Impact of Stochastically Distributed Renewable PV Generation on Distribution Network

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1 Impact of Stochastically Distributed Renewable PV Generation on Distribution Network Miroslav M. Begovic, FIEEE Dept. of ECE Texas A&M University, USA Insu Kim, MIEEE School of ECE Georgia Tech, USA Mladen Kezunovic, FIEEE Dept. of ECE Texas A&M University, USA Abstract--Small-scale and decentralized renewable distributed generation (DG) systems, especially PV systems, will soon be dispersed in distribution networks. The objective of this study is to present tools and algorithms useful for planning distribution networks enhanced by randomly distributed DG. To remove the uncertainties of the location, the capacity, and the field orientation of randomly distributed PV systems, a stochastic simulation algorithm is implemented. The stochastic simulation analyzes the impact of stochastically dispersed residential PV systems from the perspective of energy, especially regarding peak power, electricity generation costs, and emissions, and quantifies the effects of the method of variance reduction, including importance sampling. Index Terms--Distributed generation (DG), photovoltaic (PV), importance sampling, sampling of representative clusters and extreme points, and stochastic Monte Carlo simulation. I. INTRODUCTION Distributed generation (DG) systems, mostly with small capacities, are growing in distribution networks. Because of the inherent uncertainty of the solar resource input, the location, and the capacity, the conventional algorithm for power-flow analysis is not useful in determining their impact on the network. Stochastic process simulation such as a Monte Carlo method may be more useful for that purpose. One study quantifies the benefits of PV systems by examining the relationship between the generation of a PV system and peak loads from the standpoint of energy savings, loss, and emissions as well as a reliability and economic analysis [1]. A method for calculating distribution line losses and optimizing the plant size and the feeder configuration is presented in []. The effect of a PV system installed on radial and looped feeders is investigated from the perspective of power quality such as voltage, active power, and loss using MATLAB [3]. Three-phase power-flow algorithm for shipboard power systems for analysis of the effects of DG system with various capacities and with varying loads is shown in [4]. Stochastic methods model the influence of the many uncertainties by conducting random sampling. One study estimates the uncertainty of loads and the solar resource input to PV systems, in [5, 6]. A testbed consisting of 69-bus test feeder using the stochastic Monte Carlo method is used for application of the method of sampling representative clusters and boundary points of large data sets obtained from stochastic simulations [7, 8]. Recent study proposes a probabilistic power-flow algorithm to remove the uncertainty of solar input and load variations [9]. The capacity of the PV system as affected by policy incentives, personal characteristics, income level, age, race, and education is studied in [10]. The inherent high-dimensional uncertainty of the input, the location, and the capacity of the DG system are harder to study, especially from the standpoint of long-term, typically annual, operation. The high-dimensionality of uncertainty requires a very large number of simulations, possible only with a fast-performing, fine-tuned, and optimized algorithm. In this paper, we propose a stochastic simulation algorithm combined with the fine-tuned threephase power-flow algorithm [11]. Then, to reduce the number of computational steps, we explore variance reduction (including importance sampling and stratification sampling presented in [1, 13]) and the sampling of representative clusters and extreme points proposed in [7, 8]. This paper is organized as follows: Section describes the problem statement; Section 3 defines the uncertainties of stochastically dispersed DG systems; Section 4 introduces the stochastic methods of variance reduction, including importance sampling and stratification sampling; Section 5 presents the procedure of the proposed stochastic simulation algorithm; Section 6 provides a case study; Section 7 presents the results of stochastic simulations; and then Section 8 summarizes major conclusions. II. PROBLEM STATEMENT To analyze a set of PV systems on urban distribution feeders, in other words, to determine the annual effect of the energy savings, the emissions savings, and the generation cost changes of residential PV systems with a capacity of ten percent of total peak demand stochastically dispersed

