. Moreover, according to Equation (10), the q-axis current iq is negative in the purely inductive case, which is shown in Figure 4a. However, the q-axis current iq is positive in the purely resistive case, which is shown in Figure 4b. Then, in the hybrid case, the current vector should be in the middle area between these two extreme cases, which is shown in Figure 4c. As seen in Figure 4, because the amplitude of the inverter current is invariable, the larger the d-axis current component, the smaller the q-axis current component, and vice versa. Thus, as the output reactive power increases, the output active power decreases. This characteristic is more obvious in a weaker grid, where the amplitude of Zg is higher.
The output active power of the inverter Pinv shown in Figure 2 can be calculated as: Thus, according to Equations (13) and (14), the output active power of the inverter (per unit) can be derived as: Moreover, according to Equations (1) and (2), the power loss introduced by the transmission line (per unit) is derived as: Thus, according to Equations (15) and (16), the transferred active power to the grid (per unit) can be derived as: As seen in Figure 4, because the amplitude of the inverter current is invariable, the larger the d-axis current component, the smaller the q-axis current component, and vice versa. Thus, as the output reactive power increases, the output active power decreases. This characteristic is more obvious in a weaker grid, where the amplitude of Z g is higher.
The output active power of the inverter P inv shown in Figure 2 can be calculated as: Thus, according to Equations (13) and (14), the output active power of the inverter (per unit) can be derived as: Moreover, according to Equations (1) and (2), the power loss introduced by the transmission line (per unit) is derived as: Thus, according to Equations (15) and (16), the transferred active power to the grid (per unit) can be derived as: Based on Equations (15) and (17), Figure 5 presents the relationship between the inverter power P inv(pu) , grid power P grid(pu), and R g /X g ratio with different SCRs at the rated PCC voltage. Since the SCR is usually higher than 1 in practice, the situation where SCR is lower than 1 is omitted. It can be seen that a higher SCR leads to higher grid power, P grid(pu) . However, a higher R/X ratio causes lower grid power, P grid(pu) . Therefore, inductive grids with a high SCR have a high power transfer capability, while resistive grids with a low SCR have a limited power transfer capability.
Based on Equations (15) and (17), Figure 5 presents the relationship between the inverter power Pinv(pu), grid power Pgrid(pu), and Rg/Xg ratio with different SCRs at the rated PCC voltage. Since the SCR is usually higher than 1 in practice, the situation where SCR is lower than 1 is omitted. It can be seen that a higher SCR leads to higher grid power, Pgrid(pu). However, a higher R/X ratio causes lower grid power, Pgrid(pu). Therefore, inductive grids with a high SCR have a high power transfer capability, while resistive grids with a low SCR have a limited power transfer capability. Moreover, in the resistive grid (Rg/Xg > 1) case, the higher the SCR, the lower the output power of the inverter, Pinv(pu). It seems counterintuitive. This is because more reactive power has to be supplied to maintain the PCC voltage so that the output active power of the inverter is limited. This indicates that it is uneconomical to use reactive power to compensate the PCC voltage in resistive grids because much of the potential active power capacity is sacrificed. Aiming at this problem, tuning the PCC voltage within a small range can reduce reactive power consumption and increase the power transfer capability in resistive grids, which will be analyzed in the next section.

Maximum Power Transfer Capability of Grid-Connected Systems at Varying PCC Voltages
In the previous section, the situation where the PCC voltage equals the rated value was evaluated. A more general situation where the PCC voltage varies within a small range around the rated value will be analyzed in this section.
By substituting Equations (2) and (8) into Equation (4), a standard quadratic equation for the q-axis inverter current iq can be obtained. Then, the positive solution of the quadratic equation is derived as Equation (18). Because the control law in the voltage PI controller is only suitable for the positive solution, the negative solution does not exist in the actual control system. Hence, it is disregarded here. Figure 5. Relationship between inverter power P inv(pu) , grid power P grid(pu) , and R g /X g ratio with different SCRs at the rated PCC voltage.
Moreover, in the resistive grid (R g /X g > 1) case, the higher the SCR, the lower the output power of the inverter, P inv(pu) . It seems counterintuitive. This is because more reactive power has to be supplied to maintain the PCC voltage so that the output active power of the inverter is limited. This indicates that it is uneconomical to use reactive power to compensate the PCC voltage in resistive grids because much of the potential active power capacity is sacrificed. Aiming at this problem, tuning the PCC voltage within a small range can reduce reactive power consumption and increase the power transfer capability in resistive grids, which will be analyzed in the next section.

