Stochastic PV model for power system planning applications

IET Renewable Power GenerationVolume 13, Issue 16 p. 3168-3179 Research ArticleFree Access Stochastic PV model for power system planning applications Ameena Saad Al-Sumaiti, Corresponding Author Ameena Saad Al-Sumaiti ameena.alsumaiti@ku.ac.ae Advanced Power and Energy Center, Electrical and Computer Science Department, Khalifa University, Abu Dhabi, United Arab EmiratesSearch for more papers by this authorMohammed Hassan Ahmed, Mohammed Hassan Ahmed Electrical and Computer Engineering Department, University of Waterloo, Waterloo, Ontario, CanadaSearch for more papers by this authorSergio Rivera, Sergio Rivera Electrical and Electronics Engineering Department, Universidad Nacional de Colombia, Bogota, ColombiaSearch for more papers by this authorMohammed Shawky El Moursi, Mohammed Shawky El Moursi Advanced Power and Energy Center, Electrical and Computer Science Department, Khalifa University, Abu Dhabi, United Arab EmiratesSearch for more papers by this authorMohamed M.A. Salama, Mohamed M.A. Salama Electrical and Computer Engineering Department, University of Waterloo, Waterloo, Ontario, CanadaSearch for more papers by this authorTareefa Alsumaiti, Tareefa Alsumaiti Geography and Urban Planning Department, United Arab Emirates University, Al-Ain, United Arab EmiratesSearch for more papers by this author Ameena Saad Al-Sumaiti, Corresponding Author Ameena Saad Al-Sumaiti ameena.alsumaiti@ku.ac.ae Advanced Power and Energy Center, Electrical and Computer Science Department, Khalifa University, Abu Dhabi, United Arab EmiratesSearch for more papers by this authorMohammed Hassan Ahmed, Mohammed Hassan Ahmed Electrical and Computer Engineering Department, University of Waterloo, Waterloo, Ontario, CanadaSearch for more papers by this authorSergio Rivera, Sergio Rivera Electrical and Electronics Engineering Department, Universidad Nacional de Colombia, Bogota, ColombiaSearch for more papers by this authorMohammed Shawky El Moursi, Mohammed Shawky El Moursi Advanced Power and Energy Center, Electrical and Computer Science Department, Khalifa University, Abu Dhabi, United Arab EmiratesSearch for more papers by this authorMohamed M.A. Salama, Mohamed M.A. Salama Electrical and Computer Engineering Department, University of Waterloo, Waterloo, Ontario, CanadaSearch for more papers by this authorTareefa Alsumaiti, Tareefa Alsumaiti Geography and Urban Planning Department, United Arab Emirates University, Al-Ain, United Arab EmiratesSearch for more papers by this author First published: 25 November 2019 https://doi.org/10.1049/iet-rpg.2019.0345Citations: 22AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Planning photovoltaic (PV) power systems integration into the grid necessitates accurate modelling of renewable power generation. Global solar irradiance, weather temperature and PV power losses due to overheating specifically in hot regimes are major factors contributing to PV power generation uncertainty. This study targets demonstrating the effectiveness of deploying advanced five parameter probabilistic distribution ‘Wakeby’ for modelling PV uncertain power generation, measured as a function of such factors, in power system planning applications. The impact of different approaches for incorporating weather temperature on PV energy estimation is studied. Wakeby-Monte Carlo Simulation for PV power data training with an emphasis on MCS stopping criteria for such advanced distribution is presented. The model is tested and verified in 31-bus distribution system to demonstrate its effectiveness over other literature uncertainty modelling approaches when planning integration of PV systems' integration into the grid to minimise the grid losses cost. Real PV power measurements are utilised as benchmark verifying the accuracy and suitability of the presented uncertainty modelling approach. Simulation results demonstrate a small error of $4.7 in the expected annual cost of grid losses when deploying Wakeby model compared to the benchmark case and that error can vary significantly when deploying other PV models. Nomenclature , b, ç, d, factors relating the parameters of Wakeby distribution as explained in Section 2.3.1 area area of PV module (m2) c representative cluster number of PV power at the probability of occurrence CPL annual cost of energy losses ($) cost annual cost per unit of energy loss DC1 first central moment about zero mean of the Wakeby data generated from the Monte Carlo simulation (MCS) DC2 second central moment of the Wakeby data generated from the MCS DR2 second raw moment of the Wakeby data generated from the MCS for (xi−e) DR3 third raw moment of the Wakeby data generated from the MCS for (xi−e) DR4 fourth raw moment of the Wakeby data generated from the MCS for (xi−e) ech error in