# Developing a Mathematical Model for Wind Power Plant Siting and Sizing in Distribution Networks

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## Abstract

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## 1. Introduction

_{PCC}) of a distribution network connected WPP, with a set point voltage ranging from 95% to 105% of the grid rated voltage [4]. Furthermore, the step-V

_{PCC}variation in response to the changes in wind power penetration should generally be maintained at less than 3% [5]. Nevertheless, two main reasons have incurred challenges in meeting the requirements for grid code: the provisions of WPP design regarding the placement of the Point of Common Coupling (PCC), and the limitations in WPP capability to control the terminal voltage [6].

_{PCC}) and the Short Circuit Capacity (SCC) are important parameters that affect V

_{PCC}stability [7,8]. Short Circuit Ratio (SCR), defined as the grid’s SCC divided by the power generated by WPP (P

_{wind}) [9], is another determining factor. A SCR greater than 20 signifies that the stability requirements of the grid codes have been satisfied [10]. At a given point of a distribution system, SCC is proportional to the square of the system nominal voltage and inverse of the magnitude of system short circuit impedance (Z

_{sc}) seen at this point [11]. Generally, the wind velocity is high at sites located far from the distribution substation. Therefore, WPPs are usually connected to the distribution systems through long lines, which makes the Z

_{sc}value measured at the PCC large and the values of SCC and SCR small. The SCR value in many grid-connected WPPs is less than 10. The small values of the SCR leads to substantial difficulties to keep the V

_{PCC}profile and step-V

_{PCC}variations in the limits of steady-state standard defined by the grid codes [12]. Hence, it is critical to determine the optimal size of WPP and select the best site for the interconnection of the WPP to the grid in a way that the grid code requirements, in regards to steady-state voltage stability, are satisfied.

_{PCC}and the system SCR. The proposed relation enables to predict the voltage value at a given distribution feeder and find the optimal size and interconnection site of WPP while ensuring that the voltage regulation requirements are provided without carrying out complex calculations and the computational modelling of test systems. However, the author did not take into account all the steps required for validating the developed relations. For instance, the analysis was performed on a test system having SCR in the range of 0 and 2.5. According to the guidelines provided by Australia Energy Market Operator (AEMO) [25], the wind turbine generator may trip when SCR < 2, indicating that the SCR value should not be less than 2 in actual systems. This demonstrates that the numerical relations developed in [26] cannot be applied in practice. In addition, the main disadvantage of the aforementioned work is that it does not consider the relation between the V

_{PCC}and X/R

_{PCC}ratio. As discussed earlier, in distribution networks, the V

_{PCC}stability has a high dependency on X/R

_{PCC}. Therefore, the validation of the equation proposed in [26] is adversely affected due to the lack of consideration of the relation between V

_{PCC}and X/R

_{PCC}.

- A simple WPP sizing and allocation approach: As discussed, the existing DG sizing and allocation methods suffer from complex calculations, modelling, and simulation issues. Therefore, a simple approach that does not require the analytical or simulation model of the distribution system is still a noticeable gap in the literature
- A holistic mathematical relation for WPP sizing and siting: The problems of existing DG placement and sizing methods in regard to calculating large and complex dimensional matrices and modelling the entire test system can be removed using a holistic mathematical model between steady-state V
_{PCC}and the key parameters of distribution systems. However, only one reference [26], addressed this issue by developing a mathematical relation between V_{PCC}and SCR. The work did not consider the necessary factors needed for validating the results, such as consideration of X/R as one of the parameters of the proposed equation or the realistic range of SCR ratio.

_{PCC}, SCC and X/R

_{PCC}and P

_{wind}. However, in [27], the relation between the aforementioned parameters were presented using power–voltage and reactive-power–voltage curve characteristics and the work has not proposed a mathematical relation between the parameters. The analysis studies carried out in [27] was further developed in our work in [28] by proposing an initial voltage stability mathematical model to show the relation between V

_{PCC}and key distribution network connected to Induction Generator (IG)-based WPP. However, the model presented in [28] lacks the investigation of WPPs that are based on the Double Fed Induction Generator (DFIG), which is one of the most common type of WTGs. In addition, the mathematical model developed in [28] has not considered all the steps required for validating the developed relations such as verifying the model using test systems that are different from those used for developing the model.

- Propose a novel voltage stability mathematical model demonstrating the mathematical relations between V
_{PCC}, P_{wind}, SCC, and X/R_{PCC}. - Validate the accuracy of the proposed analytical model in predicting three important voltage stability criteria at a given connection point of a distribution network penetrated by wind power, including: V
_{PCC}profile, step variation of V_{PCC}due to the change of P_{wind}(ΔV_{PCC}), and the WPP maximum permissible size ensuring the grid code requirements (P_{max}), using standard test systems.

_{wind}, and the PCC parameters. A Genetic Algorithm (GA)-based approach was then used to identify the coefficients of the developed equations. The accuracy of the proposed equations was then evaluated using different scenarios involving a wide range of operating conditions. The methodology proposed is based on all critical parameters that impact the voltage stability at potential distribution system WPP interconnection point. Therefore, the model enables to predict the important voltage stability criteria at a given PCC site. Using the predicted criteria, the design engineers can select the optimal site and size of WPPs in distribution grids, ensuring that the grid code requirements in regard to the voltage stability are provided.

_{PCC}profile and ΔV

_{PCC}are in the standard range defined by the grid codes given the SCC and X/R

_{PCC}values. In the proposed mathematical model, the key PCC parameters, i.e., SCC, X/R

_{PCC}, are the only unknown of the developed equations. These parameters are usually available as the baseline characteristics of the distribution grid connected WPPs or can be easily calculated at any point looking back to the distribution substation. Hence, the proposed equations enable WPP siting and sizing without the need to solve complex and time-consuming computational tasks, which is the main advantage of the presented method over the existing approaches.

## 2. Developing Mathematical Formulations

_{PCC}, WPP size and inherent PCC characteristics of a distribution system, including: the SCC and X/R

_{PCC}ratio through voltage stability analysis studies. This was primarily conducted by investigating how V

_{PCC}behaves in response to changes in X/R

_{PCC}for networks with different SCC and SCR ratios. Analysis was preformed based on two types of generators most commonly used in wind turbines: the Induction Generator (IG) and the Double Fed Induction Generator (DFIG). Different steps followed for developing the proposed mathematical model have been presented in the sub-sequent sections.

#### 2.1. Methodology

**Modelling the test systems and measuring the SCR ratio for each system:**Four distribution system models defined by the Institute of Electrical and Electronics Engineers (IEEE) were simulated and considered to provide data required for developing the mathematical relations. After modelling the test systems using MATLAB/Simulink (version 2019 developed by MathWorks, Natick, MA, USA) the short circuit current value was obtained at the PCC of each system. The I

_{sc}values were obtained and then the grid’s SCC was calculated using (1). Finally, the SCR value was given by (2) regarding the amount of power injected by the wind farm in each test system.