2 throughout the urban distribution network, we propose the following objective function: θ = EPEM [,,, andc], (1) Minimize Var[ θ ], () where P = the annual power generation of residential PV systems in kwh/year/; E, M, and C = the annual energy savings in kwh/year/, emissions savings in kg/year/ or gallons/year/, and the generation cost changes in $/year/ of PV systems. III. DESIGN OF UNCERTAINTY To determine the objective function defined in (1), the following uncertainties are analyzed: location, generation capacity, and orientation of the PV system. A. PV Systems First, the capacity of the individual PV system for residential customers may be affected by the size of the dwelling, various incentives, rebates, government subsidies, rooftop sizes, and daily, monthly, or annual load consumption patterns. Therefore, PV system capacity is not unique for individual residential customers. We analyzed the distribution of PV system capacities collected from 114,066 residential customers in California in 014 [14]. A conveniently flexible distribution for importance sampling is a normal distribution, so we assume that the PV system capacities are consistent with a truncated normal distribution. Distribution of the azimuth angles of individual residential roofs is not known, and in the absence of better information, we model uniformly distributed azimuth angles. For a south-facing PV systems, which indicates an azimuth angle of 180, we use a truncated normal distribution with a mean of 180 and a standard deviation determined by the range rule for azimuth angles. PV module shipments for the residential sector in Georgia in 010 were 99.15% crystalline silicon (c-si) type and a 0.85 % thin film type [15]. The inherent uncertainties of the capacity and the field orientation of stochastically dispersed residential PV systems, we therefore propose the following uncertainties: TABLE I The uncertainty of stochastically dispersed residential PV systems Sample Random Variable Value Distribution Size System capacity (inverter) in kw [14] Azimuth angle in [16] 1 kw~0 kw in 1 kw intervals 135 ~ 5 in 0.1 intervals Tilt angle in Typical roof pitch [16, 17] angles PV module material 99.15% c-si type and [15] 0.85% thin-film type B. Monte Carlo Simulation Truncated normal = N(µ=5kW, approximate σ) Truncated normal = N(µ=180, σ=range/4) or Uniform = U(135,5 ) Typical roof pitch angles with equal probabilities 99.15% c-si type and 0.85% thin-film type In this paper, we propose a stochastic Monte Carlo simulation with following steps: Step 1. Initialization. It calculates the total number of s on a test feeder, selecting the type of PV system, and representative commercial inverters with capacities of 1 kw to 0 kw. Step. Random sampling. It generates normally distributed random numbers for random variables such as system capacity, azimuth angle, and tilt angle, for PV modules installed on M individual residential roofs. Each generated random sample corresponds to a value of a random variable provided in TABLE I. Step 3. Selection of s. It assumes that PV systems with randomly generated modules and inverters are installed on certain roofs of individual houses on the distribution feeder. Then it randomly selects N s with their own PV systems that possess a total capacity of ten percent of peak load demand on the same feeder, and estimates the annual generation of the N PV systems based on hourly interval operation modeled during one calendar year. Step 4. Calculation of annual power flow. It calculates the set of hourly power flow simulations on the test feeder enhanced by N randomly dispersed residential PV systems. Step 5. Termination and normalization. It repeats steps 3 and 4 for to complete the simulation period (determined in the next section) and assesses the annual effects of both reductions in energy consumption and emissions and changes in the costs of electricity generation of N residential PV systems stochastically dispersed throughout the feeder. C. Stochastic Simulation Period The selection of N residential PV systems in operation during one year in M s, presented in steps 3 and 4 of the previous section, is a combination problem of M C N. After selecting up to 10,000 samples from the combination problem, or performing 10,000 annual simulations, we plot in Fig. 1 the standard deviation (error) of the sample mean as percentage of the annual PV production in kwh/year/. Central limit theorem indicates that the standard error decreases with the order of the inverse square root of the sample size, 1,000 annual Monte Carlo simulations show a standard deviation error od sample means equal to 3. percent ( 1/ 1, 000 ). Fig. 1. Standard deviation error is reduced by 1/ Sample Size. IV. STOCHASTIC METHODS The proposed stochastic simulation estimates the expected

