## 1. Introduction

Wind power is one of the most sustainable, abundant and cost-effective energy sources [

1,

2]. A large portion of wind power is being injected into distribution systems through small wind power plants (WPPs). According to voltage regulation requirements defined by grid codes in various countries, such as Australia, the UK and Canada, the interconnection of WPPs to distribution networks must ensure that the steady-state voltage at the point of common coupling (PCC) is maintained between 95% and 105% of the rated grid voltage [

3]. At a given distribution network connected WPP, the steady-state voltage at the PCC (V

_{PCC}) is significantly impacted by short circuit capacity (SCC), short circuit ratio (SCR) and overall system impedance angle ratio seen at that site expressed by the X/R

_{PCC}. These parameters are explained as follows:

SCC: The amount of power that flows through a specified point when a short-circuit fault occurs at that point is expressed by SCC. The value of SCC depends on rated voltage (V

_{rated}) and short-circuit impedance (Z

_{sc}) and is given as in (1) [

4].

SCR: The ratio between the grid’s SCC and the power injected by WPP is given by SCR. At the PCC bus of a distribution system connected to WPP, SCR quantifies the bus strength against the power quality issues caused by the wind power penetration. The value of SCR is calculated, as shown in (2) [

4].

Given that wind turbine generators are generally installed on areas located far from the distribution substation, e.g., hilltops and close by the ocean, the output electrical power is transmitted to the grid through long lines. This results in large short-circuit impedance (Z

_{sc}) and small SCC and SCR [

1,

5]. Typically, the SCR value is less than 10 in distribution grid-connected WPPs. The small range of SCR, in turn, causes high voltage variations and power Quality issues at the PCC [

4]. Hence, there is a tradeoff between the value of SCR and the voltage stability in distribution systems connected to WPP.

X/R

_{PCC}: The grid impedance angle ratio seen at the PCC bus is defined by the X/R

_{PCC}. The value of the X/R

_{PCC} is determined by the ratio of Thevenin equivalent reactance and Thevenin equivalent resistance seen from that specified point [

4]. The internal reactance of distribution lines is small, making the equivalent X/R value seen at the PCC small. The majority of existing approaches proposed for mitigating the voltage stability issues through reactive power compensation are applicable to power transmission networks where the X/R ratio is large [

6]. Hence, these methods are not appropriate for distribution networks.

Given the significance of the three aforementioned parameters in the PCC voltage stability in distribution systems connected to WPP, designers should select an optimal PCC site where the values of SCC and X/R_{PCC} ensure the V_{PCC} stability requirements defined by the grid codes. In addition, given the relation between wind power penetration and SCR, engineers need to define the maximum power that can be injected by WPP, ensuring that V_{PCC} is maintained within the standard range, i.e., 0.95 pu < V_{PCC} < 1.05 pu.

Different approaches have been proposed in the literature for siting and sizing of distributed generators (DGs) in distribution networks. Authors in [

7] applied analytical methods for sizing and siting of DGs to minimize power losses in the system. Such analytical methods require calculating the system bus impedance matrix, the inverse of bus admittance matrix and Jacobean matrix. Given the large size of distribution networks, calculating these matrices is computationally demanding [

1]. In [

8], A. Keane and M. O’Malley proposed linear programming (LP)-based DG allocation method for harvesting maximum DG energy and minimizing the voltage variations in an Irish 38-KV seven-bus radial distribution network. Mixed-integer nonlinear programming (MINLP)-based method has been studied in [

9] to determine the optimal combination of different renewable DGs with minimum power loss in an IEEE-RTS 41-bus test system. Similarly, dynamic programming (DP) has been utilized in [

10] for optimal allocation of DGs for power-loss reduction and reliability improvement in a 9-bus radial test distribution system. The main drawback of the methods proposed in [

8,

9,

10] is that the methods rely on simulation of the whole distribution system, which is a complicated and time-consuming task and requires specifications of each system component [

11].