_{rated}is the nominal voltage of the distribution feeder, I

_{sc}is the short circuit current at the PCC and P

_{wind}is the amount of power generated by WPP.

**Plotting V**For each test feeder modeled in the previous step, the X/R

_{PCC}-X/R_{PCC}characteristics:_{PCC}ratio was measured and varied under a fixed SCC and SCR network. The X/R

_{PCC}ratio was changed throughout the course of the analysis by changing the X/R ratios of the distribution lines. Then, V

_{PCC}value was recorded taking note of the variations in response to the change of the overall X/R

_{PCC}ratio. The V

_{PCC}versus X/R

_{PCC}characteristic has been plotted for each test feeder using the obtained data.

**Developing alternative mathematical functions modelling the relations between V**The voltage behaviour, with respect to changes in the X/R

_{PCC}, X/R_{PCC}, SCC and P_{wind}:_{PCC}ratio, was then analysed for all test systems to mathematically correlate V

_{PCC}, X/R

_{PCC}, P

_{wind}and SCC. At the end of this step, the general forms of alternative equations, which can be used to describe the relation between the aforementioned parameters were developed. Then, the coefficients of the equations were identified using an Artificial Intelligence (AI)-based approach.

**Determining the best fit function:**The accuracy of each alternative equation was investigated in order to identify the best fit equations.

**Developing the final form of the functions:**Finally, it was investigated how the functions presented in the previous step can be further developed for different operating conditions.

#### 2.2. Modelling Test Systems

_{PCC}and hence develop a formulation with higher validity. The modelling of two test systems, referred to as Test 1 and Test 2, relied on the 37-bus IEEE distribution network (Figure 1), while the modelling of other two test systems, referred to as Test 3 and Test 4, was based on the 9-bus IEEE distribution network (Figure 2). The PCC was considered to be at Bus 9 in the 9-bus system and at Bus 6 in the 37-bus system. The size of the WPP and the length of the lines are different amongst the four test systems, resulting in different values of SCC and SCR.

_{initial}, has a significant impact on the V

_{PCC}stability after the WPP connection. Developing the mathematical relations for a constant V

_{initial}value hinders the application of the relations for any distribution network, as loading conditions and various voltage regulator set-point values impact the V

_{initial}value. In this study, it was firstly assumed that V

_{initial}is around 0.98 pu for each test distribution feeder. Therefore, the mathematical relations were firstly developed regarding V

_{initial}= 0.98 pu. Then, in Section 2.6, the proposed relations were further developed such that it satisfies a wide range of V

_{initial}values. The adoption of various V

_{initial}values in the proposed methodology was to reduce the uncertainty due to the load deviations from the values used in the test systems.

_{sc}, SCC, P

_{wind}, and SCR, for each test system.

_{wind}is seen for Test 4 with the greatest SCR value, which is due to the small P

_{wind}. This results in the PCC point in Test 4 being the stiffest among other test systems.

#### 2.3. Characteristics of Voltage-X/R Ratio at the PCC

_{PCC}-X/R

_{PCC}data obtained from simulation models. The reason for using simulation models for obtaining V

_{PCC}-X/R

_{PCC}data is the fact that the development of mathematical relations with an appropriate accuracy requires a large number of data points. The lack of such a data base adversely impacts the accuracy of the mathematical relations. The use of simulation models enables to provide a large number of data by carrying out sensitive analysis through changing the X/R

_{PCC}ratios and monitoring V

_{PCC}taking note of the variations against the change of the X/R

_{PCC}ratio. The X/R

_{PCC}is an inherent characteristic of a power system and cannot be dynamically changed in an actual network. Therefore, having access to a large quantity of V

_{PCC}-X/R

_{PCC}data points obtained from actual systems is a complicated process.

_{PCC}ratio, analysed in this study, was derived from an analysis of actual distribution networks performed in [29]. Reginato et. all [29] carried out a research study concerned with an analysis of the effect of X/R

_{PCC}and the inverse of SCR, the so-called integration level (p), on V

_{PCC}using data from actual systems. The analysis was performed to determine a potentially viable X/R

_{PCC}range, ensuring the grid code requirements with regard to the steady-state V

_{PCC}stability for a given integration level. The analysis results were presented using p vs X/R

_{PCC}graphs, while 0.95 pu < VPCC < 1.05 pu. Although the authors in [29] did not present the value of V

_{PCC}for a given X/R

_{PCC}and p, the realistic range of X/R

_{PCC}has been identified for different integration levels using data obtained by actual systems. In the V

_{PCC}-X/R

_{PCC}characteristics presented in this section, the range of X/R

_{PCC}and the corresponding SCR value are based on the p-X/R graphs presented in [29] to decrease the issues related to the uncertainty of the data obtained from simulation models, i.e., V

_{PCC}- X/R

_{PCC}data points, in case of real application.

_{PCC}/X/R

_{PCC}curve characteristics, the X/R

_{PCC}ratio of the test systems in Section 2.2 was changed to observe the V

_{PCC}values for each X/R

_{PCC}with SCC and SCR values of the test system being constant. The data points collected were utilised to plot the characteristic of V

_{PCC}against the X/R

_{PCC}ratio. The results are presented in Figure 3.

_{PCC}around 2, the V

_{PCC}values for all four systems are nearly equal. This observation is in accordance with [29], which states that the lowest variation in voltage happens at X/R

_{PCC}= 2 irrespective of the SCR value in an IG-based WPP, which is another piece of evidence to confirm that the methodology results are based on realistic cases.

_{PCC}(X/R

_{PCC}< 2). This is particularly apparent in small SCR ratios. For example, from Figure 3, it can be seen that the PCC voltage of Test 1 with an SCR ratio equal to 3 is greater than the maximum acceptable voltage level defined by the grid codes (V

_{PCC}> 1.05 pu) when the X/R

_{PCC}ratio is smaller than 0.5. This is in contrast with Test 4 with the maximum SCR ratio equal to 7, where the low values of the X/R

_{PCC}ratio do not cause a noticeable increase in the voltage. When dealing with large X/R

_{PCC}ratios, Figure 3 demonstrates that the voltage variation due to changes in wind power penetration is negligible at the connection point of a DFIG-based WPP with X/R

_{PCC}> 2. Furthermore, Figure 3 shows that voltage at the PCC of IG-based WPP is small when the X/R

_{PCC}ratio is large (X/P

_{PCC}> 2). The variations in voltage according to a rise in X/R

_{PCC}ratio is more remarkable at the weaker test feeders. In Figure 3, Test 1 shows a V

_{PCC}value less than the minimum limit of the allowable range, i.e., V

_{PCC}< 0.95 pu, when X/R

_{PCC}is around 2.5. The voltage for Test 4 is shown to be within the standard range, even in X/R

_{PCC}ratios greater than 5.

