FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 35, No 1, March 2022, pp. 71-92 https://doi.org/10.2298/FUEE2201071P Β© 2022 by University of NiΕ‘, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper POSSIBILISTIC UNCERTAINTY ASSESSMENT IN THE PRESENCE OF OPTIMALLY INTEGRATED SOLAR PV-DG AND PROBABILISTIC LOAD MODEL IN DISTRIBUTION NETWORK Shradha Singh Parihar, Nitin Malik The NorthCap University, Gurgaon, India Abstract. To integrate network load and line uncertainties in the radial distribution network (RDN), the probabilistic and possibilistic method has been applied. The load uncertainty is considered to vary as Gaussian distribution function whereas line uncertainty is varied at a fixed proportion. A voltage stability index is proposed to assign solar PV-DG optimally followed by application of PSO technique to determine the optimal power rating of DG. Standard IEEE 33- and 69-bus RDN are considered for the analysis. The impact of various uncertainties in the presence of optimally integrated solar PV-DG has been carried out on 69-bus network. The results obtained are superior to fuzzy-arithmetic algorithm. Faster convergence characteristic is obtained and analyzed at different degree of belongingness and realistic load models. The narrower interval width indicates that the observed results are numerically stable. To improve network performance, the technique takes into account long-term changes in the load profile during the planning stage. The significant drop in network power losses, upgraded bus voltage profile and noteworthy energy loss savings are observed due to the introduction of renewable DG. The results are also statistically verified. Key words: distribution network, distributed generation, optimal integration, uncertainties, Interval arithmetic, Gaussian distribution function 1. INTRODUCTION 1.1. Motivation and Literature review The distribution network is ill-conditioned because of low X/R ratio and its radial structure. Thus, the conventional approaches like Newton-Raphson, Gauss-seidel, etc, for solving power flow (PF) problem in the transmission network fails to converge in many cases in distribution network. To compute bus voltage and power flow values, deterministic PF algorithm requires precise network (N/w) load and generation data. It does not Received February 23, 2021; received in revised form April 12, 2021 Corresponding author: Nitin Malik The NorthCap University, Gurgaon, India E-mail: nitinmalik77@gmail.com 72 S. S. PARIHAR, N. MALIK contribute to optimal planning and operation of the N/w, as it finds the PF results for specific N/w configuration and operating conditions only at a given instant. The emphasis is on integrating distributed generation (DG) into the distribution N/w due to socioeconomic, environmental, and technical constraints. DG is a decentralized generation of electric power in distribution N/w using non-renewable (turbine, engine, etc.) or renewable (small/micro/mini hydropower, wind, solar, fuel cell, geothermal, etc.) resources. As the output of these technologies depicts stochastic behavior, hence, capable to introduce significant uncertainty in total power production. In real, the networks are complex due to the presence of non-linearity and incapability in expressing N/w variables in very precise terms which can be simplified by either allowing some degree of uncertainty or making assumptions about the distribution N/w. The input parameters (N/w load, line, and transformer data) are considered to be fixed while performing power grid operations, however they are not in practise. Because of the erroneous calculation of reactance and resistance due to conductor ageing and temperature variation, there is ambiguity in N/w line data. The load uncertainties are caused due to incorrect estimation in load demand (load forecasting). Changes in climate and water runoff induce uncertainty in hydropower plants. Temperature sensitivity in a fuel cell creates uncertainty since temperature change has