the h central moment (h = {1st, 2nd}) eRf error in the f raw moment (f = {3rd, 4th}) E empirical cumulative distribution f order of the raw moment that can be either 3 or 4 ff probability distribution function F theoretical cumulative distribution FF fill factor G conductance of the line in the IEEE-31 bus distribution test system in per unit GC1 first central moment of the general extreme value distribution GC2 second central moment of the general extreme value distribution h order of the central moment that can be either 1 or 2 i, j, u, l, o order of the PV power data point such that i>j > u > l > o Imp current of PV module at maximum power point (A) Isc short-circuit current of PV module (A) , bus number in the 31- bus distribution test system KS Statistic Kolmogorov–Smirnov statistic m month of the year n total number of PV power data points n(i) rank of the PV power data point number of data points generated from the MCS N number of modules total number of buses the 31-bus distribution test system NCOT nominal open-circuited cells temperature (°C) given TNCOT, wind velocity of 1 m/s and λ@NCOT Pmaximum maximum PV system power output while neglecting system losses (W) Pnew-maximum new PV system power output after major system losses are accounted for (W) PPV Pnew-maximum converted to MW PR probability t time (hour) of the day T total number of hours in the year Tambient ambient air temperature (°C) Tcell operating PV cell temperature (°C) TNCOT ambient temperature of 20°C Tstc PV cell temperature at standard test conditions (°C) V voltage magnitude in per unit Vmp PV module voltage at the maximum power point (V) Voc PV module open-circuit voltage (V) WC1 first central moment WC2 second central moment WR3 third raw moment WR4 fourth raw moment αcurrent PV module current–temperature coefficient as per the manufacturer (%/°C) Bvoltage voltage–temperature coefficient as per the manufacturer (%/°C) λ global solar irradiance (W/m2) λ@NCOT solar irradiance of 800 W/m2 at the cell surface ζpower power–temperature coefficient as given by the manufacturer (%/°C) η PV system efficiency (%) Ω factor used to calculate the impact of system losses on PV power generation (η = 1 + Ω) ΩDC cable loss factor used to find losses due to DC cable loss (%) Ωmismatch loss factor used to determine the mismatches effect on losses (%) and thus system efficiency Ωsoiling loss factor used to determine soiling effect on losses (%) Ωsolar radiation loss factor used to determine losses due to solar radiation reflection from glass on PV surface (%) Ωtemperature derating factor used to determine losses due to module temperature derating (%) Gamma function of α, β, γ, δ, ɛ five parameters of Wakeby distribution K, µ, σ three parameters of general extreme value distribution θ voltage angle 1 Introduction In response to limited fossil fuel reserves, investment in photovoltaic (PV) renewable energy is a sustainable option for supplying electricity [1]. Planning solar energy projects [2-4] requires reliable knowledge of a long-term climate measurement as the main factors influencing PV power generation are not only the uncertainty in solar irradiance but also the ambient air temperature and the array structure [5]. The unanticipated variations in PV power generation can contribute significantly to the cost of operation of an electric network [6]. Therefore, the accuracy of PV power prediction is necessary. Solar irradiance forecast [7] or errors in PV cell temperature forecast [8] was used for predicting future PV power; however, key factors governing such a power production (open-circuit voltage and short-circuit current) were not accounted for. Given the importance of the variant factors influencing the PV power generation, this section will provide an overview of literature work focusing on the probabilistic modelling aspects of PV power and factors driving the estimation of such a power generation. Then, a five-parameter Wakeby distribution (WBD), recommended in this work as better fit, is introduced. After that, the objectives of this paper and its organisation are presented. 