#### 2.4. Developing Alternative Functions

_{PCC}-X/R

_{PCC}characteristics shown in the previous section. The relations were developed for the values of X/R

_{PCC}that imposes Power Quality (PQ) concerns and voltage regulation problems at the PCC. Therefore, both small and large X/R

_{PCC}ratio ranges were considered for developing the formulation for the IG-based WPP. Furthermore, only small X/R

_{PCC}ratios were considered for expressing the mathematical relations for the DFIG-based WPP.

#### 2.4.1. General Form of Alternative Functions for IG-Based WPPs

_{PCC}-X/R

_{PCC}data points presented in Figure 3. Two key features in Figure 3 were considered in developing the equations, which are presented in Table 2.

- Function 1—Exponential function$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}={\mathrm{Z}}_{0}+{\mathrm{Z}}_{1}\times {\mathrm{e}}^{\left(-\mathsf{\delta}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)}$$
- Function 2—Polynomial with an order of 2$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}={\mathrm{Y}}_{0}-{\mathrm{Y}}_{1}\times \left(\mathsf{\gamma}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)+{\mathrm{Y}}_{2}\times {(\mathsf{\gamma}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}})}^{2}$$
_{0}, Z_{1}, Y_{0}, Y_{1}and Y_{2}, are positive coefficients. γ and δ represent the exponential and polynomial decay coefficients, respectively.

_{PCC}-X/R

_{PCC}characteristic curves given by (3) and (4) have similar patterns for all X/R

_{PCC}ranges and different decay coefficients. Hence, both functions can be applied to model the relation between V

_{PCC}and X/R

_{PCC}. Figure 4 shows the general V

_{PCC}-X/R

_{PCC}curve of the considered alternative functions for different values of decay coefficients.

_{PCC}ratio. Therefore, it can be concluded that SCR and the decay coefficients are directly related when X/R

_{PCC}is small (X/R

_{PCC}< 2), expressed by (5) and (6).

_{1}and K

_{2}are considered positive.

_{PCC}. Referring to the second characteristic explained in Table 1, small SCR values decrease voltage at the feeder with a large X/R

_{PCC}ratio. Hence, SCR and decay coefficients have an inverse relationship when X/R

_{PCC}is large (X/R

_{PCC}> 2). This relationship is given by (7) and (8):

_{3}and K

_{4}are considered positive.

_{s0}, Z

_{s1}, Y

_{s0}, Y

_{s1}and Y

_{s2}are positive coefficients in the equations representing the small X/R

_{PCC}range (X/R

_{PCC}< 2) and Z

_{l0}, Z

_{l1}, Y

_{l0}, Y

_{l1}and Y

_{l2}are positive coefficients in the equations representing the large X/R

_{PCC}range (X/R

_{PCC}> 2).

#### 2.4.2. General Form of Alternative Functions for DFIG-Based WPPs

_{PCC}variation against changes in X/R

_{PCC}follows a similar pattern in both IG- and DFIG-based WPPs when X/R

_{PCC}< 2. Hence, the V

_{PCC}-X/R

_{PCC}curves plotted for DFIG and IG are similar to each other in the small X/RPCC ratio, where DFIG cannot efficiently regulate the voltage. This signifies that the alternative functions applied for the DFIG-based WPP are similar to those developed for the IG-based WPP with small X/R

_{PCC}. The general forms of the functions are expressed in (13) and (14).

_{s0}, W

_{s1}, W

_{s0}, U

_{s0}, U

_{s1}, U

_{s2}and K

_{5}are positive coefficients.

#### 2.4.3. Determining the Coefficients Values Using GA

_{PCC}-X/R

_{PCC}curve was considered as the output parameter. The main idea was to find the coefficient of each alternative equation in a way that the curve predicted and plotted by the equation has the lowest error with respect to the corresponding reference V

_{PCC}-X/R

_{PCC}curves shown in Figure 3. The error was calculated using Standard Deviation (SD) formulas.

**IG-based WPP with X/R**_{PCC}< 2:$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=0.9867+0.0912\times {\mathrm{e}}^{\left(-0.29\times \mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)}$$$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=1.068-0.015\times \left(\mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)+0.001\times {(\mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}})}^{2}$$**IG-based WPP with X/R**_{PCC}> 2:$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=0.788+0.195\times {\mathrm{e}}^{\left(-0.24\times \frac{1}{\mathrm{S}\mathrm{C}\mathrm{R}}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)}$$$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=0.9813-0.0427\times \left(\frac{1}{\mathrm{S}\mathrm{C}\mathrm{R}}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)+0.002\times {(\frac{1}{\mathrm{S}\mathrm{C}\mathrm{R}}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}})}^{2}$$**DFIG-based WPP with X/R**_{PCC}< 2:$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=0.99+0.101\times {\mathrm{e}}^{\left(-0.347\times \mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)}$$$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=1.102-0.03\times \left(\mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)+0.002\times {(\mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}})}^{2}$$

#### 2.5. Finding the Fit Functions

_{PCC}-X/R

_{PCC}data points was obtained for Equations (15)–(20). As stated in Section 2.4.3, the output parameter for GA objective functions was considered to be the error between reference and predicted values and was calculated by SD formulas. In addition to SD, the Mean of Relative Error (MRE) and the Mean of Absolute Error (MAE), as two frequently used criteria, were considered for evaluation of the error in this section.

_{PCC}-X/R

_{PCC}curve characteristics were calculated using SD, MRE and MAE formulas, as shown in (21) to (23).

_{PCC}is expressed as V

_{k}, and its predicted values are expressed as $\widehat{{\mathrm{V}}_{\mathrm{k}}}$. Moreover, the total number of reference values is given as m.

_{PCC}-X/RPCC values have been presented in Table 8 for each alternative function and test system. Furthermore, the difference between the reference V

_{PCC}-X/R

_{PCC}curve characteristics obtained by the simulation results and the V

_{PCC}-X/R

_{PCC}curve characteristics plotted by the developed alternative equations can be compared for each test system using graphical representation, as shown in Figure 6a–l.

_{PCC}< 2 for all test cases, the evaluation criteria calculated for (15) is less than the criteria obtained for (16). This is confirmed in Figure 6a–d, where the curve for (15) shows a smaller error than the curve for (16) compared with the reference curve. Analysis of IG-based WPP with X/R

_{PCC}> 2 for all test cases based on Figure 6e–h and Table 8 reveals a slightly higher accuracy for (18) compared to (17).

_{initial}of 0.98 pu. Nevertheless, the initial voltage varies from the default value because of the loading conditions and values of voltage regulator set-point. Consequently, another set of equations is required to calculate the voltage stability criteria at a certain test system with any initial voltage value.