Heuristics methods have been commonly used in optimal distributed generation placement (ODGP) because of their simplicity, generality, flexibility and superiority in solving optimization problems [

12]. Ali et al. in [

13] investigated four DG sizing and siting methods based on simulated annealing (SA), variable search environment descending (VSED), genetic algorithm (GA) and hybrid genetic algorithm (HGA) to minimize the power loss and improve the voltage profile in IEEE standard 34 bus test distribution. Similarly, particle swarm optimization (PSO) has been applied for sizing and sitting of DGs in [

14] to improve voltage profile and minimize the cost of power losses in four different bus systems: 12-, 15-, 33- and 69-bus system. Ant colony optimization (ACO) has been utilized in [

12] to determine the optimal sizing and placement of multiple DGs using a 69-bus distribution system. Artificial bee colony (ABC) algorithm has been proposed for optimal placement and sizing of DGs in [

15] for improvement of voltage profile in IEEE 33, 69 and 229 bus system. In addition, other heuristics methods, such as harmony search (HS), differential evolution (DE), Tabu search (TS) etc., have been applied for DG optimal sizing and siting. These methods can deal with large and complex ODGP and provide a near-optimal solution. However, similar to the previous analytical methods discussed earlier, the heuristic methods also rely on simulation of a whole distribution system, which is complex and time-consuming [

11]. Moreover, the accuracy of the heuristic methods depends on the tuning of optimization parameters, such as crossover and mutation in GA [

13], acceleration constants (c1, c2) in PSO [

14], etc. Improper tuning of these parameters may lead to higher computational effort and adversely affect the accuracy of the prediction [

4]. In addition, using analytical approaches and artificial intelligence (AI)-based methods for WPP siting and sizing produces unrealistic results as the reactive power exchanged between the grid and WPP is considered to be zero in these methods [

4].

To address the aforementioned issues related to using analytical and AI-based methods, it is required to simplify the WPP sizing and siting in distribution systems using more efficient and accurate approaches. As a suitable approach for WPP sizing in the distribution systems, the author in [

5] developed a mathematical relation between V

_{PCC} and SCR for a test system with 0 ≤ SCR ≤ 2.5. Referring to [

5], V

_{PCC} can be taken as a quadratic function of SCR. However, the equation proposed in [

5] did not consider the relation between V

_{PCC} and X/R

_{PCC} ratio. Given the significant effect of the X/R

_{PCC} ratio on V

_{PCC} stability, the lack of consideration of the relationship between these parameters adversely impacts the accuracy and validation of the relation proposed in [

5]. In addition, in the majority of actual distribution networks, the SCR value is more than 2.5 [

16]. Given that the mathematical model has been tested for 0 ≤ SCR ≤ 2.5, the validity of the proposed relation in [

5] for a system with SCR > 2.5 is ambiguous. The aforementioned issues concerned with the mathematical relation proposed in [

5] were addressed and removed by a more comprehensive mathematical model proposed in our previous work presented in [

1]. The model expressed the mathematical relation between the V

_{PCC} variation, SCR and X/R

_{PCC} ratio for various test distribution networks connected to induction generator (IG) and doubly-fed induction generator (DFIG)-based WPPs. For IG-based WPPs, two mathematical relations were developed regarding the range of the X/R

_{PCC}: an exponential function for WPPs with the X/R

_{PCC} < 2 and a quadratic function for WPPs with the X/R

_{PCC} > 2. Furthermore, for DFIG-based WPPs, a mathematical relation was developed considering that the X/R

_{PCC} < 2. The mathematical method presented in [

1] is one of the most valuable and comprehensive approaches expressing the relationships between V

_{PCC} and the main PCC parameters of distribution network connected WPPs. Such a mathematical model enables the prediction of the key V

_{PCC} stability criteria, including V

_{PCC} profile, step-V

_{PCC} variation and maximum permissible size of WPP. Taking advantage of the predicted V

_{PCC} parameters, the design engineers can easily find the best bus for the interconnection of a WPP without carrying out complex and time-demanding computational tasks and simulating the test systems. However, the results obtained in [

1] demonstrated that the accuracy of the mathematical relations is adversely impacted when SCR and X/R

_{PCC} ratios are small. In addition, for IG-based WPPs, the accuracy of the proposed relations is low when the X/R

_{PCC} is around 2. Hence, although the method proposed in [

1] simplifies the WPP siting and sizing process compared to the other existing methods, its accuracy is impacted by small SCR and X/R

_{PCC} ratios, which, in turn, limits the method applicability. To address this issue and increase the prediction accuracy, the mathematical model proposed in [

1] was replaced by a decision tree algorithm-based method in this paper. Therefore, in this work, a decision tree algorithm method was developed to model the relation between V

_{PCC} variation (dV

_{PCC}), SCR and X/R

_{PCC}. The input parameters of the proposed decision tree-based model are SCR and X/R

_{PCC,} which are the baseline characteristics of distribution feeders and easily available in any power system network. Using the values of input parameters, the model precisely predicts the P