1.1 Probabilistic modelling aspects of PV power A variety of research works had been conducted to model the solar irradiance and its uncertainty as it is the main driver of the PV power generation needed in many power system applications. Estimating the solar irradiance time horizons and the daily energy by clustering the earlier daily values using the principal component method was described in [9]; while the study in [10] focused on clustering solar irradiance monthly-hourly data to guide practical problem solving based on population parameters. Most researchers modelled the solar irradiance uncertainty as a beta distribution [8, 11, 12]. Others employed a statistical model of the daily global solar irradiance [13] or probabilistic distributional analysis of the solar irradiance through major well-known distributions: exponential, Weibull, gamma, normal, log normal, beta, or geometric [14]. For example, normal distribution was considered in [15] for uncertainty modelling of solar irradiance when optimally planning the distributed generation (DG) in a grid incorporating a risk-based approach. Also, Niknam et al. [16] deployed the normal distribution to model the parameter uncertainty of the solar PV system when studying the effect of producing hydrogen and recovering thermal energy on managing micro-grids’ energy. In summary, a probabilistic beta solar irradiance distribution is suitable for estimating PV power independently of temperature when the solar energy is a function of both the surface area (m2) and the solar irradiance (W/m2) [11]. Yet, such a model fails to resemble reality, in which temperature is a major factor in the actual performance of a PV module as indicated in [17] and discussed next. The relationship between the daily solar irradiance and the daily maximum temperature variation was shown in [18]. Monte Carlo simulation (MCS) was used in [19] to model the random behaviour of solar irradiance, and then PV power was calculated using the generated solar irradiance data along with the mean temperature values. No allowance was made for the probabilistic nature of the temperature. On the other hand, both the solar irradiance and the ambient air temperature were used in [20] to find a power probability density function suitable for short-term forecasting. In this case, a Bayesian autoregressive time-series model was used to predict the clearness index, and an MCS was employed to assess the power distribution function for a random sampling of the clearness index. Still, using the nonlinear relationship between the clearness index and power to identify one representative power value can lead to a huge error due to the difficulty in prioritising which distribution parameter effectively describes this single value. Given this analysis, more elaboration on how the clearness index is incorporated into the literature to account for the PV power prediction is discussed next. Solar irradiance is an important factor that affects the amount of power generation from the PV system. Therefore, measuring the effective amount of solar irradiance that reaches the earth is necessary for accurate prediction of PV power. The clearness index defines the fraction of solar irradiance reaching the earth surface [21]. The clearness index exhibits uncertainty and can vary over the day and from one day to another. In the literature, many attempts have been shown to model the variability in the clearness index. In [21], daily clearness index patterns were developed through various clustering techniques for designing PV systems. Al-Saadi et al. [22] focused on a unimodal parameterised shape to model the uncertainty in the clearness index used to measure the PV power output. In [23], the uncertainty in the clearness index was modelled through Graham's algorithm. Moreover, modelling the horizontal solar irradiance for every clearness index was reported in [24]. In contrast, modelling the uncertainty in the clearness index is unnecessary when considering the global solar irradiance data as the latter is a function of it as presented in [25]. In summary, considering the clearness index is effective when it is used to determine the global solar irradiance data that can be used for PV power calculation. PV power generation is uncertain in nature and can vary with both the solar irradiance and temperature variations as elaborated earlier. In the literature, several researchers tried to model the uncertainty in PV power generation. For example, the randomness of PV power was modelled in [8] using two independent probabilistic distributions, beta distribution to represent the uncertainty in solar irradiance and normal distribution to model the error in the predicted cell temperature. Such an independent modelling of the two variables is not extremely true as [26] had emphasised on the dependency of the cell temperature on the solar irradiance. Tabone and Callaway [27] highlighted the importance of modelling the uncertainty in PV power generation to include the significant effect of its low probability events on planning reserve requirement in the power system. An intent to represent the PV power uncertainty by a non-parametric Kernel distribution was presented in [28]. Such a representation demands a large sample size, a selection of kernel function type, an estimation of the function bandwidth, a validation process, and optimisation methods resulting in extensive computations. This model is not valid when the time step is to be reduced [28]. Moreover, fitting the data using kernel