**Research Article** | **Open Access**

**Optimization of PV response under the uncertainty of load and irradiance during the peak**

*Kamel A. Alboaouh**

***Corresponding author:****Kamel A. Alboaouh**

PhD Student, Colorado School of Mines, Golden, CO 80401, USA; Email: alboaouh@mymail.mines.edu

**Received:** August 30^{th}, 2017; **Accepted:** September 12^{th}, 2017; **Published:** September 18^{th}, 2017

*Eng Press*. 2017; 1(1): 12-23. doi: 10.28964/EngPress-1-103

Ⓒ 2017 Copyright by Alboaouh KA. Creative Commons Attribution 4.0 International License (CC BY 4.0).

**ABSTRACT**

As the penetration of photovoltaic (PV) panels in distribution system increases, many challenges have emerged. The rapid fluctuation of irradiance during the day and the bi-directional power flow in distribution network necessitate fast mitigating actions to maintain the network voltage profile within the desired limits. In this work, we proposed a control scheme to optimize PV output power by utilizing the capabilities of the smart inverters. The proposed scheme is designed to optimize the penetration of the PV active power during the peak period of the day while maintaining the voltage of the buses within the desired limits and to minimize the losses of the system. The output active and reactive powers of the PVs are the control variables. Our model takes into account the uncertainty associated with irradiance and load demand. The proposed scheme has been tested on IEEE123 bus system. The simulation output shows the penetration level could go up to 50% of the PVs without violating the limitations of the bus voltage.

**KEYWORDS:** Smart inverters; Photovoltaic (PV); Unconventional generation; Distribution; Uncertainty; Forecast; Centralized control; MPPT.

**ABBREVIATIONS:** PV: Photovoltaic; DOE: Department of Energy; DG: Distributed Generation; VVO: Optimum Volt/Var; OLTC: On-Load Tap Changer; PCC: Point of Common Coupling; ML: Maximum Likelihood; IRLS: Iteratively Reweighted Least Squares; CVR: Conservative Voltage Reduction; ARMA: Auto-Regressive Moving Average model.

**INTRODUCTION**

The interest to integrate renewable energy resources, such as photovoltaic (PV) panels, into electrical system has increased due to their valuable contribution in mitigating global warming. Therefore, the Department of Energy (DOE) at U.S. targeted 19% penetration of solar power by 2050.1 However, PV generation is stochastic in nature and highly dependent on weather fluctuations. Consequently, many problems emerge once the penetration of PVs reaches to a certain extent, such as the violation of voltage limit at buses. According to [Wang et al]2 voltage fluctuation problems appear in the network once the PV penetration exceeded 20%.

The traditional Volt/Var control devices – such as transformer taps, capacitors banks, and voltage regulators – work based on the local measurements with a time delay response.3 This may not be an effective solution under the circumstances of heavy PV penetration due to the rapid fluctuation in insolation-solar irradiation at ground level. Therefore, to maximize PV penetration and optimize Volt/Var, a smart inverter connecting the PV panel to the grid has been proposed to control the output active and reactive power of the PV.4-6 The smart inverter has two features: 1) fast response, and 2) remote control option. Therefore, the researchers have developed centralized and decentralized coordination methods to optimize the PV output power.

Several methods7-17 have been proposed to optimize the PV output power in a decentralized fashion. The decentralized operation of PVs in [Smith et al]10 establishes a first-order-piece-wise curve, Figure 1a, to determine the amount of injected reactive power according to the instantaneous value of the terminal voltage. In [Malekpour and Pahwa]7, a reactive control scheme is proposed based on the intensity of fluctuations in the weather and the terminal voltage. Therefore, they proposed three operating states: normal state, fluctuate state, and contingency state. First, a coordination scheme determines the operating status. Then, the output reactive power would change according to the operating status to maintain the terminal voltage within the desired range.

Since the R/X ratio in distribution network is high relative to the transmission network’s ratio, the active power injected by PVs contribute significantly to overvoltage mitigation. Therefore, a droop based active power curtailment to prevent overvoltage in distribution network is proposed in [11], see Figure 1b. In [15], they combined the reactive power injection and active power curtailment to enhance PV performance, Figures 1b and 1c. Although, the curtailment of the active power mitigates overvoltage conditions significantly, it is not an attractive option economically. Therefore, researchers in [13] improved the later method by keeping the curtailed power of PVs in storage units and inject it later on in a coordinated fashion to avoid voltage violation at any bus. On the contrary, instead of using storage units, researchers in [15] used the on-site solar measurements-along with the historical data-to preform 15 seconds forecast of solar irradiation. Then, they developed an algorithm to determine the output active and reactive power of the PVs based on the forecasted data and the instantaneous value of the voltage at the terminal bus.