a stronger impact at higher current [1]. This varies the power flow in the radial distribution network (RDN). These uncertainties are modeled either using a possibilistic or probabilistic method. Monte Carlo simulation (MCS) and stochastic approach belongs to probabilistic domain whereas Interval Arithmetic (IA) and fuzzy sets comes under possibilistic methods. IA establishes a strict constraint on all feasible N/w circumstances that could have been achieved by hundreds of successive MCS; as a result, the MCS calculation time increases, making analysis more difficult. With little computational effort, IA can produce high- quality results. Probabilistic and possibilistic modelling is quantitative and qualitative in nature, respectively [2]. Probabilistic method is used where sufficient historical data of uncertain parameter or their probability density function (PDF) like load pattern, wind speed and solar irradiation [3] is easily available whereas possibilistic modelling is preferred when the data available is not sufficient for the planners and operators to establish PDF [3]. Both the methods are cooperative and their utilization provides more realistic approximation to N/w modelling. The approaches for integrating DG are generally categorized as analytical and heuristic methods. The analytical method makes use of mathematical equations to determine the optimum solution. An analytical method for evaluating DG in N/w is presented in [4] without considering cost benefits. Many numerical approaches like Kalman filter algorithm [5] and mixed integer non-linear programming [6] are applied to integrate DG optimally in the RDN. Authors in [7], demonstrated an analytical and a meta-heuristic approach to optimally allocate DG units in the RDN. Numerous evolutionary algorithms such as Grey Wolf Optimizer [8], PSO [9] and gravitational search algorithm [10] and have been employed for solving issues related to DG allocation in the RDN. An ant lion optimization algorithm [11] is demonstrated to optimally allocate DG in the RDN. Authors in [12], utilized augmented Lagrangian genetic algorithm to integrate renewable DGs for minimizing N/w losses, satisfying operational constraints without considering economic benefits. In [13], author implements an IA technique to incorporate load uncertainty in the distribution N/w considering constant power load only. A correlated interval-based Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 73 backward/forward (b/f) PF method is developed in [14] to consider uncertainties in renewable energy resources. The interval-based PF models are transformed into the optimization problem in [15] which minimizes the conservatism of the obtained interval solutions. An Affine Arithmetic method is projected in [16] to introduce N/w uncertainties of different types. A probabilistic distribution-based IA approach is presented in [17] to introduce uncertainty in load demand in conventional PF. Authors in [18], demonstrated the IA based PF analysis in the presence of load, line and DG uncertainties. The analysis has also been done with various types of DG units that are not optimally allocated. Abdelkader et al. [19] proposed a Fuzzy Arithmetic Algorithm (FAA) for incorporating uncertainties in the RDN. Triangular fuzzy number method is proposed in [20] to introduce uncertainties in the N/w but resulted in higher N/w loss. To deal with N/w uncertainties a new midpoint-radius interval-based approach has been demonstrated in [21] to eliminate the factorization of the interval Jacobian matrix. 