distribution requires the right positioning of the data making it harder to exclude any single point of the incomplete data record [29]. PV power generation data obtained from a rooftop PV system was considered in [30], and MCS was applied in a problem targeting managing the aggregator's energy in the grid accommodating PV generation. However, the number of generated scenarios was limited and assigned an equal probability. Also, the work in [31] used a typical day PV profile for managing the energy of a microgrid in a Stackelberg game, and the billing mechanism was the only factor incorporated to account for the uncertainty in PV generation as the error is usually experienced with short-term forecasting. In summary, modelling the power output of a PV system may be favoured over modelling individual components driving its generation. However, if no real measurements are available, all factors driving such a power generation are to be accounted for to have a better estimation of such a power. For accurate power system planning applications, training of such data is necessary to account for all possible scenarios of PV power generation. Another factor that governs the power output production from a solar PV system is the latter's efficiency reflecting system losses as in [20, 32]. Since energy loss due to module temperature is a component in the efficiency model of a PV system, the uncertainty in module surface temperature will result in an uncertainty in such a loss. The concept of system efficiency in [20] does not account for the variation in losses but rather sets such a loss to be fixed all times even though the temperature could vary. Although this effect was addressed in [17], the model focused on the total daily solar irradiance instead of the hourly PV power. Moreover, the temperature effect on PV power losses was discussed in [33]; however, it was measured such as a daily average temperature effect on PV system efficiency and then the power was calculated accounting for the hourly uncertainty in the clearness index rather than finding such an efficiency hourly. Moreover, Tina et al. [33] followed the assumption in [34] where the difference between the cell temperature and the average weather temperature was set fixed and also varied through a fixed temperature tolerance. Overall, the variability of PV module temperature loss is not to be neglected specifically in hot regimes as it can influence PV power prediction. 1.2 Modelling of randomness through WBD The WBD was used in the literature to model the randomness of data in various fields. It was used to model the equivalence carbon dioxide intensity of apartment buildings [35], discharges of flow [36], rainfall [37] and regions of a probabilistic distribution function for solar irradiance data [38]. Since the WBD can model heavy events in the distribution tail, such one model can accurately and effectively model the probabilistic nature of PV power. A WBD is effective for anticipating high return phases and for matching frequency analysis events to those that can be estimated with the amount of recorded data available [39]. This advanced distribution, which overcomes deficiencies of traditional distributions [40], is distinguished for its five parameters, which constitutes a significant increase compared to the two or three parameters associated with standard distributions [40] providing it with flexibility to acquire a broader variety of distributional shapes than other traditional distributions [41]. A notable feature of the WBD is mimicking of shapes of distributions that are skewed including extreme value, log-normal, and Pearson distributions [37, 41]. This category of distribution is suitable for describing a number of realistic observations because of its finite lower bound and the fact that its random samples can be easily generated from its explicit quantile function [41]. For modelling the PV system power output, it is preferred over the bimodal beta distribution utilised for modelling the solar irradiance in [19], which requires the data to be categorised into two groups such that every group has a unimodal distribution. The absence of such a requirement facilitates the application and analysis of the WBD. A further description is provided in Section 2.3.1. 