Some researchers use the correlation between the active and reactive power injected by the PV (P/Q) as a control strategy. In other words, the PV absorbs (or injects) the reactive power once a threshold value of the output active power of the PV is reached. However, the previous method is independent of the PV location on the grid. In [14], they enhanced the performance of the aforementioned method by including the location of the PV on the grid into consideration. In [8,9], they take it further by including the location of the PVs with respect to the On-Load Tap Changer (OLTC) to determine the amount of the reactive power injection by PVs. In [17], they used the historical data of the voltage at the critical buses to develop an optimized piece-wise reactive power curve of each PV versus its injected active power.

A predictive approach to counteract overvoltage by determining the reactive power injection is in [18]. They exploited the estimated Thevenin equivalent of the network as seen by the PV point of common coupling (PCC) to predict the limit of the active power injection by PV. Since R/X ratio in distribution system is high, the voltage is sensitive to the amount of active power injected by the PV. Thus, based on the predictive limitation of the active power injected by the PVs, they designed a voltage control loop to avoid overvoltage problems.

The recent increase in the awareness of distribution network opens a new perspective in optimizing PV’s output. Therefore, some researchers exploit the communication infrastructure and accessibility through remote control to enhance the optimality of PV. This approach can be extended to any kind of distributed generation (DG) to search for the optimum Volt/Var (VVO) settings in the system,19-45 including the settings of optimum settings of OLTC, switching capacitors, and voltage regulators. Nonetheless, other publications14,16,46-49 limited their work to optimize PV response with respect to the rest of the system. For instance, in [46], they optimized PV output by including the uncertainty of the forecasted information. The optimization constraints are power loss minimization and voltage violations of a balanced system. Also, they assumed the PV droop curve is linear. In [12], they optimized the output of PVs by updating the parameters of the PVs every 15 minutes. They, also, combined the local control with the remote control to adjust the shape of the curve of both the active and the reactive powers injected by PVs to optimize their output. The result would be local controller of each PV consisting of a piece-wise linear lines (Figure 1d). In [47], they proposed a centralized control of PVs. They used the method of perturb & observe to investigate the effect of the amount of the output power of the PVs on the voltage level of the critical buses. Then, they select the suitable amount of the injected power (active & reactive) of the PVs that maintain the voltage of the buses within an acceptable range. In [48,49], they proposed a centralized coordination for the PV smart inverters. To correct the voltage at a critical bus, the injected reactive power of the nearby PVs will be adjusted based on voltage sensitivity analysis. If the voltage hasn’t improved, the injected reactive power by the nearby PVs will be adjusted accordingly until they reach the desired limits. Otherwise, the active power of the nearby PVs will be curtailed gradually until the voltage at the designated bus is corrected. In [16] they proposed a fair active power curtailment of PV output based on the voltage of the critical buses.

In this work, the proposed a PV-control scheme optimizes its output (or minimizing the curtailments of the active power) during the peak period of the day – it has been assumed the PVs can receive remote commands. The peak period in this work refers to the period whenever the load demand is low and the active power injected by the PVs is high. The proposed control takes into account the uncertainty of load and irradiance. The features of our work are as follow:

1. The PV response is dependent on the statuses of the switching devices – such as voltage.

2. We broaden the PV curve to become a multi variable dependent, instead of being one variable dependent. Therefore, the power injection curve of PV is going to be versus time, instead of being versus the voltage at the terminal bus. This allows us to get the benefits of the forecasted data because they the forecasts are time dependent too.

3. The proposed scheme is suitable for both of the balanced and unbalanced operations. Also, it accounts for the location of PVs inherently.

Since the proposed model is dependent on the remote communications among the Control Center, the Metrological Center, the voltage sensors at the critical buses, and the power measurement devices at load buses, this work relies on the accuracy of states estimation and reliability of the communication medium. Therefore, we assumed the communication medium is ideal and left for future work. The simulation result showed a tremendous enhancement in the system performance – see section V.