#### 2.6. Developing the Final Form of the Functions

_{wind}versus V

_{PCC}curve was evaluated using simulation systems characterized in Table 1. The initial voltage was varied in the simulations from the default value (V

_{inital}= 0.98 pu) to plot the variations of P

_{wind}against V

_{PCC}for each new voltage value. The shape of the curves remained the same with both the default and varied initial voltages. The only difference was a downward shift for initial voltages less than 0.98 pu and an upward shift for the initial voltages greater than 0.98 pu. Defining ΔV

_{initial}= V

_{initial}- 0.98, it can be approximated that for a given initial voltage, the addition of ΔV

_{initial}to the PCC voltage value calculated by the equations in Table 9 gives the voltage value at PCC point. Hence, V

_{initial}parameter was incorporated in (15), (18) and (19) to give (24)–(26) in Table 10, respectively.

_{wind}, Equations (21)–(23) can be rewritten to calculate optimal value of wind power generation (P

_{max}) such that the voltage value at PCC point is kept in the standard range. To do so, in Equation (25) related to WPP with large X/R

_{PCC,}the voltage value at the PCC point was assumed to be 0.95 pu, which is the minimum allowable voltage level defined by the Australian grid code. In Equations (24) and (26), developed for WPP with a small X/R

_{PCC}ratio, the voltage at PCC point was assumed to be equal to the maximum limit of the admissible steady-state voltage.

_{PCC}values, the equations were rewritten in terms of P

_{wind}as a function of PCC parameters and V

_{initial}as can be found in (27) to (29) in Table 10.

## 3. The Significance of Proposed Model

_{PCC}in general terms and using graphical representation and different curve characteristics such as V

_{PCC}versus X/R

_{PCC}, V

_{PCC}versus SCR etc. For example, Kothari and et al. investigated the general relation between V

_{PCC}, X/R

_{PCC}and SCR in distribution systems connected WPP [34]. Their work presented the analysis results using the curve characteristic of V

_{PCC}versus SCR plotted for different X/R

_{PCC}ratios. For the X/R

_{PCC}= 0.5 case, the results in [34] demonstrated that the difference between the initial value of V

_{PCC}(V

_{initial}) when P

_{wind}= 0 and the V

_{PCC}value obtained for high wind power penetration is around 7%. Referring to (2), the high penetration of wind power makes the SCR value small. Hence, in [34], it was demonstrated that the difference between V

_{initial}and V

_{PCC}after WPP connection is around 7% (ΔV

_{PCC}= 7%) when X/R

_{PCC}is 0.5 and SCR is noticeably small.

_{initial}and V

_{PCC}values after WPP connection was calculated using proposed equations considering that X/R

_{PCC}= 0.5 and SCR is small. The calculations were performed assuming that SCR is slightly greater than 2, i.e., SCR = 2.5. This is because, as mentioned in Section 1, AEMO documentation indicates that an SCR < 2 should be avoided in WPPs, as it adversely impacts the voltage stability in the steady-state operation and may lead to generator tripping [25]. The comparison results have been shown in Table 11.

_{initial}and V

_{PCC}under high wind power penetration is around 7% when X/R

_{PCC}= 0.5. Although the authors in [34] investigated the general relation between V

_{PCC}, SCR and X/R

_{PCC}using graphical representation, they did not propose any mathematical relation between these parameters. This shortcoming hinders to investigate the voltage variation for any given X/R

_{PCC}and SCR values.

_{PCC}tends to increase for small X/R

_{PCC}ratios, while it decreases for large X/R

_{PCC}ratios in weak distribution networks penetrated by DGs. Furthermore, the authors in [36] showed that the voltage drop, due to an increase in X/R

_{PCC}, is more serious in distribution systems with low SCR values (weak systems) compared with that in the distribution feeders with larger SCR values (stiff systems). The results presented in [12,35,36] confirm the general characteristics of the functions given by the equations presented in Table 10. However, the authors in these works, only discussed the relation between V

_{PCC}, X/R

_{PCC}and SCR in a general manner and using graphical representation and their works do not propose a mathematical relation between these parameters.

_{PCC}and SCR for the steady-state operation of the WPPs. It was demonstrated that the relation between V

_{PCC}and SCR can be expressed through a polynomial function with an order two as shown in (30) [26].

_{PCC}, which is one of the most important characteristics of distribution networks, and V

_{PCC}. In addition, the equation was developed using data provided by an invented test system with unrealistic SCR range (0 < SCR < 2.5), which adversely impact the accuracy of (30) in real distribution systems.

_{PCC}and the key parameters of the distribution systems, i.e., the SCR and X/R

_{PCC}ratio, is still a noticeable gap. This shortcoming has been eliminated in this paper using the proposed mathematical model presented in Table 10. Furthermore, the proposed equations have been developed using data obtained by IEEE standard distribution networks.

_{wind}versus voltage at a potential distribution network WPP interconnection point using (24) and (26), considering the WTG type and the values of SCC, V

_{initial}and X/R at that point. Taking the advantage of the P

_{wind}versus voltage curve characteristic achieved by the proposed equations, the engineers can predict the V

_{PCC}value for different wind power penetration at a potential interconnection point. The predicted V

_{PCC}values enable the grid codes’ compliance check, i.e., to verify if V

_{PCC}is between 0.95 and 1.05 pu (compliance with the grid codes) or if V

_{PCC}is not maintained within 0.95 and 1.05 pu (grid codes violation), at the interconnection of WPP to a potential interconnection site. The potential site can then be selected for connecting WPP if the grid code requirements are met or identified as an inappropriate PCC point if the grid code requirements are not provided.

_{PCC}variations in response to changes in wind power penetration at a given PCC point using (24) to (26) depending on the WTG type and the X/R

_{PCC}ratio. The predicted results obtained for the step-V

_{PCC}variations enables the engineers to decide how many WTGs can be simultaneously switched on, ensuring that the step-V

_{PCC}variation satisfies the grid code requirements (ΔV

_{PCC}≤ 3%).

_{PCC}profile, step-V

_{PCC}variation and maximum permissible wind power generation, ensuring the grid code requirements.

_{initial}, X/R

_{PCC}and SCC) which are usually available as the baseline characteristics of the distribution systems or can easily be calculated at any point looking back to the distribution substation. Hence, the proposed equations enable WPP siting and sizing by promptly conducting three important voltage stability criteria at a potential distribution network interconnection point. Consequently, it removes the need to solve complex and time-consuming computational tasks or modelling the entire test distribution system, which is the main advantage of the presented over the existing WPP sizing and siting approaches. The proposed analytical model removes the need to investigate the effect of distribution network configuration and its component specifications on PCC bus voltage stability. This is due to the fact that the effect of these factors has been considered and modelled in the proposed analytical approach using SCC and X/R

_{PCC}parameters.