Maximum Power Transfer Capability of Grid-Connected Systems at Varying PCC Voltages
In the previous section, the situation where the PCC voltage equals the rated value was evaluated. A more general situation where the PCC voltage varies within a small range around the rated value will be analyzed in this section.
By substituting Equations (2) and (8) into Equation (4), a standard quadratic equation for the q-axis inverter current i q can be obtained. Then, the positive solution of the quadratic equation is derived as Equation (18). Because the control law in the voltage PI controller is only suitable for the positive solution, the negative solution does not exist in the actual control system. Hence, it is disregarded here.
where V pcc(pu) is the per-unit value of the PCC voltage, expressed as "V pcc(pu) = V pcc /V g "; K 1 and K 2 are two intermediate variables.
If the solution listed in Equation (18)   According to Equation (19), when the SCR is equal to 1, the range of V pcc(pu) is [0, 2]. When the SCR is equal to 10, the range of V pcc(pu) is [0.9, 1.1]. Hence, it is noted that the PCC voltage is naturally close to its rated value under strong grid conditions, where reactive power compensation may not be necessary. However, the PCC voltage varies easily under weak grid conditions. Thus, reactive power compensation is necessary to keep the PCC voltage close to its rated value.
When Equation (18) is substituted into Equation (8), the expression of the d-axis inverter current i d can be derived as: where the expressions of K 1 and K 2 are shown in Equation (18). According to Equations (14) and (20), the output active power of the inverter (per unit) can be derived as: According to Equations (16) and (21), the transferred active power to the grid (per unit) can be derived as: According to Equations (21) and (22), Figure 6 presents the relationship between inverter power P inv(pu) , grid power P grid(pu) , and PCC voltage V pcc(pu) with different R g /X g ratios and SCRs. Moreover, the points (A 1 , A 2 , A 3 ), (B 1 , B 2 , B 3 ), (C 1 , C 2 , C 3 ), and (D 1 , D 2 , D 3 ) in Figure 6 are the maximum value points of P grid(pu) .
As shown in Figure 6a,b, when the SCR is low, the grid power P grid(pu) drops significantly as the R/X ratio increases. For example, the maximum value of P grid(pu) decreases from point A 1 to A 2 and A 3 as the R/X ratio increases. However, as presented in Figure 6c,d, when the SCR is high, the maximum value of P grid(pu) can stay at a high level, such as points D 1 , D 2 , and D 3 . Therefore, a larger SCR is beneficial for power transmission.
Comparing the maximum power points (A 1 , A 2 , A 3 ), (B 1 , B 2 , B 3 ), (C 1 , C 2 , C 3 ), and (D 1 , D 2 , D 3 ) in Figure 6, the rated PCC voltage in the inductive grid (R g /X g = 0.1) is appropriate for power transmission. In contrast, a higher PCC voltage than the rated value in the resistive grid (R g /X g = 10) is beneficial for power transmission. Furthermore, as shown in Figure 6a, the value of P grid(pu) in the resistive grid (R g /X g = 10) is lower than 0 whatever the PCC voltage is, which means that the resistive weak grid cannot be used for power transmission. Hence, a required minimum SCR should be known in the resistive grid.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 15 PCC voltage is, which means that the resistive weak grid cannot be used for power transmission. Hence, a required minimum SCR should be known in the resistive grid. -0.5

Optimal Parameters for Maximum Power Operation in Inductive and Resistive Grids
Based on the aforementioned analysis, the maximum power transfer capability of the grid-connected system is jointly determined by the SCR and the R/X ratio as well as the PCC voltage. Therefore, the optimal values of these parameters for effective power transmission need to be investigated in a mathematical manner.
According to Equation (17), when the PCC voltage is at the rated value, the partial derivatives of Pgrid(pu) can be derived as:

Optimal Parameters for Maximum Power Operation in Inductive and Resistive Grids
Based on the aforementioned analysis, the maximum power transfer capability of the grid-connected system is jointly determined by the SCR and the R/X ratio as well as the PCC voltage. Therefore, the optimal values of these parameters for effective power transmission need to be investigated in a mathematical manner.
According to Equation (17), when the PCC voltage is at the rated value, the partial derivatives of P grid(pu) can be derived as: It can be seen from Equation (23), the partial derivative from P grid(pu) to the R/X ratio is smaller than 0, which means that as the R/X ratio decreases, P grid(pu) increases. Namely, R/X = 0 is the theoretically optimal value for power transmission. Moreover, it can be seen from Equation (24), the partial derivative from P grid(pu) to the SCR is larger than 0, which means that as the SCR increases, P grid(pu) increases. Namely, SCR = ∞ is the theoretically optimal value for power transmission. Obviously, these two optimal values, in theory, cannot be obtained in real applications. Nevertheless, a higher SCR or a lower R/X ratio is beneficial to increase the power transfer capability of the system.