This paper is organized as follows. The section titled System Model explains the proposed model and the operatable range of the PV. The section titled Estimation Of The Parameters fi And fij explains the procedure to estimate the parameters of the proposed system. The section titled Optimization Of One Sample Shows the optimization approach to find the optimum parameters. Lastly, section titled Simulation shows the simulation result.^{5}

**SYSTEM MODEL**

The distribution system is unbalanced by nature.50 On the other hand, most of the PVs in the distribution network are single phase, especially the roof top installed ones. Therefore, the PVs are a good candidate to mitigate the imbalance in distribution system if they are distributed wisely across the network.

Based on our review (see the Introduction), we can model most of the existing control schemes of PVs, without the loss of generality, as follow:

The instantaneous value of the variable X(t) is used to determine the value of the injected reactive power Q and the active power P by the PV. The variable X could be either the voltage of the terminal bus or the injected active power of the PV. The instantaneous value of X is multiplied by the function f(X) that is dependent on the value of X itself. In other words, the function f(X) is simply the discrete curve shown in Figure 1. It worth noting that [3] claimed the terminal voltage alone isn’t enough to optimize PV output.

**The Proposed Model**

In this work, we extended the model (1) to become as follow:

To determine the amount of the injected reactive power Q or the injected active power P of the PV, we select n-number of variables X that impacts the output of PV panels significantly & accessible through the communication network. The variable X(t) could be a continuous variable, such as the instantaneous load at a certain bus, or a discrete variable, such as the status of the tap position of a certain device. Unlike previous methods, the function f is no longer dependent on the value of the variable X. In this work, the function f(T) is going to be simply a constant number. However, the value of f(T) is going to be changed based on the time period T of the day. Each period ranges from 30 minutes to 1 hours and denoted as Ti-assuming the peak period of the day lasts for few hours. The centralized control center (CC) would use the knowledge of the forecasted data over the period Ti to find the optimum value of the function . Once the time period Ti starts, the new parameters of the function f would remain unchanged until the next time period Ti+1 starts.

In addition, there is a chance of having a variable Xi that is dependent on variable Xj, while the variable Xj is independent. Concurrently, the PV output (P and Q) is dependent on both of Xj and Xj. For instance, the tap position in voltage regulators is dependent on the instant load while the optimum value of P and Q is dependent on both of them. The remedy to this problem51 is to include a cross product terms, interaction terms, in the model. To avoid confusion, we will use the following terms for the rest of this paper:

1. Response variable: they are the variables of interest. In this work, the output active and reactive power of PVs are the response variables.

2. Independent variables: They are the explanatory variables X that are stochastic, from electrical point of view. Irradiance, ambient temperature, and load demand are examples of independent variables.

3. Dependent variables: they are the explanatory variables X that are dependent on the value of the independent variables. However, the Response Variable is dependent too on the Dependent variables. For instance, the tap position of OLTC is a dependent variable.

If we selected m-number of the dependent variables, the number of the independent variables would be equal to (n−m).

Any variable Xi could be applicable to more than one bus. For instance, if the variable Xi represents the irradiance, it is going to be applicable to all buses where PVs are installed. To include the bus perspective into the model (2), we denoted the ith variable at bth bus as Xib. Hence, the model (2), can be modified to become as follow;

In the model (3), the superscript a represents all the buses where the PVs are connected. The superscript b and s represent the bus number where the variables Xi and Xj are applicable.

**Operatable Range of the PVs**

The reactive power injected by the PV is bounded by the Volt-ampere ratings of the smart inverter and the output active power.36 As shown in Figure 2, the available reactive power is bounded by the amount of the injected active power. Once the active power reaches its maximum, the available reactive power is zero. To overcome this issue, the smart inverter could be overrated.7,52 As follow:

Where Qmax represents the maximum available reactive power ready to be injected by the PV. The notation and represent the apparent power ratings and the output active power of the PV respectively. The overrating factor γ helps in increasing the amount of Qmax.

**ESTIMATION OF THE PARAMETERS fi AND fij**

In this work we assumed the PVs are controlled centrally and locally. The central control uses the knowledge of the forecasted data to find the optimum parameters-namely, fi and fij – of the PV controllers over a period of time T. For instance, the central control receives the forecasted data about cloud covers, irradiance, wind, temperature, etc from the metrological center. Then, the central control perform all the necessary simulations to find the optimal settings of the controllers. Then, the central control transmits the new settings to the PVs in the distribution system. This happens once every time period T – see Figure 3.