1.2. Paper contributions From the previously published literature, it has been concluded that the combined use of the Interval Arithmetic and probabilistic load model with optimally integrated solar photovoltaic (PV) DG in distribution N/w has not been explored before. Using the hybrid possibilistic-probabilistic strategy, this research article contributes to the published literature. The optimal penetration of solar PV-DG is carried out using a novel voltage stability index (VSI) and PSO method and, thereafter, the N/w uncertainties (line and load) are introduced in RDN to demonstrate the possible states of the solution. The detailed investigation considering various realistic loads and load fluctuations with and without solar PV-DG have been presented. The presented approach is applied on well-established standard IEEE 69-bus N/w. Two case studies considering solar PV-DG without and with N/w uncertainties are analysed and the results of the IEEE 33-bus N/w are further compared to FAA to establish the effectiveness of the proposed methodology. The summarized article contributions are mentioned below a) A new VSI has been developed for the optimal placement of solar PV-DG in a RDN after which PSO is implemented to find the optimal size of DG. This independent method for integrating renewable DG gives openness and versatility to the problem. b) A combination of IA and probabilistic load model is applied and presented to attain a more realistic representation of distribution N/w modelling which paves the way for accurate results. c) All input variables (loads and line) and generation in the distribution N/w are represented as random variables. The line uncertainty variables are considered to fluctuate at a constant proportion, whereas load uncertainty is expected to change according to a Gaussian distribution function. d) The effect of N/w uncertainties and different realistic loads (industrial, residential and commercial) and their combination with optimally integrated solar PV-DG is analysed by directly incorporating them into the Interval-based b/f PF algorithm. e) The analysis of the reduction in N/w losses and the cost of annual energy loss savings (AELS) has been carried out at three load levels to aid the distribution N/w operators (DNOs) in future N/w planning. f) The attained results of the simulation study imply that the methodology proposed in this research is much more feasible and effective for designing the large-scale RDN at all load levels. 74 S. S. PARIHAR, N. MALIK 1.3. Paper outline The brief outline of the work is given as: In Section 2, the description of Interval Arithmetic is mentioned. In section 3, the modeling of N/w data and DG is presented. In the next section, the working of the PSO is explained. Section 5 explores the development of the novel VSI and the algorithm to integrate solar PV-DG optimally in the N/w. Interval- based b/f PF solution is attained in Section 6. The simulated results for two standard RDN are mentioned in Section 7 followed by conclusion in last section. 2. INTERVAL ARITHMETIC In contrast to the point estimating technique, a number can be expressed as confidence interval that can be open, closed, or a combination of both in the IA approach. A set of real numbers can be used to express an interval number. Let L and K be two separate interval numbers (of real numbers) with supporting intervals of [l1, l2] and [k1, k2]. Here, the k1, l1 and k2, l2 signifies the lower limits and upper limits (endpoints), respectively. k2-k1 and l2- l1 are the interval widths determined for intervals K and L, respectively. Addition, subtraction, multiplication, division, minimization, and maximising are all mathematical operations that may be applied to interval numbers [22]. 𝐾 + 𝐿 = [π‘˜1 + 𝑙1, π‘˜2 + 𝑙2] (1) 𝐾 βˆ’ 𝐿 = [π‘˜1 βˆ’ 𝑙2, π‘˜2 βˆ’ 𝑙1] (2) 𝐾 Γ— 𝐿 = [π‘šπ‘–π‘›. (π‘˜1 Γ— 𝑙1, π‘˜1 Γ— 𝑙2, π‘˜2 Γ— 𝑙1, π‘˜2 Γ— 𝑙2), π‘šπ‘Žπ‘₯. (π‘˜1 Γ— 𝑙1, π‘˜1 Γ— 𝑙2, π‘˜2 Γ— 𝑙1, π‘˜2 Γ— 𝑙2)] (3) 𝐾 𝐿 = 𝐾 Γ— πΏβˆ’1 (4) where πΏβˆ’1= [1/l2, 1/l1] with 0 βˆ‰ [l1, l2]. The distance between K and L is defined as 𝑑(𝐾, 𝐿) = π‘šπ‘Žπ‘₯[|π‘˜1 βˆ’ 𝑙1|, |π‘˜2 βˆ’ 𝑙2|] (5) Complex uncertainty can be obtained by representing real numbers in complex domain. The PF study utilizes the above-mentioned fundamental operations to calculate the link between uncertain variables in terms of complex interval numbers. In this research, IA is used to deal with the uncertainties in the N/w data. Therefore, reactance, resistance, bus voltage and N/w power loss are taken as interval numbers instead of a fixed value. Rather than the fixed variation discussed in section 3.2, the N/w load at a bus is assumed to fluctuate over a specified range based on a Gaussian distribution. When the load demand is changing over the interval, the number of PF computations required are lesser than the total number of repeated PF solutions. Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 75 3. MATHEMATICAL MODEL 3.1. Line variation model Fig. 1 is a one-line diagram of a branch connecting bus i-1 and bus i. Fig. 1 Single branch equivalent From Fig. 1, π‘·π’Š + π’‹π‘Έπ’Š = π‘½π’Šβˆ πœΉπ’Š. π‘°π’Š βˆ— (6) where, Vi stands for receiving-end rms bus voltage and Ξ΄i is voltage angle at bus i. The reactive and real power load fed through the ith bus are represented by Qi and Pi, respectively. Every bus, including the source bus, has an initial voltage of [1.0,1.0] +j [0.0,0.0] p.u. Since, both power and voltage are complex interval variables, the subsequent current at bus i (𝐼𝑖 ), as described in (7), is also a complex interval quantity that can be evaluated using the division operation (4). 𝐼𝑖 = [π‘ƒπ‘–π‘™π‘œ ,𝑃𝑖𝑒𝑝 ]βˆ’ j [π‘„π‘–π‘™π‘œ ,𝑄𝑖𝑒𝑝 ] π‘‰π‘–βˆ βˆ’π›Ώπ‘– (7) where π‘ƒπ‘–π‘™π‘œ , π‘„π‘–π‘™π‘œ and 𝑃𝑖𝑒𝑝 , 𝑄𝑖𝑒𝑝 , respectively, are lower and the higher limits for the real and reactive power load at the ith bus. The N/w real and reactive loss in a branch is π‘ƒπ‘™π‘œπ‘ π‘  (𝑖 βˆ’ 1, 𝑖) = (𝑃𝑖 2+𝑄𝑖 2) |𝑉𝑖| 2 . 𝑅𝑖 (8) π‘„π‘™π‘œπ‘ π‘  (𝑖 βˆ’ 1, 𝑖) = (𝑃𝑖 2+𝑄𝑖 2) |𝑉𝑖| 2 . 𝑋𝑖 (9) The branch reactance and resistance, respectively, are Xi and Ri. At a constant proportion, the line parameter's uncertainty can be introduced as π‘‹π‘–π‘™π‘œ = (1 βˆ’ %(𝑋)). 𝑋𝑖 (10) 𝑋𝑖𝑒𝑝 = (1 + %(𝑋)). 𝑋𝑖 (11) π‘…π‘–π‘™π‘œ = (1 βˆ’ %(𝑅)). 𝑅𝑖 (12) 𝑅𝑖𝑒𝑝 = (1 + %(𝑅)). 𝑅𝑖 (13) where π‘…π‘–π‘™π‘œ , π‘‹π‘–π‘™π‘œ and 𝑅𝑖𝑒𝑝, 𝑋𝑖𝑒𝑝 are the lower and the upper constraints on the N/w resistance and reactance, respectively. 76 S. S. PARIHAR, N. MALIK 3.2. Variation in Load Model In the RDN, the majority of the loads are frequency and voltage-dependent [23]. For analyzing static load, only variation in voltage is considered as deviation in frequency is not significant [24]. The load model generally chosen is complex power type, but in reality, the load is a combination of numerous load models. Therefore, this study aims at evaluating the impact of realistic load models viz. industrial, residential and commercial loads in the distribution N/w that is particularly important for DNOs in various planning scenarios. The considered load models can be expressed mathematically as [25]. 𝑃𝑖 = π‘ƒπ‘–π‘›π‘œ . ( |𝑉𝑖| |π‘‰π‘–π‘›π‘œ| ) π‘₯1 (14) 𝑄𝑖 = π‘„π‘–π‘›π‘œ . ( |𝑉𝑖| |π‘‰π‘–π‘›π‘œ| ) π‘₯2 (15) where x1 and x2 are the load exponents. π‘„π‘–π‘›π‘œ is the nominal reactive load, π‘ƒπ‘–π‘›π‘œ is the rated real load and π‘‰π‘–π‘›π‘œ is the rated bus voltage at the i th bus, respectively. In the present study, the real and reactive exponents taken for constant power load (CPL), industrial load (IL), residential load (RL) and commercial load (CML) model are 0 & 0, 0.18 & 6.00, 0.92 & 4.04 and 1.51 & 3.40, respectively [25]. As practically any type of load might present in the N/w, therefore, composite load (CL) model is considered for the analysis with 40% of CPL, 30% of IL, 20% of RL and 10% of CML [25]. The Gaussian distribution function is utilized to predict the change in N/w power load demand. The Gaussian distribution is a symmetric mean-value distribution with a bell- shape and mentioned as (16) 𝑓(𝑦𝑖) = 1 √2πœ‹πœŽ2 𝑒 βˆ’ 1 2 (π‘¦π‘–βˆ’πœ‡)Β² 𝜎² (16) Where random variables 𝜎 2 and ΞΌ are distribution parameters that represents variance and mean (expected) value of the base loads, respectively. The variance represents how much the random variable is expected to deviate from its mean value (in a certain percentage). The normalised value of the reactive or real power load at bus i of the considered network is given by yi, which can be given as in [17]. 