3 value of annual energy production and savings resulting from PV systems randomly scattered on a distribution feeder. To reduce variance in the expected value, we propose to accelerate the stochastic simulation by variance reduction techniques, including importance sampling for the system capacity and azimuth angle random variables and stratification sampling for the tilt angle random variable. A. Importance Sampling A stochastic simulation is based on random sampling. Importance sampling conducts random sampling more extensively close to more likely values of the random variable in question. We use importance sampling for the system capacity and azimuth angle random variables. To describe how importance sampling can find more accurately the expectation value, we assume that annual PV power production depends only on PV systems capacities that follow a normal distribution. Then, we examine the expected value of the random variable x by θ = Ex [ ] = xf ( xdx ), (3) where x = the random variable that contains samples of the system capacity and f ( x ) = the original probability density function of x. Implementation is based on introduction of the new probability density function, g(x), of importance sampling by xf ( x) xf ( x) θ = E[ x] = gxdx ( ) = E[ ]. (4) gx ( ) gx ( ) If a distribution that is similar but not equal to that of the original distribution, or f (x), is selected for importance sampling, then xf (x)/g(x) can be a constant and its variance zero [13]. The optimal distribution of the importance sampling is xf ( x) xf ( x) xf ( x) g optimal ( x) = constant = = θ ( ) uf ( u ) u support xf du (5) The efficiency of importance sampling depends on the choice of g(x). However, the constant in equation (5) cannot be calculated in practice because of the huge number of possible combinations or unknown original distributions. The distribution of the system capacity of residential PV systems (as shown in Error! Reference source not found.) can be approximated by the following normal distribution with mean µ and standard deviation σ : 1 ( x µ ) ( ) /σ f x = e. (6) π σ To apply importance sampling to random variable x of the system capacity, we propose a new distribution of importance sampling by the following near-normal distribution parameterized by λ : ( ) λ x µ / σ g x, λ = λe. (7) ( ) To determine optimal λ, [13] proposed an upper bound on the variance in equation (5.1) on p.83, as follows: xf ( x) xf ( x) xf ( x) Var ( θ ) = Var = E E[ ] g( x, λ ) g( x, g ( x, x f ( x) = (, ) g x λ dx θ (8) g ( x, ( x, M xf ( x) dx θ where ( M x, = max xsupportg Ú ( ) xf ( x) (, g x. (9) Optimal λ is /3 as it satisfies the conditions of the following equation: M( x, M( x, = = 0. (10) x λ B. System Capacity and Azimuth Angle Random Sampling We propose new near-normal probability density function, g(x), of importance sampling with optimal λ =/3 for the system capacity and azimuth angle random variables. To generate random numbers for the PV system capacity random variable in the stochastic simulation, we use a truncated normal distribution with a mean of 5 kw and a standard deviation of.4 kw, both of which are determined by the non-linear least-squares method presented in Error! Reference source not found.. The probability density function of the PV system capacities is presented in Fig. 4(a). In addition, we apply optimal distribution of importance sampling for the system capacities. Since this study approximates a standard deviation of.4 kw for the system capacity random variable, it selects more samples near 3 kw ( µ-0.60σ) to 7 kw ( µ+0.60σ). We initially conduct random sampling from uniformly distributed azimuth angles. South-facing PV systems, having azimuth angle of 180, maximize energy output [16]. As an approximation, we propose random sampling from normally distributed azimuth angles with a mean of 180 and a standard deviation of.5, or σ=90 /4. We use near-normal distribution of importance sampling for azimuth angle. C. Stratification Sampling In stratification sampling, random variable can be split into mutually exclusive strata by {S (i), i=1,,..,m} and then sampled from each stratum of the tilt angle of the PV system as the typical pitch angle for residential roofs depicted in TABLE II. TABLE II Typical pitch for the residential roof [17] Pitch Degree( ) Pitch Degree( ) Pitch Degree( ) Pitch Degree( ) 3/ / / / / / / / /1.6 8/ / /

4 V. IMPLEMENTATION OF THE STOCHASTIC SIMULATION ALGORITHM We implement variance reduction to reduce the computational burden. To perform the stochastic simulation of N randomly dispersed PV systems, the algorithm generates the truncated normal distributions of random variables such as the system capacity and the azimuth angle, produces the near-normal distribution of importance sampling with optimal λ for the system capacity and azimuth angle random variables, and defines the stratum of the tilt angle random variable of the PV system. Implementation of the stochastic simulation algorithm is done in MATLAB. We use PV_LIB, which is based on typical meteorological year (TMY) version 3 weather data (1991 to 005), to calculate the annual power flow of the test feeder enhanced by PV systems in hourly intervals, and use three-phase power-flow algorithm developed in [11, 1]. A. Distribution Feeder VI. CASE STUDY IEEE 13-bus test feeder [, 3]. The maximum load on the feeder, which is originally supplied by a three-phase transformer of 5 MVA, is 3,490 kw and 1,90 kvar. This study selects N s that produce a capacity of ten percent of the total peak demand, which is used for the normalization of the simulation results. B. Load Profile The load profile is a graph of the total generation of electricity that varies continuously according to customer demand, which can be classified into