Once the local controllers of the PVs received the updated values of fi and fij, the model (3) can be reduced to the following form.

Above model is going to work as a local controller over a definite period T. The parameters α and β are simply a constants. However, the constants α and β has to be selected provided that the response P and Q are optimized. The following subsections explain the estimation of the parameters α and β.

**Data Format**

Initially, the dependent and the independent variables must be selected. For the sake of explanation, the selected variables would be.

1. The independent variables: We will select two independent variables (n=2). The forecasted irradiance X1 and the forecasted load X2 would be the selected independent variables.

2. The dependent variables: We will select one dependent variables (m=1). The selected variable would be the tap position of the voltage regulator, X3.

Let us assume we have one PV installed at bus number No.7 and two loads at bus No.9 and at bus No.8. Also, one voltage regulator is installed at bus No.5. If we implement above scenario into (4) we would obtain the following;

We can re-write above equation into a matrix form as follow;

Or we can generalize above system into the following form.

P= α+β X (5)

Ultimately, for the purpose of the maximum likelihood fittings, we need to have many samples (k-samples) of all X variables and their corresponding optimum P response. The following subsection explains how one sample is build.

**Data Collection for Time Domain T**

In brief, we will use the forecasted data coming from the Metrological Center to synthesis X. Then, we will use the synthesized data X to find the optimum response P.

We start with the independent variables – load and irradiance. In the previous example, the independent variables of interest are X17, X28 and X29. The forecasted information of one day would look like as shown in Figure 4. which represents the forecasted data of both load, and irradiance. One day forecast will be divided into subperiods Ti. However, there will be an uncertainty (or forecast error) associated with each variable. To overcome the uncertainty problem, we calculate the average value ͞Xi of the independent variable Xi over the subperiod Ti.

Then, as shown Figure 4, the forecasted data of the independent variable Xi over the period T will be drawn from a Gaussian distribution with mean of ͞Xi and variance of σ2 – the value of the variance is chosen based on the engineer best judgment.

Now, k-number of samples (a numerical example in the appendix is added for the purpose of explanation) will be drawn from three Gaussian distributions as follow:

) k-samples shall be drawn from a Gaussian distribution~G(͞X17 ,σ2) These samples represent the forecasted data of irradiance at bus No.7.

2) k-samples shall be drawn from a Gaussian distribution ~G(͞X28,σ2). These samples represent the forecasted data of the load at bus No.8.

3) k-samples shall be drawn from a Gaussian distribution ~G(͞X29,σ2). These samples represent the forecasted data of the load at bus No.9.

The next step is to find the corresponding optimum response of P against each sample of the independent variables, ͞X27 , ͞X28 and ͞X29. Refer to the section titled Optimization Of One Sample for more details about the optimization constraints and algorithm.

Lastly, we can find k-number of samples of the dependent variable ͞X25 easily. This can be done by running the power flow using the independent variables, ͞X17, ͞X28 and ͞X29 along with their optimum response P as an inputs. Then, the power flow yields the converged value of the variable ͞X29 against each sample.

Now, we have k-number of samples of both P and X. The next step is to find the parameters α and β that fit to the samples we already have. The next section explains how to find the parameters α and β in details.

**Maximum Likelihood Fitting**

In the previous section, we found P and X for k-number of samples. In this section, we will find the value of α and β that fits to all of the samples. According to reference [53], the system (5) has three components.

**1) Random Component:** This component identifies the response of the system and assume a probability distribution for it. In (5), the response of the system is P.

**2) Systematic Component:** This component specifies the explanatory variables. In our case, the explanatory variables are contained in X.

**3) Link function:** This component specifies the expected value of the Random Component.

We assume the response P follows a normal distribution ~N(μ,σ2) and its link function is equal to μ. The Systematic Component contains linear predictors and interaction terms.

The parameters α and β has to be selected provided that they most likely to produce the collected samples. In other words, we are looking for the maximum likelihood (ML) estimate of the model parameters α and β. The ML fitting can be found numerically by starting with an initial guess of the parameters. Then, an algorithm try to get closer the ML estimates iteratively. The algorithm doing this job is called Iteratively Reweighted Least Squares (IRLS).53,54 Due to space limitation, IRLS algorithm is omitted.