𝑦𝑖 = 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ and 𝑦𝑖 = 𝑄𝑖 π‘„π‘–π‘›π‘œ (17) where, (14) and (15) defines 𝑃𝑖 and 𝑄𝑖 , respectively. π›Όπ‘žπ‘™ (π‘˜) and 𝛼𝑝𝑙(π‘˜) are the degree of belongingness for reactive and real power load, where k indicates a number of degree of belongingness. From the load curve illustrated in Fig. 2 the mean value of the normalized real and reactive power load is unity for the degree of belongingness 1.0. The degree of belongingness can have any value between plmax / N and plmax where, the number of points of linearization of the Gaussian curve is denoted by N and plmax is the maximum degree of belongingness. The Gaussian distribution curve for a real power load is depicted in Fig. 2 [17]. Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 77 Fig. 2 Gaussian distribution of load Equation (16) can be written as 𝛼𝑝𝑙 (π‘˜π‘‘ ) = 𝑓 [ 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ ] = 1 √2πœ‹πœŽ2 𝑒 [ 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ βˆ’πœ‡] 2 2𝜎2 ⁄ (18) From equation (18), we get 𝜎 = 0.399 for Β΅ = 1.0 and 𝛼𝑝𝑙(π‘˜π‘‘ ) = 1.0. For these values, equation (18) can be written as 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ βˆ’ 1 = ±√ βˆ’ln (𝛼𝑝𝑙(π‘˜π‘‘)) πœ‹ for 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ β‰  1 (19) Similarly, for reactive load 𝑄𝑖 π‘„π‘–π‘›π‘œ βˆ’ 1 = ±√ βˆ’ln (π›Όπ‘žπ‘™(π‘˜π‘‘)) πœ‹ for 𝑄𝑖 π‘„π‘–π‘›π‘œ β‰  1 (20) Right Hand Side of equation (19) and (20) can be specified as √ βˆ’ln (𝛼𝑝𝑙(π‘˜π‘‘)) πœ‹ = 𝛼𝐾 = √ βˆ’ln (π›Όπ‘žπ‘™(π‘˜π‘‘)) πœ‹ (21) Thus, (19) can be rewritten as 𝑃𝑖 π‘ƒπ‘–π‘›π‘œ = 1 Β± 𝛼𝐾𝑑 (22) 𝑃𝑖 = π‘ƒπ‘–π‘›π‘œ (1 Β± 𝛼𝐾𝑑 ) (23) where Β± sign gives a lower and upper constraints of the N/w load at bus i. π‘ƒπ‘–π‘™π‘œ = π‘ƒπ‘–π‘›π‘œ (1 βˆ’ π›Όπ‘˜π‘‘ ) (24) 𝑃𝑖𝑒𝑝 = π‘ƒπ‘–π‘›π‘œ (1 + π›Όπ‘˜π‘‘ ) (25) π‘„π‘–π‘™π‘œ = π‘„π‘–π‘›π‘œ (1 βˆ’ π›Όπ‘˜π‘‘ ) (26) 𝑄𝑖𝑒𝑝 = π‘„π‘–π‘›π‘œ (1 + π›Όπ‘˜π‘‘ ) where π‘˜π‘‘ =1, 2..N (27) Linearization at different π‘˜π‘‘ values in equations (24)-(27) results in π‘˜π‘‘ discrete load intervals in closed form. For the analysis purpose, the linearization is carried out at three 78 S. S. PARIHAR, N. MALIK different points which results in three distinct load intervals (D-regions) as shown in Fig. 2 and given below 𝐷1 β†’ {π‘ƒπ‘–π‘›π‘œ , π‘ƒπ‘–π‘›π‘œ } point interval for π‘˜π‘‘ =1 (28) 𝐷2 β†’ {π‘ƒπ‘–π‘›π‘œ [1 βˆ’ 𝛼2], π‘ƒπ‘–π‘›π‘œ [1 + 𝛼2]} for π‘˜π‘‘ =2 (29) 𝐷3 β†’ {π‘ƒπ‘–π‘›π‘œ [1 βˆ’ 𝛼3], π‘ƒπ‘–π‘›π‘œ [1 + 𝛼3]} for π‘˜π‘‘ =3 (30) Equations (28)-(30) shows that the D1, D2 and D3 are in bound form. Therefore, an IA operation has been implemented to introduce these variations in the power flow. 3.3. DG Modelling The generator bus has been characterised as a continuous negative PQ load for the small size DG resources, implying that they run in constant power mode. According to the IEEE 1547 Standard [26] the DGs are not meant for regulating the voltage at the buses as they may conflict with the utilities' existing distribution voltage regulating schemes [27]. The total N/w load gets reduced by the power generated by the connected DG. The solar PV- DG injects real power at unity power factor. The resultant load at bus i at which solar-PV DG has been placed will be π‘ƒπ‘Ÿπ‘’π‘– = 𝑃𝑖 βˆ’ π‘ƒπ‘ π‘œπ‘™π‘Žπ‘Ÿ π‘ƒπ‘‰βˆ’π·πΊπ‘– (31) where, π‘ƒπ‘ π‘œπ‘™π‘Žπ‘Ÿ π‘ƒπ‘‰βˆ’π·πΊπ‘– represents real power injected by the solar PV-DG at bus i. 4. PSO ALGORITHM PSO is a stochastic technique in which each particle in a search space alters its state. In a d-dimensional hyperspace, the updated particle velocity and position are expressed as: 𝑣𝑝𝑑 𝑛+1 = 𝑀𝑝𝑣𝑝𝑑 𝑛 + 𝑐1π‘Ÿπ‘Žπ‘›π‘‘1(𝑝𝑏𝑒𝑠𝑑𝑝𝑑 βˆ’ 𝑆𝑝𝑑 𝑛 ) + 𝑐2π‘Ÿπ‘Žπ‘›π‘‘2(𝑔𝑏𝑒𝑠𝑑𝑝𝑑 βˆ’ 𝑆𝑝𝑑 𝑛 ) (32) 𝑆𝑝𝑑 𝑛+1 = 𝑆𝑝𝑑 𝑛 + 𝑣𝑝𝑑 𝑛+1 (33) where, 𝑆𝑝𝑑 𝑛 and 𝑣𝑝𝑑 𝑛 shows the particle’s current position and velocity at nth iteration, respectively. 