## 4. Validation Results

_{PCC}stability criteria, including: V

_{PCC}, ΔV

_{PCC}, and P

_{max}for a potential connection site in a distribution network, given the V

_{initial}value. In this respect, the accuracy of the proposed functions was investigated using two case studies. Case study 1 provides initial validation studies by assessing the accuracy of the proposed mathematical model using the four test systems presented in Section 2 with different X/R ratios and V

_{initial}values. Then, in Case study 2, the functions developed were applied for a new test system based on a 30-bus IEEE distribution network.

#### 4.1. Case Study 1—Validation for Different Vinitial Values

_{initial}values. Therefore, the effect of load variations and any other factors that change operating voltage at a given distribution feeder before the WPP connection has been modelled using V

_{initial}parameter. V

_{initial}has a profound effect on the P

_{wind}versus V

_{PCC}characteristic and on the steady-state voltage stability after the connection of WPP. In this regard, the proposed mathematical model should ideally work for any V

_{initial}value to ensure that the equations developed can be applied for a wide range of distribution networks with different loading conditions. Hence, as an initial assessment of the accuracy of the proposed mathematical model, the equations presented in Table 10 are verified for different V

_{initial}values. The validation study has been performed using the four test systems applied for developing the mathematical model with details presented in Section 2.2. However, the validation studies were performed considering that the loading conditions and V

_{initial}values of the four test systems are different from the default value (V

_{initial_default}= 0.98 pu) considered in Section 2.2.

_{initial}values are different from that considered for developing the proposed equations. Therefore, the mathematical model is validated using new operating conditions. In addition, the proposed model is validated for various X/R

_{PCC}ratios.

_{PCC}< 2, two scenarios for the IG-based WPP with X/R

_{PCC}> 2 and two scenarios for the DFIG-based WPP. Table 12 shows the test system, X/R

_{PCC}ratio and V

_{initial}value considered in each scenario. Furthermore, the SCC value and topology of each test system have been re-written in Table 12.

_{wind}versus V

_{PCC}curve characteristic obtained by the simulation results is compared with the P

_{wind}versus V

_{PCC}curve characteristic given by the corresponding proposed equation presented in Table 10. The results are as shown in Figure 7a–f.

_{PCC}profile and ΔV

_{PCC}.

_{PCC}value, the largest error between the reference and predicted results occurs in Scenarios 1 and 5 when wind power is not connected (P

_{wind}= 0). Referring to Table 12, both the SCC and X/R

_{PCC}ratio are small in Test 1 and 5. Therefore, the accuracy of the proposed mathematical model is slightly impacted when both the SCC and X/R

_{PCC}ratio are small. However, as shown in Figure 7a,e, the highest error is 1%. In the other scenarios considered, the error between the simulation and predicted results is less than 0.5%.

_{PCC}, the simulation results in Figure 7a–f demonstrate that the proposed equations accurately verify the compliance or violation of ΔV

_{PCC}with respect to the grid code requirements. As defined by the grid codes, the ΔV

_{PCC}value must not exceed 3%. Comparing the curve characteristics shown in Figure 7a–f, it can be observed that both simulated and predicted results demonstrate that the highest ΔV

_{PCC}occurred in Scenario A, where the SCC and X/R

_{PCC}are both small. According to Figure 7a, both simulation and predicted results show that the V

_{PCC}variation in response to an increase in P

_{wind}from 0 to 3 MVA is greater than 3%. Therefore, the results confirm that ΔV

_{PCC}violates the grid code requirement when only one 3 MVA generator is connected to the grid. However, for the same increase in P

_{wind}, ΔV

_{PCC}does not exceed the standard range in other scenarios. In the case of scenarios with large X/R

_{PCC}(Scenarios 3 and 4), both simulation and predicted results presented in Figure 7c,d demonstrate that ΔV

_{PCC}is more serious in Scenario 3, where the SCC value of the test system considered is smaller.

_{PCC}, projecting the correct grid code compliance or violation outcome. Therefore, two voltage stability criteria (V

_{PCC}value and step-V

_{PCC}variation) could be predicted using the proposed mathematical model with an appropriate approximation for all scenarios considered for the 9-bus and 37-bus test systems.

_{max}, which is possible to be predicted from the curve characteristics presented in Figure 7a–f. However, because there is no straight way to estimate the value of P

_{max}using PV characteristics, Equations (27) to (29) were utilized for the estimation of the P

_{max}value in Scenarios 1 to 6. For each scenario, the maximum permissible wind power generation obtained from both simulation and equations is illustrated in Figure 8.

_{PCC}are small. The error is much less, around 0.5 MW, in scenarios with greater SCC or X/R

_{PCC}> 2. Therefore, the IG-based WPP with a small X/R

_{PCC}ratio slightly affects the accuracy of the predictions by the mathematical model. For DFIG-based scenarios (Scenarios 5 and 6), the highest error is related to Scenario 5 where the SCC value is smaller than another scenario. Therefore, small SCC values slightly impact the accuracy of the equation proposed for DFIG-based WPP. However, as presented in Scenario 5 in Figure 8, the highest error is less than 1 MW (percentage error = 13%). Therefore, in both IG- and DFIG-based Scenarios, the worst-case accuracy is 87%. It can be seen from the results that the values of P

_{max}calculated from the presented mathematical model well agree with the values obtained from simulation with only a slight error margin.

#### 4.2. Case Study 2—Validation for a New Test System

_{initial}values, various SCC and X/RPCC ratios, compared to the models used for developing the proposed equations. The new system is based on the 30-bus IEEE distribution network simulated in MATLAB, as shown in Figure 9.

_{PCC}and V

_{initial}. For each scenario, the type of WPP, PCC site location, X/R

_{PCC}ratio, grid’s SCC and V

_{initial}value are as shown in Table 13.

_{wind}versus V

_{PCC}was obtained using simulation results. In addition, the equations proposed for predicting V

_{PCC}profile and ΔV

_{PCC}, i.e., (24) to (26), were used to plot the characteristic of P

_{wind}versus V

_{PCC}. Comparing the simulation and predicted results enable one to investigate the accuracy of the developed functions in projecting the V

_{PCC}value and step-V

_{PCC}variation for different wind power penetration. The results for IG-based scenarios (Scenarios A and B) and DFIG-based scenario (Scenario C) are as shown in Figure 10a,b, respectively.