**OPTIMIZATION OF ONE SAMPLE**

In this section, we explain how to find the optimum response (e.g., P) corresponding to one sample of the independent variables (e.g., X17, X28, and X29). The problem formulation would be as follow.

**Objective Function**

We have the following optimization problem to be solved:

The term ℘1 denotes the power losses in the main feeder line and the laterals. It is formulated as follow;

Lij represents the losses in the line (s) between bus i and j. The term ℘2 represents the switching steps of the switching capacitors, voltage regulators, and OLTC. In this paper, the number of switching steps are calculated with respect to the previous switching state.

The swOLTC represents the number of the change in tap position of OLTC with respect to the previous switching state. The same principle is applied for the terms and but for the capacitor banks and the voltage regulators, respectively.

The term ℘3 represents the active power injected to the distribution network. In (6), the term ℘3 is multiplied by minus sign to maximize the active power penetration of the PV panels. The ℘3 is formulated as follow:

Pi represents the active power injected by the PV panel installed at ith bus.

The conservative voltage reduction (CVR) are represented in the optimization function as ℘4. It is formulated as follow;

The voltage at ith bus is denoted as Vi, in p.u. The term ℘4 is applied to a selected set of buses which are going to be a remote buses with respect to the infinite bus and directly connected to the load. The term ℘4 is a non-negative number because the voltage Vi is constrained to be above 0.95, see next section.

**Constraints**

The optimization objectives of the previous section are constrained to the following conditions.

1) The maximum and minimum voltage on all buses: 0.95 ≤

VAll.Buses ≤1.05

2) The line ampacity limits.

3) Maximum and minimum limits of PVs active and reactive power due to its rated values and harmonics limitations.7

4) The maximum switching steps of the OLTC, the voltage regulators, and the switching capacitors.

5) The maximum capacity of the installed electric devices, such as the transformers, induction motors, generators, etc.

**Control Variables**

The fast response of the PVs to curtail active power or injecting/absorbing reactive power make them a good candidates to reduce system’s losses. In this work, the PV devices will be the main contributors in optimizing the distribution network.

**SIMULATION**

The proposed method has been simulated on IEEE123-bus system. The PV panels has been added to the IEEE123 according to the load pattern. For instance, the customers who draws large amount of active power are considered rich and able to install PV panels to reduce their electric bills. The selected buses to install PV panels are shown in Table 1. Along with the rated PV power. According to [Wang et al],2 the PV penetration problems emerge beyond 20% penetration. In this work, the penetration is about 20 % (with respect to the total connected load). However, the ratio of the input power from PVs to the instantaneous load varies across the day. Therefore, the 20% penetration of PV active power to the distribution network isn’t an accurate indicator unless it represents the instantaneous data.

**Solar Data
**

Both of the measured and forecasted data of solar irradiance are adopted from sources [Sexauer]55 [Sexauer and Mohagheghi]56. The measured data of solar irradiance has been collected from the Solar Radiation Research Laboratory of NREL located at Golden, between 2006 and 2011. The data is processed to detect cloud events. Then, the appropriate distribution has been fitted to the data. The forecasted data were formulated by modulating the extraterrestrial irradiance by a stochastic cloud events. The number of cloud events were randomly generated to follow a Poisson distribution with Lambda of 7.68 for summer season. The inter-event waiting time is modelled as an exponential distribution with mean of 29.18 minutes for summer season. The variation of irradiance were modelled using the Auto-Regressive Moving Average model (ARMA) with five lags.

In this work, we selected a moderately clouded day during summer time. The forecasted data of the selected day are shown in Figure 5a and the measured data are shown in Figure 5b.

**Load Data**

According to [Sexauer and Mohagheghi]56 the typical load profile of residential house follows a two peaks distribution, as shown in Figure 6. The first peak has a mean of μ1 and the second peak has a mean of μ2. We used the two peak curve to synthesised a load data of the IEEE 123 test system. The data are synthesized according to the method described in the section titled Data Collection for Time Domain T.

**Model**

The period Ti is selected to be 1 hour. Then, a hundred samples has been drawn from the forecasted load and irradiance for each time period Ti. Then, for each sample, we used the genetic algorithm to find the optimum response of all PVs. Once the optimum response against each sample is found, the terminal voltage of each PV and the tap position of the voltage regulators are recorded. The inverter is 120% overrated (γ=0.2).