1.3 Paper objectives and structure The goal of this paper, in comparison to the aforementioned literature [7-34], is to first present and validate a stochastic PV power modelling framework and is second to demonstrate the effectiveness of using an advanced stochastic PV model in a more accurate power system planning application. A comparison of the effect of the deployed model with respect to literature PV models on minimising the cost of grid losses is conducted. More specifically, the problem of location and sizing of DG units, specifically PV, in a 31-bus distribution system for minimising such a cost under uncertainty of PV generation is revisited. This paper is organised as follows: Section 2 provides an overview of the PV modelling approach. Section 3 presents the deployment of an MCS to regenerate PV power data with an emphasis on the stopping criteria for data regeneration. Section 4 investigates the impact of data size on the precision of PV power estimation. Section 5 proves the accuracy of the applied methodology for data regeneration through MCS. Section 6 focuses on estimating the PV energy considering different approaches for incorporating the temperature, an important factor influencing PV power generation. Finally, Section 7 provides a power system planning application where the effectiveness of the stochastic Wakeby PV model is compared to the current state-of-the-art in modelling the PV power output. 2 PV modelling framework Effective planning of a power system with PV power penetration necessitates evaluating the system operation under all possible scenarios of PV power generation to account for the latter's uncertainty. This paper recommends an advanced distribution ‘Wakeby’ to model such uncertainty and compare such a model with the current state-of-the-art for evaluating the randomness of factors influencing the PV power generation. This section provides an overall discussion on the PV modelling framework, described in Fig. 1. The framework is subject to using codes in MATLAB software, excel and statistical test tool. Fig. 1Open in figure viewerPowerPoint PV modelling framework 2.1 Hourly PV power measurements The 1 MW PV power measurement stage uses seven years hourly climate data of global solar irradiance (inherently accounting for clearness index) and ambient air temperature for Amritsar city in India [42] as well as module characteristics as in Table 1 given that the PV system is made of two types of modules, 225 PV module and 240 PV module [43]. Table 1. PV power model parameters Parameter Value Parameter Value Bvoltage [43] −0.365%/°C TNCOT [19] 20°C αcurrent [43] 0.035%/°C λ@NCOT [19] 800 W/m2 Vmp [43] 28.63 V for 225 PV Ϛpower [43] −0.45%/°C 29.60 V for 240 PV Imp [43] 7.93 A for 225 PV Ωmismatch loss −97.6% 8.12 A for 240 PV Isc [43] 8.52 A for 225 PV ΩDC cable loss −98.7% 8.58 A for 240 PV Voc [43] 37.50 V for 225 PV Ωsolar radiation loss −97% 37.50 V for 240 PV NCOT [43] 45°C Ωsoiling loss −95.5% λ@stc [ 44] 1000 W/m2 Tstc [44] 25°C The PV power (Pnew maximum) can be measured according to (1)–(8) [24, 44-47] after accounting for the effect of weather and manmade losses in [44, 48] on system performance. To elaborate further on this concept, the manufactured PV system expected power output (Pmaximum), given in (1), is described by the PV module characteristics. The module characteristics include the fill factor in (2) that describes how square is the current–voltage characteristic curve of the PV solar cell [49], module open-circuit voltage (Voc) in (3), module short-circuit current (Isc) in (4), and the number of PV modules forming the PV system (N) (1) (2) (3) (4) The PV system may experience major system losses degrading the PV power generation from Pmaximum to Pnew maximum as given below: (5) The PV system losses are accounted for through the concept of PV system efficiency. The concept of system efficiency implies the impact consideration of solar radiation loss, mismatch loss, DC cable loss, soiling loss and losses due to temperature derating on PV power generation [44]. These impacts are considered in two forms subject to the current PV cell temperature, given by (6), in comparison to that temperature at the standard test condition as described in (7a) and (7b). In more details, if the PV cell temperature is less than or equal to its temperature at the standard test condition (7a), then the impact of the major system losses is accounted for except for the impact of temperature derating. Such an impact is given by (8) and is considered when the PV cell temperature is greater than the standard test condition (7b). (6) (7a) (7b) (8) 2.2 Probability distribution fitting to PV power data The maximum likelihood, L-moments, least-square and method of moments are alternative methods that can be applied to fit probabilistic distributions to PV power data. The reasons behind the variety of such methods are further explained in [50]. Sixty probabilistic distributions are fitted for the hourly PV power data. Some of the distributions are displayed in Fig. 2. The best model for describing the data nature is selected based on goodness of fit test, which finds the distance between the distribution and the power data. This distance and the threshold value are then compared. The threshold value is based on the size of the data set and the probability