_{wind}versus V

_{PCC}characteristics enable an investigation on validating whether the proposed functions can accurately foresee ΔV

_{PCC}and project compliance or violation of the grid codes. For the scenario with the lowest SCC and small X/R

_{PCC}ratio (Scenario A with SCC = 18 and X/R

_{PCC}= 0.4), both simulation and predicted results presented in Figure 10a demonstrate that ΔV

_{PCC}violates the grid code requirements (ΔV

_{PCC}> 3%) when only one 3 MVA generator is connected to the grid. Furthermore, both the simulation and predicted curves plotted for another IG-based scenario (Scenario B with SCC = 46 and X/R

_{PCC}= 3.5) demonstrate that ΔV

_{PCC}is less than 3% for different wind power penetration. In the case of DFIG-based scenario, both simulation and predicted results in Figure 10b demonstrate that ΔV

_{PCC}satisfies the standard range if the maximum increase in P

_{wind}is 6 MW, for example, when P

_{wind}increases from 0 to 3 MVA (one 3 MVA generator is connected to the grid) or from 3 to 9 MVA (two 3 MVA generators are simultaneously connected to the grid). However, ΔV

_{PCC}violates the grid code requirements when three or more 3 MVA generators are simultaneously connected to the grid. Hence, the results demonstrate that the proposed equations are accurate in predicting ΔV

_{PCC}for different wind power penetration levels.

_{max}, the simulation results shown in Figure 10a,b were used to determine the maximum value of wind power generation ensuring 0.95 < V

_{PCC}< 1.05. The simulation results were then compared with the values given by (27) to (29). Figure 11 presents the simulation and predicted values of P

_{max}for Scenarios A to C.

_{max}value for the IEEE 30-bus test system.

## 5. Conclusions

_{PCC}, SCC, the overall system X/R

_{PCC}ratio and P

_{wind}. The equations were developed using the V

_{PCC}-X/R

_{PCC}data points given by four test systems with different SCC and SCR values. The test systems were modelled and simulated based on IEEE standard networks. For each test system, a voltage stability hypothesis was developed based on the SCC and X/R

_{PCC}ratio measured at a potential PCC bus. In this respect, for each test system, the V

_{PCC}-X/R

_{PCC}characteristic was obtained and plotted under a specific SCR value considering the realistic range of X/R

_{PCC}and V

_{PCC}. The analysis studies were performed based on two types of WTGs commonly used in the WPPs, including: IG and DFIG. Alternative mathematical approximations were formulated for each WTG type and the corresponding V

_{PCC}-X/R

_{PCC}characteristic. The coefficients of the developed formula were determined using the GA optimization method. The accuracy of the alternative equations was then tested using the error evaluation criteria to determine the most accurate equations for each WPP type. The equations with the lowest error were then developed to calculate V

_{PCC}for any V

_{initial}values. Finally, six equations were proposed to calculate the three critical voltage stability criteria at a given PCC point, including the V

_{PCC}profile, step-V

_{PCC}variation in response to changes in wind power penetration and the maximum permissible wind power, ensuring that the steady-state V

_{PCC}requirements defined by the grid codes would be satisfied. The validation studies conducted for the V

_{PCC}profile demonstrated that the highest error amongst the all scenarios considered was less than 1%. Furthermore, validation studies confirmed the accuracy of the proposed method in predicting the step-voltage variation grid codes compliance check, i.e., to verify ΔV

_{PCC}≤ 3% (compliance with the grid codes) or ΔV

_{PCC}≥ 3% (grid codes violation). For maximum permissible wind power, the verification results showed that the accuracy of the proposed relations was slightly impacted in IG- and DFIG-based WPPs with small SCC and X/R

_{PCC}ratio. However, the worst-case accuracy amongst the all scenarios investigated was around 87%. Hence, the validity of the equations developed was confirmed through comparison with simulation results, which showed a highly accurate result for the voltage stability criteria at PCC points with various SCC and X/R

_{PCC}values.

_{PCC}regions, (X/R

_{PCC}< 2 and X/R

_{PCC}> 2). As another extension to this work, the proposed formulation can be further developed as a single function, such as a polynomial function with a high rank, which satisfies the whole X/R region and removes the need for dividing the X/R region into two parts. This may enhance the applicability of the mathematical model for estimating V

_{PCC}at feeders with X/R

_{PCC}around 2. However, the accuracy of such a function must be compared with the equations developed in this work to ensure that the error is not high. Moreover, the proposed model was developed and validated using IEEE distribution system models. Although, the IEEE standard models have been commonly used in electrical engineering studies, the validation of the proposed mathematical relations using actual cases is important. From a practical perspective, the application of the proposed model to the actual cases requires data obtained from real world distribution systems, such as the values of the X/R ratio and SCC measured at the buses of test system, which is not currently available to the authors. Hence, as part of future work, the authors are considering applying the obtained functions to actual cases to further complement this research.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

IG | Induction generator |

DFIG | Double fed induction generator |

GA | Genetic Algorithm |

PCC | Point of Common Coupling |

PQ | Power Quality |

P_{wind} | Power generated by wind power plant |

SCC | Short circuit capacity |

SCR | Short circuit ratio |

V_{PCC} | Voltage at the point of common coupling |

V_{initial} | Voltage at distribution feeder before the connection of wind power plant |

WPP | Wind power plant |

X/R_{PCC} | Short circuit impedance angle ratio seen at the point of common coupling |

## Appendix A

Parameter | Unit | Value |
---|---|---|

Nominal power (P_{n}) | MVA | 47.5 |

Frequency (f_{n}) | Hz | 50 |

Primary winding phase to phase voltage (V_{1}) | kV | 120 |

Primary winding resistance (R_{1}) | p.u. | 0.0027 |

Primary winding inductance (L_{1}) | p.u. | 0.08 |

Secondary winding phase to phase voltage (V_{2}) | kV | 22 |

Secondary winding resistance (R_{2}) | p.u. | 0.0027 |

Secondary winding inductance (L_{2}) | p.u. | 0.08 |

Magnetization resistance (R_{m}) | p.u. | 500 |

Magnetization inductance (L_{m}) | p.u. | 500 |

Parameter | Unit | Value |
---|---|---|

Nominal power (P_{n}) | MVA | 3 |

Frequency (f_{n}) | Hz | 50 |

Line to line voltage (V) | kV | 575 |

Stator resistance (R_{s}) | p.u. | 0.004843 |

Stator leakage inductance (L_{s}) | p.u. | 0.1248 |

Rotor reactance referred to stator (R_{r’}) | p.u. | 0.004377 |

Rotor leakage inductance referred to stator (L_{r’}) | p.u. | 0.1791 |

Magnetizing inductance (L_{m}) | p.u. | 6.77 |

Parameter | Unit | Value |
---|---|---|

Pitch angle controller gains (K_{p}, K_{i}) | - | 5, 25 |

Maximum pitch angle | Degree (°) | 45 |

Parameter | Unit | Value |
---|---|---|

Nominal power (P_{n}) | MVA | 4 |

Frequency (f_{n}) | Hz | 50 |

Primary winding phase to phase voltage (V_{1}) | kV | 22 |

Primary winding resistance (R_{1}) | p.u. | 0.00084 |

Primary winding inductance (L_{1}) | p.u. | 0.025 |

Secondary winding phase to phase voltage (V_{2}) | kV | 575 |

Secondary winding resistance (R_{2}) | p.u. | 0.00084 |

Secondary winding inductance (L_{2}) | p.u. | 0.025 |

Magnetization resistance (R_{m}) | p.u. | 500 |

Magnetization inductance (L_{m}) | p.u. | Infinite |

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**Figure 3.**V

_{PCC}-X/R

_{PCC}characteristic for each test system for both the Induction Generator (IG) and the Double Fed Induction Generator (DFIG).