The model selected for each PV is shown in (8) where Pi and Qi represent the injected active and reactive power of the PV installed at ith bus. The chosen explanatory variables are.

1. The terminal voltage Vi of the PV installed at ith bus. The voltage values are in p.u.

2. The three phase active power P47 , P48, P49, P65, and P76 consumed consumed at buses number 47, 48, 49, 65, and 76 respectively. The value of the power in the model is calculated by taking the ratio of the instantaneous power to the maximum size of the load connected the designated bus – Pi(t)/Pimax.

3. The tap position τ25c of the voltage regulator connected to phase C at bus number 25.

4. The tap position τ160c of the voltage regulator connected to phase C at bus number 160.

5. The irradiance I at the ith bus.

The Matlab command “fitglm” were used to do the fittings. Since we have 42 PVs, 46 coefficients and 12 hours the number of the obtained coefficients is about 23184. Due to space limitation, the values of the coefficients are omitted (it is a table containing 23000 cells approximately).

**RESULT AND DISCUSSION**

The plot of the losses between 6 AM to 7 PM in Figure 7. The losses are minimal during peak period where the ratio of PV-to-Load is high. In the same figure, an index is added to indicate forecasting accuracy. The forecasting accuracy index (FAI) is obtained using (7). In (7), both of the active power of the load Lp and the reactive power of the load Lq are used to determine FAI, in addition to the irradiance I.

The proposed method proved to be highly dependent on the accuracy of the forecasted data. In Figure 7, once the forecasting accuracy index start fluctuating (poor forecast) between 3 PM and 7 PM, the losses of the system increases dramatically. This tells us, the proposed scheme yields a better result as long as the forecast error margin is low.

The total demanded load during the day fluctuates and the same thing applies to the output active power by the PVs. Therefore, despite the penetration level is about 20%, the ratio (R) of the injected active power of the PVs to the demanded load during the day will not remain 20%. The peak period of the day is the time when R is high. In our case, the peak period happened to be between 11 AM and 3 PM. In our simulation, the R reaches 0.53 (53%) around 1 PM. Figure 8 shows the value of R during the peak period of the day. Despite the ratio R ranges between 38% to 53%, no voltage violation of any bus is recorded. This means that the proposed scheme allows PV penetration to reaches 53% while voltage of all buses remain within limit (±5%).

Although, no voltage violation is recorded, some of the active power of PVs are curtailed. Figure 9 shows the amount of the curtailed active power of the PVs during the peak period. At worst scenario, 2.8% of the PVs active power is curtailed. This tells us that the proposed scheme minimize the active power curtailment even if the penetration reaches 50%.

**Figure 9:** The curtailed active power of the PVs during the peak period to maintain bus voltage within limits & to reduce losses. The horizontal axis represents the time slot of the day. The vertical axis on the left represent the amount of the curtailed active power of the PVs in kW. The vertical axis the right represent the curtailed active power in % relative to the total rated power of the PVs (658 kW).

**CONCLUSION**

This paper describes a central control scheme of the PVs during peak times. The peak time is defined in this work as the period when the ratio of the PV power to the demanded load is high. In unbalanced distribution system, the proposed scheme yields a better optimization result as long as the forecast error is minimal. The optimum response of the PVs is determined based on the status of many variables that are selected by the users. The selection of variables is limited to the existing infrastructure of the system, such as the existing Remote Terminals Units (RTU), availability of the communication medium, etc.

The proposed scheme is a time dependent. This feature-time dependency-allows the user to get the benefit of the forecasted data, because forecasting is a time dependent too. The simulation shows no bus-voltage violation under heavy PV penetration. The active power curtailment were low, about 2.8% at worst case. However, the curtailment of the active power isn’t equalized (fair curtailment) among PV.

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**APPENDIX**

The purpose of the following example is to explain the meaning of the following statement “k-number of samples is drawn from a Gaussian distribution ~G(0,7)”.

Example: Let us assume we have a set of data (any set of data) that contains N number of elements. The data set satisfies the following three conditions:

1) The average of all data is 0

2) The standard deviation of the data is 7.

3) The shape of the histogram plot of the data looks similar to the shape of the Gaussian distribution.

If we have a data set satisfying the above conditions, we can say the data set follows a Gaussian distribution~G(0,7). Now, let us draw a k-number of elements randomly from the above data set, such that k