_{wind}-dV

_{PCC} characteristic, which can then be used for optimal WPP siting and sizing. The decision tree algorithm is one of the supervised learning algorithms and can be implemented for regression and classification problems [

17]. The accuracy of the decision tree algorithm in predicting output parameters is enhanced by training decision trees with a large training data set [

17]. In this study, the X/R

_{PCC}-dV

_{PCC} data points were initially obtained using simulation test systems with different SCR values. Later on, the simulation results were extended to enlarge the training data set. The extended data were then used to develop the decision tree algorithm-based model. The proposed decision tree-based model enables to plot P

_{wind} versus dV

_{PCC} and provides the design engineer with insightful information to carry out an initial predictive assessment on the key power quality parameters at the PCC of WPPs, including V

_{PCC} profile, and maximum permissible power can be injected into the distribution network (P

_{wind}_max). Taking advantage of the power quality parameters predicted by the proposed decision tree algorithm, WPP planning engineers can easily estimate the optimal size of WPP and select the most appropriate site for the interconnection of WPP to distribution networks where the voltage stability requirements defined by the grid codes are provided with very high accuracy. Hence, the main contribution of this work to the existing knowledge is to simplify the WPP sizing and siting analysis as well as achieving a noticeable higher accuracy compared to the similar methods recently published in the literature. The aims of this study were to:

Develop a novel voltage stability decision tree algorithm-based model predicting the key power quality components at a given PCC bus, i.e., V_{PCC} and P_{wind}, based on the values of SCR and X/R_{PCC} seen at that bus;

Simplify the siting and sizing of IG- and DFIG-based WPPs in weak distribution network;

Increase the prediction accuracy compared to the voltage stability mathematical model presented in [

1].

The paper structure is as follows:

Section 2 outlines and discusses the methodology and different steps followed to develop the decision tree algorithm-based model.

Section 3 presents the validation results obtained and compares the accuracy of the proposed model with similar previous techniques.

Section 4 explains the significance and novelty of the work and its application in predicting the key voltage stability criteria and analyzing the WPP siting and sizing. Finally,

Section 5 summarizes the highlights of this work.

## 4. Significance of the Proposed Decision Tree-Based Model

The decision tree-based model proposed in this study encompasses the advantages of the similar methods proposed for simplifying the WPP sizing and siting, while it significantly provides higher accuracy. Referring to the results shown in the previous section, the proposed model enables to accurately predict the P_{wind}-dV_{PCC} characteristic for any X/R_{PCC} ratio and SCC and SCR values. Consequently, for a potential WPP interconnection site, design engineers can calculate the V_{PCC} profile given the V_{initial} value using (3) and plot the P_{wind} versus V_{PCC} profile characteristic.

For example,

Figure 12 and

Figure 13 show the P

_{wind}-V

_{PCC} characteristic for one of the IG-based scenarios (Scenario 4 in

Table 3) and one of the DFIG-based scenarios (Scenario 2 in

Table 4), respectively. The V

_{initial} value at the PCC of test systems used for the scenarios considered in

Figure 12 and

Figure 13 is 1 pu and 0.98 pu, respectively.

As mentioned in

Section 1, the V

_{PCC} profile must be maintained between 95% and 105% of the network nominal voltage to satisfy the steady-state voltage stability requirements defined by the grid codes [

3]. Therefore, after plotting the V

_{PCC}-P

_{wind} characteristics for the potential WPP interconnection sites, designers and planners can determine the best PCC site, where the X/R

_{PCC} and SCC values ensure that the grid code requirements are concerned with the magnitude of steady-state V

_{PCC} are met.

In addition, the prediction of the P

_{wind}-dV

_{PCC} characteristic using the proposed model enables to estimate the maximum permissible size of WPP, called P

_{wind}_max, ensuring that the steady-state V

_{PCC} requirements defined by the grid codes would be satisfied. For example, from

Figure 12 and

Figure 13, the P

_{wind}_max values at the PCC of the test system considered are 3.6 MW and 5.2 MW, respectively. The results presented in

Figure 12 and

Figure 13 confirm that the predicted P

_{wind}_max gained by the proposed decision tree algorithm literally tracks the reference P

_{wind}_max values obtained by the simulation models.