residential, commercial, industrial, agricultural and pumping, and large industrial customer demand. This study collected load profile data from an actual utility in kw in hourly in 007, which span 8,760 hours in one year [4]. Because of the lack of available reactive power data, we randomly generate reactive power consumption maintaining a power factor over All the individual loads of each customer type in the distribution network vary in a timecorrelated fashion. The maximum total active power of the feeder throughout the year is 3, kw. C. Energy Mix For a utility, the benefits of using DG are less demand for energy production during critically high peak levels, and a reduction of transmission and distribution losses. We assume that PV systems are installed in Atlanta, GA, USA. We use the energy mix of the state of Georgia as the input data of the case study. TABLE III presents the detailed data of the energy mix from December 010 [5]. The data, supplied by Georgia Power Company, pertain to the fuel type, energy mix, and electricity costs. Since the levelized cost of electricity (LCOE) of PV is rapidly decreasing, this study uses the average levelized cost of electricity of PV entering new service in the United States in 019 [6]. TABLE III Energy mix of the state of Georgia [5, 6] Type Capacity in kw Fuel Energy Mix Cost in $/MWh Base 0~ Nuclear 1% 6.6 Intermediate ~3, Coal 67% 45.3 Gas and Gas (10%) and 3,18.057~3, Peak Hydro Hydro (%) % of peak PV 10% VII. RESULTS OF SIMULATION A. Energy Savings and Generation Cost Changes of N Randomly Dispersed PV Systems To analyze energy savings and changes in generation costs of randomly dispersed PV systems, we perform three stochastic simulations of 1,000 years each. The unknown random variables are system capacity, azimuth angle, tilt angle, and the module type used in PV system. TABLE III shows an annual statistical summary of energy savings and generation cost changes, which indicates that PV systems with a ten percent capacity of the total peak demand reduce the consumption of energy from MWh/year/ to MWh/year/ annually (change rates are from -3.8% to -7.1%). A a reference scenario, we use the interval arithmetic method, which often expresses the extent of uncertainty, to the proposed stochastic simulation. Typically, a south-facing PV system tilted at the latitude of the location on which it is installed receives maximum insolation [16, 18]. Therefore, an interval of the tilt angle random variable presented in TABLE III is [18.63, ] or [33.63, ] (in which the latitude of the Atlanta Hartsfield International Airport in the TMY3 data set is ). An interval for the azimuth angle random variable is [135, 180 ] or [180, 5 ] (in which 180 indicates an azimuth angle facing south for maximum power production). A range of system capacities is [1kW, 0kW]. Since the costs of generating electricity of the PV system, $/MWh, are still higher than those of the other generation types, the PV system effectively increases the costs of generating electricity from.05 $/MWh to $/MWh. This does not include some of the environmental consideration nor the pricing which addresses the safety issues of nuclear generation, which have recently been widely discussed. B. Ecological Impact of N Dispersed PV Systems To analyze the ecological impact of the PV system, we calculate the emission coefficients of each fuel type by averaging the amount of emissions emitted by each fuel type consumed for generating electricity divided by the amount of electricity generated from each fuel type in the United States from 1989 to 008 [7]. TABLE V shows that while nuclear generation does not emit greenhouse gases and toxic pollutants, mining uranium, the enrichment process, and transportation of the fuel emit carbon dioxide [8]. The coefficients of water consumption by each fuel type are

5 determined by the amount of water required for cooling, which only accounts for the amount of evaporation [9, 30]. TABLE III Annual statistical summary of the energy savings and the generation cost changes of dispersed PV systems Variance Reduction Method Energy from Non-solar Plants Energy Savings Unit Generation Cost $/MWh MWh/year/ Reference (No PV) Reference θ Min (Interval θ Arithmetic) θ Max Stochastic (Truncated Normal) θ Min θ θ Max θ Min Importance θ Sampling θ Max Change in % -3.8%~-7.1% - +5.%~+9.9% TABLE V Emission coefficients of various fuel types Emissions Water Fuel CO SO NO kg/kwh kg/kwh kg/kwh Gallons/kWh Coal Gas Nuclear Hydroelectricity TABLE III presents an annual statistical summary of the ecological impact of N stochastically dispersed PV systems that decrease CO from 3.97 to tons/year/, SO from to kg/year/, NO from 9.80 to kg/year/, and water consumption from to kgal/year/. C. Reduction of Variance and Stochastic Simulation Speed We estimate the power output of N residential PV systems that produce a capacity of ten percent of total peak demand in hourly intervals and are stochastically dispersed across the Atlanta area as the urban distribution area and calculates the annual power flow of the test feeder enhanced by N stochastically dispersed