of rejecting what might seem to be a poor distribution for describing the data when it actually provides a good fit [51] as described next. Fig. 2Open in figure viewerPowerPoint Comparison between a proposed probabilistic distribution and those used in the literature for PV power output data at 9 am in April (example) 2.3 Kolmogorov–Smirnov (KS) test The choice of the best distribution describing the PV power is based on the KS test of goodness of fit [52]. Given that the data points are arranged in an increasing order (x1, x2, …, xn), an empirical cumulative distribution underlying the KS test is given by (9) [52], where n(i) is the rank of the data point, and n is the total number of data points, provided that the first point in the arranged set has the order zero. The maximum vertical difference between the theoretical cumulative distribution function and the empirical cumulative distribution function denotes the KS statistic given by (10) [52]. The distributions fitting data are ranked in order based on this statistic. The best fit for PV power data will have the lowest KS Statistic value. This test is favoured over others as it does not assume a data distribution [53], as with student's t-test, and does not require a specific sample size as with chi-square test [52] (9) (10) 2.4 Chosen probabilistic distribution Sixty probabilistic distributions are fitted for the hourly power data. 85% of the time, the WBD serves as the best model. This is further explained by the example in Table 2 and Fig. 2. Thus, the WBD is chosen to represent the hourly probabilistic behaviour of the PV power. Under rare circumstances, a general extreme value distribution (GEVD) is the best distribution describing PV power and chosen as the best fit when PV power regeneration from the next stage is hard to attain by MCS due to tolerance specification as further explained in Sections 2.4.2 and 3. Table 2. Ranking of April's PV power probabilistic distributions based on KS statistic (example: 9 am–12 pm) Parameter 9 am 10 am 11 am 12 pm Test/rank KS test Rank KS test Rank KS test Rank KS test Rank Wakeby 0.04402 1 0.07883 1 0.04601 1 0.07785 1 enhanced beta 0.14689 9 0.22279 13 0.19534 14 0.1916 16 gen. extreme value 0.07608 4 0.14982 5 0.14376 6 0.15735 11 normal 0.22689 22 0.28604 25 0.26311 24 0.2451 24 Weibull 0.26734 38 0.30691 35 0.2671 30 0.26298 31 Gamma 0.2545 35 0.31168 37 0.28418 36 0.26292 30 log normal 0.28768 42 0.33269 42 0.30766 41 0.29404 40 Rayleigh 0.31183 49 0.35495 47 0.35096 49 0.38655 50 2.4.1 Five parameters WBD model The WBD is characterised with its five parameters compared to the other three parameters probabilistic distributions. The three parameters distributions are deficient in having enough kurtosis for any specific skew [40]. In contrast, the five parameters WBD can overcome this problem [40] as it is flexible [39] compared to the other various probabilistic distributions as explained earlier in Section 1.6. Sometimes, data does not follow a specific distribution all the time and in some attempts as in [19], the data is divided into groups and two distributions are used rather than one. If the whole data can be modelled through the five parameters distribution; however, the decision is made to have three-parameter distribution fitting it, the whole curve will be distorted [40]. Therefore, WBD is effective and more accurate in modelling the randomness of the data. The WBD fitting data through L-moments is expressed by the transformation in (11) [54], where . This transformation can be rewritten as in (12) [40] because the essential elements for plotting the distribution are relations among parameters, where , b = β, ç = γ/δ, d = δ, and . The location of the distribution peak is affected by the values of α, β, and ɛ, while the height of the peak is affected by δ and γ. The probability density function (f(x)) of the WBD is given by (13), where is defined as shown in (14) [54, 55]. Further explanations on how to estimate parameters of this distribution can be found in [37]. (11) (12) (13) (14) To determine the parameters of the WBD, its L-moments based on PV power data are to be determined as given by the left-hand side of (15)–(19) according to [37, 56]. In this case, xi, xj, xu, xl and xo are the ordered PV power data in ascending order, respectively [56]. The five L-moments have relationships with the parameters of the WBD shown in the right-hand side of these equations [37]. Solving the equations through Newton–Raphson iteration as declared by Park et al. [37] can result in finding the values of these parameters (15) (16) (17) (18) (19) 2.4.2 General extreme value distribution When MCS is used for regenerating the PV system power output data, the error between the distribution that describes the data and the distribution that fits the generate