Aside from the SCR and the R/X ratio, the PCC voltage is another key parameter affecting the maximum power transfer capability of the system. As aforementioned, the rated PCC voltage in an inductive grid is appropriate for power transmission, while a higher PCC voltage than the rated value in a resistive grid is beneficial for power transmission. Thus, according to Equation (22), the curves for maximum P grid(pu) with different R g /X g ratios, SCRs, and PCC voltages are shown in Figure 7. Figure 7a presents the inductive grid impedance case, where the rated PCC voltage is preferable to be used. Figure 7b shows the resistive grid impedance case, where the power curves at two PCC voltages are compared. Obviously, a higher PCC voltage is better than the rated PCC voltage.
It can be seen from Equation (23), the partial derivative from Pgrid(pu) to the R/X ratio is smaller than 0, which means that as the R/X ratio decreases, Pgrid(pu) increases. Namely, R/X = 0 is the theoretically optimal value for power transmission. Moreover, it can be seen from Equation (24), the partial derivative from Pgrid(pu) to the SCR is larger than 0, which means that as the SCR increases, Pgrid(pu) increases. Namely, SCR = ∞ is the theoretically optimal value for power transmission. Obviously, these two optimal values, in theory, cannot be obtained in real applications. Nevertheless, a higher SCR or a lower R/X ratio is beneficial to increase the power transfer capability of the system.
Aside from the SCR and the R/X ratio, the PCC voltage is another key parameter affecting the maximum power transfer capability of the system. As aforementioned, the rated PCC voltage in an inductive grid is appropriate for power transmission, while a higher PCC voltage than the rated value in a resistive grid is beneficial for power transmission. Thus, according to Equation (22), the curves for maximum Pgrid(pu) with different Rg/Xg ratios, SCRs, and PCC voltages are shown in Figure 7. Figure 7a presents the inductive grid impedance case, where the rated PCC voltage is preferable to be used. Figure 7b shows the resistive grid impedance case, where the power curves at two PCC voltages are compared. Obviously, a higher PCC voltage is better than the rated PCC voltage. Based on the above analysis, Pgrid(pu) in Equation (22) can be considered a key performance index to quantify the performance of the system at different Vpcc, SCRs, and R/X values. Namely, if Pgrid(pu) is within [0. 8,1], the performance of the system is good. If Pgrid(pu) is within [0.5, 0.8], the performance is medium. If Pgrid(pu) is lower than 0.5, the performance of the system is bad. Hence, the suitable Vpcc, SCRs, and R/X values can be found according to this definition of performance.
It was reported in [21] that the R/X ratio of the high voltage line is around 0.3, while the R/X ratio of the low voltage line is around 7 in real applications. Thus, as shown in Figure 7a, when the R/X ratio is 0.3, an SCR higher than 2 is required in order to make Pgrid higher than 0.8 pu (good performance). Moreover, as shown in Figure 7b, when the R/X ratio is 7, an SCR higher than 10 and a PCC voltage higher than 1.05 pu are required in order to make Pgrid higher than 0.5 pu (medium performance).
In order to apply the above research to practical projects, a case study on wind generation is discussed, as follows. Question: How to increase the penetration level from 20% to 40% for wind generation connected to an inductive grid with Rg/Xg = 0.3 and SCR = 3? If the penetration level is increased from 20% to 40%, the rated power SN will be doubled, and the SCR will become 3/2. As shown in Figure 7a, when the Rg/Xg is 0.3, an SCR higher than 2 is preferable. Thus, the grid impedance |Zg| should be reduced to 3/4. For example, increasing the voltage level can reduce grid impedance. Moreover, since the low inertia in Figure 7. Curves for maximum P grid(pu) with different R g /X g ratios, SCRs, and PCC voltages. (a) Inductive grid impedance case; (b) resistive grid impedance case.
Based on the above analysis, P grid(pu) in Equation (22) can be considered a key performance index to quantify the performance of the system at different V pcc , SCRs, and R/X values. Namely, if P grid(pu) is within [0. 8,1], the performance of the system is good. If P grid(pu) is within [0.5, 0.8], the performance is medium. If P grid(pu) is lower than 0.5, the performance of the system is bad. Hence, the suitable V pcc , SCRs, and R/X values can be found according to this definition of performance.