PV systems in hourly intervals, and examines the expectation value of their annual energy savings. TABLE IV shows that the expectation value and the variance of the annual energy savings of N stochastically dispersed PV systems in the reference scenario using the interval arithmetic method are kwh/year/ and kwh/year/, respectively. To reduce the variance of the annual energy savings of the PV systems, we apply importance sampling to the simulation involving the uncertainties of system capacity and azimuth angle. As a result, importance sampling reduces the variance of the annual energy savings of N dispersed residential PV systems from kwh/year/ to kwh/year/ with a variance reduction ratio of The reduced variance indicates that the expectation value more closely approaches the true value. TABLE III Ecological impact of N stochastically dispersed PV systems Variance CO SO NO Water Reduction Method tons/year/ kg/year/ kg/year/ kgal/year/ Reference (No PV) Reference θ Min (Interval θ Arithmetic) θ Max Stochastic (Truncated Normal) Importance sampling θ Min θ θ Max θ Min θ θ Max Change in % -9.5% ~ -15.6% -10.6% ~ % -10.6% ~ % -4.1% ~ -7.5% TABLE IV Expectation and variance of annual energy savings of N stochastically dispersed residential PV systems Period Annual PV Energy Savings Cluster Time Efficiency Scenario θ σ VRR Year kwh/year/ - - Hour - Reference 1, No (Interval Arithmetic) Truncated Normal 1, Yes Importance Sampling 1, Yes To calculate 1,000 annual power flow simulations in hourly intervals, the proposed algorithm takes an average of seconds per power-flow calculation on an average desktop computer. Therefore, without the acceleration, the stochastic simulations require 8,760 1, sec hours. To reduce the extremely heavy computational burden, a stochastic simulation time of hours, the power flow of the test feeder is calculated using the technique of representative clusters and extreme points [8]. As a result, the sampling of the representative clusters and extreme points reduces the simulation time of hours to hours at an efficiency of VIII. CONCLUSION The objective of this study is (1) to propose tools and algorithms useful for planning, design, and analysis of distribution feeders enhanced by PV (photovoltaic) systems. Another objective is to design a framework for streamlining the future development and smooth integration of stochastic renewable DG systems in smart grids [11, 1]. We have implemented a stochastic simulation algorithm combined with an algorithm for three-phase power-flow analysis based on the backward and forward sweep method. A few methods of variance reduction were applied, including optimal nearnormal distribution of importance sampling, and the sampling of representative clusters and extreme points. We use the IEEE 13-bus test feeder as a case study that incorporates load profile data collected from an actual utility in kw in hourly intervals in 007 and PV systems installed in the Atlanta area. The results of the simulations of a case study have shown that an urban distribution network enhanced by stochastically distributed PV systems can provide some peak shaving (as

6 shown in Error! Reference source not found.), reduce energy needs from conventional sources (presented in TABLE III), and reduce the pollutants and greenhouse gases by using less fuel from coal- and gas-fired plants (described in TABLE III). In fact, if (1) the power output of PV systems is estimated by solar radiation and meteorological data of the Atlanta area in TMY (typical meteorological year) data sets in hourly intervals and () the PV systems are stochastically dispersed throughout the IEEE 13-bus test feeder as an example of the urban distribution network with a capacity equal to ten percent of total peak demand, they can reduce the amount of energy produced from non-solar plants, particularly by about % in this case study (at the expense of higher costs of generating electricity resulting from high PV system costs) in TABLE III. In this study, they can also reduce about % of CO, % of SO, % of NO, and % of water consumption released by non-solar generating plants. The proposed methodology can be viewed as a framework for analyzing the impact of renewable DG systems, particularly PV systems, on distribution feeders. In addition, this study has shown that the method of sampling representative clusters and extreme points effectively reduces a stochastic simulation time. However, we do not target here commercial- and utility-scale PV systems, whose prices are expected to be lower than those of the smaller dispersed systems, and can relatively easily be incorporated in similar, suitably reconfigured stochastic simulation framework. I. REFERENCES [1] D. S. Shugar, "Photovoltaics in the utility distribution system: The evaluation of system and distributed benefits," Photovoltaic Specialists Conference, [] T. Hoff and D. S. Shugar, "The value of grid-support photovoltaics in reducing distribution system losses," IEEE Transactions on Energy Conversion, vol. 10, pp , [3] N. Srisaen and A. 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