𝑝 = 1,2, … 𝑁𝑠 where Ns represents the swarm size. The acceleration coefficients for the IInd and Ist particles are c2 and c1, respectively. Random numbers in the interval [0,1] are rand1(.) and rand2(. ). pbestpd and gbestpd are particle personal best and the global best position, respectively. The particle p’s inertia weight (𝑀𝑝) is given as 𝑀𝑝 = π‘€π‘π‘šπ‘Žπ‘₯ βˆ’ (π‘€π‘π‘šπ‘Žπ‘₯βˆ’π‘€π‘π‘šπ‘–π‘› ) π‘›π‘šπ‘Žπ‘₯ . 𝑛 (34) where, π‘€π‘π‘šπ‘–π‘› and π‘€π‘π‘šπ‘Žπ‘₯ are the minimum and the maximum inertia weight value, respectively. π‘›π‘šπ‘Žπ‘₯ and n are the maximum and current iteration number, respectively. 5. DEVELOPMENT OF NOVEL VSI AND OPTIMAL ALLOCATION OF SOLAR PV-DG A novel VSI is proposed to site DG optimally and is derived by substituting the value of Ii from (6) in Vi, we get 𝑉𝑖 βˆ π›Ώπ‘– = π‘‰π‘–βˆ’1∠0 βˆ’ [(𝑅𝑖 + 𝑗𝑋𝑖 ). ( π‘ƒπ‘–βˆ’π‘—π‘„π‘– π‘‰π‘–βˆ βˆ’π›Ώπ‘– )] (35) Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 79 By multiplying (35) with 𝑉𝑖 ∠ βˆ’ 𝛿𝑖 on both sides, we obtain 𝑉𝑖 2 = π‘‰π‘–βˆ’1𝑉𝑖 ∠ βˆ’ 𝛿𝑖 βˆ’ (𝑅𝑖 + 𝑗𝑋𝑖 )(𝑃𝑖 βˆ’ 𝑗𝑄𝑖 ) (36) 𝑉𝑖 2 + [𝑃𝑖 𝑅𝑖 + 𝑄𝑖 𝑋𝑖 + 𝑗(𝑃𝑖 𝑋𝑖 βˆ’ 𝑄𝑖 𝑅𝑖)] = π‘‰π‘–βˆ’1𝑉𝑖 cos 𝛿𝑖 βˆ’ π‘—π‘‰π‘–βˆ’1𝑉𝑖 sin 𝛿𝑖 (37) On segregation of real and imaginary part of (37), we obtain 𝑉𝑖 2 + 𝑃𝑖 𝑅𝑖 + 𝑄𝑖 𝑋𝑖 = π‘‰π‘–βˆ’1𝑉𝑖 cos 𝛿𝑖 (38) 𝑃𝑖 𝑋𝑖 βˆ’ 𝑄𝑖 𝑅𝑖 = βˆ’π‘‰π‘–βˆ’1𝑉𝑖 sin 𝛿𝑖 (39) Substituting 𝑋𝑖 from (39) in (38), we obtain 𝑉𝑖 2 + 𝑃𝑖 𝑅𝑖 + 𝑄𝑖 . ( π‘„π‘–π‘…π‘–βˆ’π‘‰π‘–βˆ’1𝑉𝑖 sin 𝛿𝑖 𝑃𝑖 ) = π‘‰π‘–βˆ’1𝑉𝑖 cos 𝛿𝑖 (40) 𝑉𝑖 2 βˆ’ π‘„π‘–π‘‰π‘–βˆ’1𝑉𝑖 sin 𝛿𝑖 𝑃𝑖 βˆ’ π‘‰π‘–βˆ’1𝑉𝑖 cos 𝛿𝑖 + 𝑃𝑖 𝑅𝑖 + 𝑅𝑖𝑄𝑖 2 𝑃𝑖 = 0 (41) 𝑉𝑖 2 + (βˆ’ π‘„π‘–π‘‰π‘–βˆ’1 sin π›Ώπ‘–βˆ’π‘ƒπ‘–π‘‰π‘–βˆ’1 cos 𝛿𝑖 𝑃𝑖 )𝑉𝑖 + 𝑅(𝑃𝑖 + 𝑅𝑄𝑖 2 𝑃𝑖 ) = 0 (42) For bus voltages to be stable, (42) must have real roots, i.e. discriminant > 0 that resulted in the proposed VSI for the given branch and can be articulated as in (44) ( βˆ’π‘„π‘–π‘‰π‘–βˆ’1 sin π›Ώπ‘–βˆ’π‘ƒ2π‘‰π‘–βˆ’1 cos 𝛿𝑖 𝑃𝑖 )2 βˆ’ 4𝑅𝑖 (𝑃𝑖 + 𝑅𝑖𝑄𝑖 2 𝑃𝑖 ) ≀ 0 (43) 4𝑅𝑖.𝑃𝑖 2 (π‘„π‘–π‘‰π‘–βˆ’1 sin 𝛿𝑖+π‘ƒπ‘–π‘‰π‘–βˆ’1 cos 𝛿𝑖) 2 . (𝑃𝑖 + 𝑄𝑖 2 𝑃𝑖 ) ≀ 1 (44) The bus voltage determined from the PF solution is utilised to calculate VSI for each branch that lies in [0,1] range. Any value of proposed VSI nearing 0 shows stable operation, in contrary, VSI value approaching 1 indicates that the bus is gradually leading towards instability. The constraints taken for the analysis: a) Power balance: 𝑃𝐺 = 𝑃𝐷 + π‘ƒπ‘™π‘œπ‘ π‘  (45) 𝑄𝐺 = 𝑄𝐷 + π‘„π‘™π‘œπ‘ π‘  (46) where 𝑄𝐺 and 𝑃𝐺 shows the reactive and real power generated. 𝑄𝐷 and 𝑃𝐷 stands for reactive and real load demand on the network. b) Voltage constraint: 0.95 𝑝. 𝑒 ≀ 𝑉𝑖 ≀ 1.05 𝑝. 𝑒 (47) c) Current constraint: πΌπ‘π‘Ÿπ‘Žπ‘›π‘β„Ž ≀ πΌπ‘‘β„Žπ‘’π‘Ÿπ‘šπ‘Žπ‘™ (48) where, πΌπ‘π‘Ÿπ‘Žπ‘›π‘β„Ž and πΌπ‘‘β„Žπ‘’π‘Ÿπ‘šπ‘Žπ‘™ shows the branch current and its thermal limit, respectively. d) DG power generation constraint: 0 ≀ π‘ƒπ‘ π‘œπ‘™π‘Žπ‘Ÿ π‘ƒπ‘‰βˆ’π·πΊπ‘– ≀ βˆ‘π‘ƒπΏπ‘œπ‘Žπ‘‘ (49) where βˆ‘π‘ƒπΏπ‘œπ‘Žπ‘‘ is the total real power load in the network. e) Substation capacity: 0 ≀ 𝑃𝑔 𝑖 ≀ 𝑃𝑔(π‘šπ‘Žπ‘₯) i ∈ slack (50) 80 S. S. PARIHAR, N. MALIK 0 ≀ 𝑄𝑔 𝑖 ≀ 𝑄𝑔(π‘šπ‘Žπ‘₯) (51) where 𝑄g(max) and Pg(max) represents the maximum value of reactive and real power generation, respectively. 𝑄𝑔 𝑖 and 𝑃𝑔 𝑖 shows the reactive and real generated power at the slack bus, respectively. The pseudo-code for the optimal integration of solar PV-DG for the deterministic case in the RDN is mentioned below: Step I: Run the PF program to calculate the bus magnitude and its phase angle, branch current, N/w power losses using direct PF method [28] of RDN for the base case. The following iterative formula is used to determine the solution [𝑉𝑖 𝑛] = [𝑉𝑖 0] + [𝐡𝐢𝐡𝑉][𝐡𝐼𝐡𝐢][𝐼𝑖 π‘›βˆ’1] (52) where, BCBV stands for Branch Current-to-Bus Voltage and BIBC stands for Bus Incidence-to-Branch current matrix. The initial voltage (𝑉𝑖 0) is 1.0 +j 0 p.u. 𝐼𝑖 π‘›βˆ’1 is the branch current at n-1 iteration and 𝑉𝑖 𝑛 is the bus voltage at nth iteration at bus i. Step II: Determine the cost of annual energy loss [29] using π΄π‘›π‘›π‘’π‘Žπ‘™ π‘π‘œπ‘ π‘‘ π‘œπ‘“ π‘’π‘›π‘’π‘Ÿπ‘”π‘¦ π‘™π‘œπ‘ π‘ π‘’π‘  = (βˆ‘ π‘ƒπΏπ‘œπ‘ π‘  (𝑖 βˆ’ 1, 𝑖) NB i=2 𝑇. 