It was reported in [21] that the R/X ratio of the high voltage line is around 0.3, while the R/X ratio of the low voltage line is around 7 in real applications. Thus, as shown in Figure 7a, when the R/X ratio is 0.3, an SCR higher than 2 is required in order to make P grid higher than 0.8 pu (good performance). Moreover, as shown in Figure 7b, when the R/X ratio is 7, an SCR higher than 10 and a PCC voltage higher than 1.05 pu are required in order to make P grid higher than 0.5 pu (medium performance).
In order to apply the above research to practical projects, a case study on wind generation is discussed, as follows. Question: How to increase the penetration level from 20% to 40% for wind generation connected to an inductive grid with R g /X g = 0.3 and SCR = 3? If the penetration level is increased from 20% to 40%, the rated power S N will be doubled, and the SCR will become 3/2. As shown in Figure 7a, when the R g /X g is 0.3, an SCR higher than 2 is preferable. Thus, the grid impedance |Z g | should be reduced to 3/4. For example, increasing the voltage level can reduce grid impedance. Moreover, since the low inertia in wind generation systems may cause frequency instability issues, gridforming control techniques on inverters are recommended to support the grid frequency. Moreover, energy storage systems (ESSs) can also be used to smooth the wind energy and realize an uninterrupted supply of power.

Simulation Verification
In order to verify the effectiveness of the theoretical analysis, a 30 kW grid-connected inverter simulation model with classical vector current control was established in Matlab/Simulink. The control method is shown in Figure 8. To avoid the influence of highfrequency harmonics, the average model of the inverter was used. Firstly, an inductive grid case (R g /X g = 0.3) with low SCRs was carried out. Then, a resistive grid case (R g /X g = 7) with SCR = 10 was performed. The key parameters of the grid-connected inverter are listed in Table 1. Moreover, the parameters of grid impedance in the inductive grid and resistive grid cases are summarized in Table 2.
wind generation systems may cause frequency instability issues, grid-forming control techniques on inverters are recommended to support the grid frequency. Moreover, energy storage systems (ESSs) can also be used to smooth the wind energy and realize an uninterrupted supply of power.

Simulation Verification
In order to verify the effectiveness of the theoretical analysis, a 30 kW grid-connected inverter simulation model with classical vector current control was established in Matlab/Simulink. The control method is shown in Figure 8. To avoid the influence of highfrequency harmonics, the average model of the inverter was used. Firstly, an inductive grid case (Rg/Xg = 0.3) with low SCRs was carried out. Then, a resistive grid case (Rg/Xg = 7) with SCR = 10 was performed. The key parameters of the grid-connected inverter are listed in Table 1. Moreover, the parameters of grid impedance in the inductive grid and resistive grid cases are summarized in Table 2.    The simulation results of the inductive grid case are shown in Figure 9, where the power reference P ref , the output active power of inverter P inv , and the transferred active power to the grid P grid are presented, respectively. Moreover, I inv is the amplitude of the inverter current, and i d and i q are the d-axis and q-axis components of the inverter current. It can be seen that as the power reference increases, the current amplitude I inv increases. When I inv reaches 1 pu, i d is limited; thus, P inv is also limited. In the steady state, the maximum value of P grid represents the maximum power transfer capability of the grid-connected inverter. As shown in Figure 9a, when the SCR is equal to 1, the maximum value of P grid is around 0.70 pu. Moreover, as shown in Figure 9b, when the SCR is equal to 2, the maximum value of P grid is around 0.85 pu. These simulation results agree well with the maximum power operating points for R g /X g = 0.3, as shown in Figure 7a. The simulation results of the inductive grid case are shown in Figure 9, where the power reference Pref, the output active power of inverter Pinv, and the transferred active power to the grid Pgrid are presented, respectively. Moreover, Iinv is the amplitude of the inverter current, and id and iq are the d-axis and q-axis components of the inverter current. It can be seen that as the power reference increases, the current amplitude Iinv increases. When Iinv reaches 1 pu, id is limited; thus, Pinv is also limited. In the steady state, the maximum value of Pgrid represents the maximum power transfer capability of the grid-connected inverter. As shown in Figure 9a, when the SCR is equal to 1, the maximum value of Pgrid is around 0.70 pu. Moreover, as shown in Figure 9b, when the SCR is equal to 2, the maximum value of Pgrid is around 0.85 pu. These simulation results agree well with the maximum power operating points for Rg/Xg = 0.3, as shown in Figure 7a.  