**Figure 4.**General V

_{PCC}-X/R

_{PCC}characteristic curve given by (3) and (4) for different decay coefficients.

**Figure 6.**V

_{PCC}-X/R

_{PCC}characteristics plotted by the developed alternative equations. (

**a**) V

_{PCC}-X/R

_{PCC}curves obtained by (15) and (16) for Test 1; (

**b**) V

_{PCC}-X/R

_{PCC}curves obtained by (15) and (16) for Test 2; (

**c**) V

_{PCC}-X/R

_{PCC}curves obtained by (15) and (16) for Test 3; (

**d**) V

_{PCC}-X/R

_{PCC}curves obtained by (15) and (16) for Test 4; (

**e**) V

_{PCC}-X/R

_{PCC}curves obtained by (17) and (18) for Test 1; (f) V

_{PCC}-X/R

_{PCC}curves obtained by (17) and (18) for Test 2; (

**g**) V

_{PCC}-X/R

_{PCC}curves obtained by (17) and (18) for Test 3; (

**h**) V

_{PCC}-X/R

_{PCC}curves obtained by (17) and (18) for Test 4; (

**i**) V

_{PCC}-X/R

_{PCC}curves obtained by (19) and (20) for Test 1; (

**j**) V

_{PCC}-X/R

_{PCC}curves obtained by (19) and (20) for Test 2; (

**k**) V

_{PCC}-X/R

_{PCC}curves obtained by (19) and (20) for Test 3; (

**l**) V

_{PCC}-X/R

_{PCC}curves obtained by (19) and (20) for Test 4.

**Figure 7.**P

_{wind}versus V

_{PCC}characteristic for validation scenarios considered for the 9-bus and 37-bus test systems. (

**a**) Scenario 1; (

**b**) Scenario 2; (

**c**) Scenario 3; (

**d**) Scenario 4; (

**e**) Scenario 5; (

**f**) Scenario 6.

**Figure 8.**Maximum permissible wind power generation ensuring 0.95 < V

_{PCC}< 1.05 for Scenarios 1 to 6.

**Figure 10.**P

_{wind}versus V

_{PCC}curves for validation scenarios considered for the 30-bus test system. (

**a**) Scenarios A and B; (

**b**) Scenario C.

**Figure 11.**Maximum permissible wind power generation ensuring 0.95 < V

_{PCC}< 1.05 for Scenarios A to C.

Case Study | Topology | I_{sc} (kA) | SCC (MVA) | P_{wind} (MW) | SCR |
---|---|---|---|---|---|

Test 1 | IEEE 37-bus system | 0.71 | 27 | 9 | 3 |

Test 2 | IEEE 37-bus system | 1.05 | 40 | 9 | 4.5 |

Test 3 | IEEE 9-bus system | 1.42 | 54 | 9 | 6 |

Test 4 | IEEE 9-bus system | 0.55 | 21 | 3 | 7 |

**Table 2.**Two main features considered for developing mathematical equations for IG-based Wind Power Plants (WPPs).

Feature | Description |
---|---|

Feature 1 | Voltage at feeders with a low X/R_{PCC} ratio is increased as SCR is decreased. |

Feature 2 | Voltage at feeders with a high X/R_{PCC} ratio is decreased as SCR is decreased. |

Parameter | Value/Type |
---|---|

Generation | 300 |

population size | 200 |

Selection | stochastic uniform |

Crossover | Scattered |

Mutation | constraint dependent |

Alternative Equation | Objective Function Name | Input of Objective Function | Output of Objective Function |
---|---|---|---|

(9) | Func1 | Coefficients of (9), i.e., Z_{s0}, Z_{s1} and K_{1} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (9) and the reference curves presented in Figure 3 for IG-based WPP and X/R_{PCC} < 2 |

(10) | Func2 | Coefficients of (10), i.e., Y_{s0}, Y_{s1} and Y_{s2} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (10) and the reference curves presented in Figure 3 for IG-based WPP and X/R_{PCC} < 2 |

(11) | Func3 | Coefficients of (11), i.e., Z_{l0}, Z_{l1} and K_{3} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (11) and the reference curves presented in Figure 3 for IG-based WPP and X/R_{PCC} > 2 |

(12) | Func4 | Coefficients of (12), i.e., Y_{l0}, Y_{l1} and Y_{l2} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (12) and the reference curves presented in Figure 3 for IG-based WPP and X/R_{PCC} > 2 |

(13) | Func5 | Coefficients of (13), i.e., W_{s0}, W_{s1} and K_{5} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (13) and the reference curves presented in Figure 3 for DFIG-based WPP and X/R_{PCC} < 2 |

(14) | Func6 | Coefficients of (14), i.e., U_{s0}, U_{s1} and U_{s2} | Error between the V_{PCC}-X/R_{PCC} graphs predicted by (14) and the reference curves presented in Figure 3 for DFIG-based WPP and X/R_{PCC} < 2 |

Coefficient | Z_{s0} | Z_{s1} | K_{1} | Y_{s0} | Y_{s1} | Y_{s2} |

Value | 0.9867 | 0.0912 | 0.29 | 1.068 | 0.015 | 0.001 |

Coefficient | Z_{l0} | Z_{l1} | K_{3} | Y_{l0} | Y_{l1} | Y_{l2} |

Value | 0.788 | 0.195 | 0.24 | 0.9813 | 0.0427 | 0.002 |

Coefficient | W_{s0} | W_{s1} | K_{5} | U_{s0} | U_{s1} | U_{s2} |

Value | 0.99 | 0.101 | 0.347 | 1.102 | 0.03 | 0.002 |

**Table 8.**Error between the reference V

_{PCC}-X/R

_{PCC}graphs and the graphs plotted by the alternative equations.