𝐸 ) $ (53) where NB is the number of buses, T is the annual time duration (8760 hrs) and E is cost of energy (0.06 $/kWh). Step III: Evaluate VSI using (44). Select bus i as the most sensitive bus to place DG if a branch between bus i-1 and i has the greatest VSI value. Step IV: Set PSO parameters (swarm size, inertia weights, acceleration coefficients) for minimizing real power loss (RPL). Step V: Set iteration counter (n) to 0. Step VI: With random velocities and placements on the dimension as pbest, the values of solar PV-DG size are created (between 0 and βˆ‘ system loads (continuous)). Step VII: After installing DG at the location as obtained in step III, repeat the PF algorithm for each particle. Calculate RPL for the randomly initialised particles if all constraints are within limits. Otherwise, discard the infeasible solution. Step VIII: The solar PV-DG size giving minimum RPL value is opted as gbest and its corresponding position is considered as the particle best position. Step IX: The particles’ velocity, position and the weight are updated utilizing (32), (33) and (34), respectively. Step X: If maximum iterations (nmax) are reached, jump to Step XI. Otherwise, the counter is incremented and steps IV through X are repeated. If the newly obtained particle position is superior to the prior pbest and gbest, new pbest and gbest will be generated. Step XI: The best location denotes optimal solar PV-DG sizes, while the corresponding number denotes the lowest total RPL. Step XII: Determine annual power loss savings after calculating the cost of energy losses in the presence of DG using (53). Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 81 6. INTERVAL-BASED B/F PF SOLUTION METHODOLOGY The following steps are used to determine the value of the bus voltage and N/w losses: Step I: Read N/w data. Step II: Determine the degree of belongingness 𝛼𝑝𝑙 (π‘˜π‘‘ ) and π›Όπ‘žπ‘™ (π‘˜π‘‘ ) for N intervals. Step III: Complex interval numbers 𝑉𝑖 𝑛 and 𝑉𝑖 0 can be written as 𝑉𝑖 𝑛 = 𝐴1 + 𝑗𝐴2 and 𝑉𝑖 0 = 𝐡1 + 𝑗𝐡2 where 𝐴1, 𝐴2, 𝐡1 and 𝐡2 are all interval numbers. The voltage start for all buses is [1.0,1.0] +j [0.0,0.0] p.u. At first, N/w losses are set to zero. The iteration count and slack bus angle are initialized to zero. Step IV: Calculate the equivalent real power load at the bus using (31) after optimal siting and sizing of solar PV-DG as explained in step III and step XI of section 5, respectively. Step V: The closed bounded interval of line and load data is determined from (10) through (13) and (24) through (27), respectively for the various degree of belongingness. Step VI: For the complex nature of the load, update the bounded interval of real and reactive power load with the use of (14) and (15). Step VII: Form BIBC, BCBV and distribution PF matrices. Step VIII: Determine the currents and voltages at each bus using (7) and (52) using subtraction, addition, division and multiplication operation of the complex interval numbers as described in section 2. Step IX: The voltage difference between two successive iteration can be given as 𝑉𝑖 𝑛 βˆ’ 𝑉𝑖 0 = max[𝑑(𝐴1, 𝐡1), 𝑑(𝐴2, 𝐡2 )] (54) where 𝑑(𝐴1, 𝐡1) and 𝑑(𝐴2, 𝐡2) is calculated using (5). If max[𝑑(𝐴1, 𝐡1), 𝑑(𝐴2, 𝐡2 )] < 10βˆ’4 at all the buses then jump to step X, else jump to step V. Step X: Use (8) and (9) to calculate N/w power losses. Step XI: Print the results for specific value of 𝛼𝑝𝑙(π‘˜π‘‘ ) and π›Όπ‘žπ‘™ (π‘˜π‘‘ ). Step XII: If π‘˜π‘‘ =N terminate the program otherwise increment k and go to step II. 7. RESULTS AND DISCUSSION To demonstrate the performance of the IA-based PF technique in the presence of various realistic loads and solar PV-DG, IEEE distribution test networks of varying complexity and size are simulated to show its robustness. The complete N/w data for 33- and 69-bus N/w has come from [30] and [31], respectively. For both networks under consideration, the base kV and MVA are 12.66 and 100, respectively. The bus feeders of IEEE N/w were tested on MATLAB. A voltage error tolerance of 0.0001 p.u is considered for all the test cases acknowledged in this work. The