The simulation results of the resistive grid case are shown in Figure 10, focusing on the power reference Pref, the output active power of the inverter Pinv, and the transferred active power to the grid Pgrid. Figure 10 shows that as the power reference increases, the current amplitude Iinv increases. When Iinv reaches 1 pu, id is limited and Pinv is also limited. In Figure 10a, the PCC voltage is controlled as 1 pu. It can be seen that iq is quite large but id is small, so that the output active power of the inverter Pinv is limited (as low as 0.2 pu), and the grid power Pgrid is only 0.1 pu. In contrast, as shown in Figure 10b, when the PCC voltage is controlled as 1.05 pu, id is not so small. Thus, the output active power of the inverter Pinv is 0.7 pu and the grid power Pgrid is only 0.6 pu. These simulation results are in accordance with the maximum power operating points for Rg/Xg = 7 and SCR = 10, as shown in Figure 7b. The simulation results of the resistive grid case are shown in Figure 10, focusing on the power reference P ref , the output active power of the inverter P inv , and the transferred active power to the grid P grid . Figure 10 shows that as the power reference increases, the current amplitude I inv increases. When I inv reaches 1 pu, i d is limited and P inv is also limited. In Figure 10a, the PCC voltage is controlled as 1 pu. It can be seen that i q is quite large but i d is small, so that the output active power of the inverter P inv is limited (as low as 0.2 pu), and the grid power P grid is only 0.1 pu. In contrast, as shown in Figure 10b, when the PCC voltage is controlled as 1.05 pu, i d is not so small. Thus, the output active power of the inverter P inv is 0.7 pu and the grid power P grid is only 0.6 pu. These simulation results are in accordance with the maximum power operating points for R g /X g = 7 and SCR = 10, as shown in Figure 7b.
It can be seen in Figures 9 and 10 that there is a steady-state error on the power P inv when the current reaches the limiting value. In fact, this steady-state error is equal to (1-P inv(pu) ). According to the expression of P inv(pu) in Equation (21), the steady-state error can be calculated accurately. The power steady-state errors in different cases are compared in Table 3. It can be seen that for the inductive grid case, the rated PCC voltage has a relatively small power steady-state error. As the R/X ratio increases, a higher PCC voltage has a smaller power steady-state error, which can also be observed from Figure 6d. Moreover, when V pcc(pu) is equal to 1, the impact of the R/X ratio and the SCR on P inv(pu) are shown in Figure 5. It can be seen in Figure 5 that there is a maximum power point on each constant SCR curve. When R/X ratios are smaller than the R/X ratios at the maximum power points, P inv(pu) increases as SCR increases. However, when R/X ratios are larger than the R/X ratios at the maximum power points, P inv(pu) decreases as SCR increases. Even so, the R/X ratio at each maximum power point is not the optimal one because the optimal R/X ratio should be determined by P grid(pu) .   It can be seen in Figures 9 and 10 that there is a steady-state error on the power Pinv when the current reaches the limiting value. In fact, this steady-state error is equal to ). According to the expression of Pinv(pu) in Equation (21), the steady-state error can be calculated accurately. The power steady-state errors in different cases are compared in Table 3. It can be seen that for the inductive grid case, the rated PCC voltage has a relatively small power steady-state error. As the R/X ratio increases, a higher PCC voltage has a smaller power steady-state error, which can also be observed from Figure 6d. Moreover, when Vpcc(pu) is equal to 1, the impact of the R/X ratio and the SCR on Pinv(pu) are shown in Figure 5. It can be seen in Figure 5 that there is a maximum power point on each constant SCR curve. When R/X ratios are smaller than the R/X ratios at the maximum power points, Pinv(pu) increases as SCR increases. However, when R/X ratios are larger than the R/X ratios at the maximum power points, Pinv(pu) decreases as SCR increases. Even so, the R/X ratio at each maximum power point is not the optimal one because the optimal R/X ratio should be determined by Pgrid(pu).

Conclusions
This paper investigates the maximum power transfer capability of grid-connected inverters, which is jointly determined by the SCR, the R/X ratio of grid impedance, and the PCC voltage amplitude. The maximum power curves in the inductive grid and resistive grid cases, with different SCRs and PCC voltages, are illustrated and benchmarked. It is revealed that increasing the SCR or reducing the R/X ratio of grid impedance can increase the maximum power transfer capability of the system. In order to have higher power transfer capability, SCR > 2 and R/X < 1 are recommended to be used. Moreover, under resistive grid conditions, a higher PCC voltage than the rated value is beneficial for power transmission. Based on this research, a suitable SCR can be designed for high-