WPP Type | X/R_{PCC} Range | Test System | Equation | SD | MAE | MRE |
---|---|---|---|---|---|---|

IG-based WPP | X/R_{PCC} < 2 | 1 | 15 | 0.0264 | 0.026 | 0.0254 |

- | - | - | 16 | 0.0324 | 0.0314 | 0.0305 |

IG-based WPP | X/R_{PCC} > 2 | 1 | 17 | 0.015 | 0.014 | 0.0144 |

- | - | - | 18 | 0.0144 | 0.013 | 0.0137 |

DFIG-based WPP | X/R_{PCC} < 2 | 1 | 19 | 0.0038 | 0.0035 | 0.0034 |

- | - | - | 20 | 0.0068 | 0.006 | 0.0058 |

IG-based WPP | X/R_{PCC} < 2 | 2 | 15 | 0.0155 | 0.0145 | 0.0143 |

- | - | - | 16 | 0.0247 | 0.0223 | 0.0218 |

IG-based WPP | X/R_{PCC} > 2 | 2 | 17 | 0.0102 | 0.0091 | 0.0095 |

- | - | - | 18 | 0.0097 | 0.0087 | 0.0091 |

DFIG-based WPP | X/R_{PCC} < 2 | 2 | 19 | 0.0014 | 0.0012 | 0.0012 |

- | - | - | 20 | 0.0103 | 0.0093 | 0.009 |

IG-based WPP | X/R_{PCC} < 2 | 3 | 15 | 0.0117 | 0.0116 | 0.0115 |

- | - | - | 16 | 0.0318 | 0.0274 | 0.0267 |

IG-based WPP | X/R_{PCC} > 2 | 3 | 17 | 0.0055 | 0.0047 | 0.0049 |

- | - | - | 18 | 0.0052 | 0.0044 | 0.0046 |

DFIG-based WPP | X/R_{PCC} < 2 | 3 | 19 | 0.0036 | 0.0031 | 0.003 |

- | - | - | 20 | 0.0361 | 0.0234 | 0.0223 |

IG-based WPP | X/R_{PCC} < 2 | 4 | 15 | 0.0122 | 0.0121 | 0.012 |

- | - | - | 16 | 0.2876 | 0.2523 | 0.194 |

IG-based WPP | X/R_{PCC} > 2 | 4 | 17 | 0.0051 | 0.005 | 0.0052 |

- | - | - | 18 | 0.0046 | 0.0045 | 0.0047 |

DFIG-based WPP | X/R_{PCC} < 2 | 4 | 19 | 0.0088 | 0.0065 | 0.0064 |

- | - | - | 20 | 0.0327 | 0.0225 | 0.0215 |

WPP Type | X/RPCC Range | Proposed Equation |
---|---|---|

IG-based WPP | X/R_{PCC} < 2 | (15) |

IG-based WPP | X/R_{PCC} > 2 | (18) |

DFIG-based WPP | X/R_{PCC} < 2 | (19) |

Equation | Equation Number | Application |
---|---|---|

$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}=0.0912{\mathrm{e}}^{\left(-0.29\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\times \frac{\mathrm{S}\mathrm{C}\mathrm{C}}{{\mathrm{P}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}}\right)}+{\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}+0.0067$$
| Plotting PV curve for IG WPP with X/R_{PCC} < 2. | |

$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}={\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}+0.0013-0.0427\times (\mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\times \frac{{\mathrm{P}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}}{\mathrm{S}\mathrm{C}\mathrm{C}})+0.002\times {(\mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\times \frac{{\mathrm{P}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}}}{\mathrm{S}\mathrm{C}\mathrm{C}})}^{2}$$
| Plotting PV curve for IG WPP with X/R_{PCC} > 2. | |

$${\mathrm{V}}_{\mathrm{P}\mathrm{C}\mathrm{C}}={\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}+0.01+0.101\times {\mathrm{e}}^{\left(-0.347\times \mathrm{S}\mathrm{C}\mathrm{R}\times \mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\right)}$$
| Plotting PV curve for DFIG WPP with X/R_{PCC} < 2 | |

$${\mathrm{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.29\times \frac{\mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\times \mathrm{S}\mathrm{C}\mathrm{C}}{\mathrm{L}\mathrm{n}\left(\frac{0.0912}{1.0433-{\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}\right)}$$
| IG WPP sizing with X/R_{PCC} < 2 | |

$${\mathrm{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{\mathrm{S}\mathrm{C}\mathrm{C}}{\mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}}\times \left(10.675-\sqrt{500\times \left(0.95-{\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}\right)+113.35}\right)$$
| IG WPP sizing with X/R_{PCC} > 2 | |

$${\mathrm{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.347\times \frac{\mathrm{X}/{\mathrm{R}}_{\mathrm{P}\mathrm{C}\mathrm{C}}\times \mathrm{S}\mathrm{C}\mathrm{C}}{\mathrm{L}\mathrm{n}\left(\frac{0.101}{1.04-{\mathrm{V}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}}\right)}$$
| DFIG WPP sizing with X/R_{PCC} < 2 |

WPP Type | SCR | X/RPCC | Proposed Equation | ΔVPCC (%) |
---|---|---|---|---|

IG-based WPP | 2.5 | 0.5 | (24) | 7 |

DFIG-based WPP | 2.5 | 0.5 | (26) | 7.5 |

Scenario | WPP Type | Test System | Topology | SCC (MVA) | X/R_{PCC} | V_{initial} |
---|---|---|---|---|---|---|

1 | IG-based WPP | Test 1 | IEEE 37-bus system | 27 | 0.3 | 0.99 |

2 | IG-based WPP | Test 3 | IEEE 9-bus system | 54 | 0.3 | 1 |

3 | IG-based WPP | Test 2 | IEEE 37-bus system | 40 | 3 | 0.98 |

4 | IG-based WPP | Test 4 | IEEE 9-bus system | 21 | 4 | 0.99 |

5 | DFIG-based WPP | Test 1 | IEEE 37-bus system | 27 | 0.5 | 1 |

6 | DFIG-based WPP | Test 2 | IEEE 37-bus system | 40 | 0.4 | 0.97 |

Scenario | WPP Type | PCC Bus | SCC (MVA) | X/R_{PCC} | V_{initial} |
---|---|---|---|---|---|

A | IG-based WPP | Bus 26 | 18 | 0.4 | 0.98 |

B | IG-based WPP | Bus 13 | 46 | 3.5 | 0.985 |

C | DFIG-based WPP | Bus 7 | 68 | 0.4 | 0.98 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alizadeh, S.M.; Sadeghipour, S.; Ozansoy, C.; Kalam, A.
Developing a Mathematical Model for Wind Power Plant Siting and Sizing in Distribution Networks. *Energies* **2020**, *13*, 3485.
https://doi.org/10.3390/en13133485

**AMA Style**

Alizadeh SM, Sadeghipour S, Ozansoy C, Kalam A.
Developing a Mathematical Model for Wind Power Plant Siting and Sizing in Distribution Networks. *Energies*. 2020; 13(13):3485.
https://doi.org/10.3390/en13133485

**Chicago/Turabian Style**

Alizadeh, Seyed Morteza, Sakineh Sadeghipour, Cagil Ozansoy, and Akhtar Kalam.
2020. "Developing a Mathematical Model for Wind Power Plant Siting and Sizing in Distribution Networks" *Energies* 13, no. 13: 3485.
https://doi.org/10.3390/en13133485