Referring to

Section 1, most works published in the literature regarding DG siting and sizing rely on the simulation of whole distribution networks with complicated structures and/or carrying out complex and time-consuming computational tasks. The authors’ previous work proposed an efficient mathematical method for simplifying WPP siting and sizing by removing the need to simulate the whole test system and conduct complex calculations [

1]. However, the accuracy of the proposed mathematical method was limited. For the IG-based WPPs, the accuracy of the mathematical model in predicting the P

_{wind}-V

_{PCC} curve characteristic is reduced as the SCR is decreased or the value of the X/R

_{PCC} moves toward 2. In addition, for the DFIG-based WPPs, the accuracy of the mathematical model in predicting P

_{wind}-V

_{PCC} characteristics is low if the PCC site has a small X/R ratio. The proposed decision tree-based model addressed the aforementioned issues by simplifying the WPP sizing and siting through predicting P

_{wind}-V

_{PCC} characteristics with noticeably high accuracy. Similar to the mathematical model developed in [

1], the model proposed in this paper requires only two PCC parameters, i.e., X/R

_{PCC} and SCC, to predict the P

_{wind}-V

_{PCC} characteristics. The predicted P

_{wind}-V

_{PCC} characteristic can be used for optimal WPP sizing and siting. Given that the X/R

_{PCC} and SCC are the baseline characteristics of a distribution feeder, their values are generally available or can easily be calculated using fundamental power system analysis methods. More importantly, the verification results shown in

Section 3 demonstrated that the proposed decision tree-based model eliminates the issues concerned with the limited accuracy of the mathematical model presented in [

1] by providing a negligible prediction error for any SCR and X/R

_{PCC} ratios.

## 5. Conclusions

In this study, a novel decision tree algorithm-based model was developed to predict the voltage behavior in response to the wind power injection at a potential feeder for connecting IG and DFIG-based WPPs in a distribution network. For this purpose, the proposed model enables to plot wind power versus PCC voltage (P_{wind}-V_{PCC}) characteristic at a given potential interconnection point using the distribution system baseline parameters seen at that point, including SCC and X/R_{PCC}. Taking advantage of the plotted P_{wind}-V_{PCC} characteristic, design engineers can carry out an initial predictive assessment on the critical voltage stability criteria, including V_{PCC} value and P_{wind}_max, to determine the optimal WPP connection site and its maximum permissible size ensuring the grid code requirements. The proposed model simplifies the siting and sizing of WPPs by removing the need to simulate the whole distribution system and performing computational calculations, which is one of the main advantages of the proposed model over the majority of existing approaches. In addition, the proposed model was benchmarked against one of the latest mathematical methods developed for simplifying WPP sizing and siting to affirm its accuracy in predicting the P_{wind}-V_{PCC} characteristic and voltage stability criteria.

For the IG-based WPPs, the verification results gained using the mathematical method demonstrated that the error between the predicted and reference results was large when SCR is small (SCR ≤ 4). However, the largest error between the reference characteristics and the corresponding curves plotted by the proposed decision tree algorithm was ignorable even when the PCC site was weak, and its SCR was smaller than 4. In addition, for the IG-based WPPs, the prediction accuracy of the mathematical model in predicting P_{wind}-dV_{PCC} curve characteristics is around 2.5% when the X/R_{PCC} ratio tends to 2, whereas the curves predicted by the proposed decision tree algorithm precisely track the reference characteristics when the X/R_{PCC} ratio is around 2.

For the DFIG-based WPPs, the proposed model sorted out the issue of the mathematical model regarding low prediction accuracy when the X/R ratio is small. In this respect, the highest prediction error between the reference results and the data predicted by the mathematical model was around 1% when the X/R_{PCC} ratio is around 0.5, while the proposed model provided an accuracy of almost 100% over the whole range of the X/R_{PCC} ratio.

Generally, the verification studies demonstrated the proposed decision tree-based model is superior to the previous similar methods. In this study, the test systems used for the verification analysis were based on IEEE standard distribution network models, which other researchers widely use for conducting power system analysis. The proposed model was developed considering a number of practical factors to increase the accuracy of the proposed model for actual applications. This includes developing the proposed model using the real-world range of the X/R_{PCC} and reducing the uncertainty due to the load deviations by considering the V_{initial} parameter. In addition, the validation of the presented model using actual systems is important and will be addressed in future studies to further complement this research. The practical verification of the proposed model requires the values of V_{PCC}, X/_{RPCC} and SCC obtained from an actual distribution network. However, the authors did not have access to such values. In addition, simulation and modeling the real-world distribution systems require using professional engineering software, such as PSS/e, which is not currently available to the authors. Therefore, as one of the extensions to this research, the authors intend to validate the proposed model using an actual case where a wind power plant is being proposed for further integration.