piecewise segmentation of annual load profile in light, nominal and heavy load level is assumed 50%, 100% and 160% of the rated N/w load [32] with an annual hourly duration of 1000, 6760, and 1000 hours [11], respectively. To confirm the efficacy of the suggested methodology, the simulated network performance is compared to previously published findings for deterministic parameters for the same base voltage and the load model. In Table 1, the power flow results at various realistic loads for 33-bus RDN are compared to the published literature. The N/w real and reactive power losses are 5.5% and 5.9% of their respective load for CPL model, whereas, for CL model the real and reactive 82 S. S. PARIHAR, N. MALIK losses reduces to 4.7% and 5.04%, respectively. The convergence is also faster when compared to that of [25]. Table 1 Power flow result at various load models for 33-bus RDN Proposed method [25] Type of load CPL IL RL CML CL CPL IL RL CML CL Total RPL (kW) 202.66 161.28 158.54 153.57 174.21 202.68 161.69 159.33 154.93 174.19 Total reactive power loss (kVAr) 135.13 107.20 105.31 101.94 115.89 135.23 107.56 105.92 102.94 115.97 Number of iterations 4 2 2 2 2 3 4 4 4 3 Assuming constant annual load with only one type of load model is a misnomer because the N/w load profile is highly affected by various type of load model and time variations, hence light load, nominal load and heavy load levels are considered. The Vmin and network power losses attained for IEEE 69-bus N/w at different load levels for CPL and CL model are mentioned in Table 2. For 69-bus N/w, the power losses and the convergence characteristics obtained for deterministic case are compared at a nominal load in Table 3 to exemplify the capability of the proposed method. From Table 2, it is inferred that the reduction in load has a positive effect on N/w bus voltage profile, while increment in load aggravates it, in both cases. For 69-bus RDN, the bus voltage profile attained with CL and CPL model is given in Fig. 3. The results demonstrate that the effect of CL model on N/w performance is over-represented as compared to CPL model. As the N/w power loss reduction varies unproportionate to the network load, thus, it is better to provide generalized equations for the N/w power losses using a curve fitting. The generalized power loss equations for CPL and CL model are as follows: For CPL, π‘ƒπ‘™π‘œπ‘ π‘  (π‘˜π‘Š) = 332.11πœ† 2 βˆ’ 151.68πœ† + 44.37 (55) π‘„π‘™π‘œπ‘ π‘  (π‘˜π‘‰π΄π‘Ÿ) = 147.98πœ† 2 βˆ’ 64.88πœ† + 18.98 (56) For CL, π‘ƒπ‘™π‘œπ‘ π‘  (π‘˜π‘Š) = 149.22πœ† 2 + 20.51πœ† βˆ’ 3.71 (57) π‘„π‘™π‘œπ‘ π‘  (π‘˜π‘‰π΄π‘Ÿ) = 69.28πœ† 2 + 9.24πœ† βˆ’ 1.73 (58) where, Ξ» represents load level. The general expressions from (55) to (58) are useful for DNOs in future power generation planning. Table 2 Results of 69-bus N/w at various load levels with no solar PV-DG CPL CL Load level Light Nominal Heavy Light Nominal Heavy Vmin (p.u.) 0.9567 0.9092 0.8446 0.9597 0.9211 0.8757 RPL (kW) 51.56 224.80 651.88 43.85 166.02 411.11 Reactive power loss (kVAr) 23.54 102.09 294.02 20.21 76.79 190.41 Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 83 Table 3 Comparative analysis of CL model at nominal load and without DG for 69-bus RDN Fig. 3 Voltage profile of 69-bus RDN with CPL and CL model Validation of novel VSI To authenticate the index, the N/w load (P and Q both) is subjected to the random variation between 0 and 160% of the total base load. For 69-bus N/w, branch 60 with 0.0286 value is determined to have the largest VSI value. As a result, the bus 61 is regarded the most vulnerable bus beyond critical loading and is investigated as the load increases. Fig. 4 displays the VSI variation at the critical bus at various loading conditions. As VSI displays a linear variation to N/w load increment, it can be utilized for the accurate prediction of the voltage stability in RDN as also concluded in [32]. Fig. 4 Variation in the value of VSI at various load levels in 69-bus RDN 7.1. Analysis of N/w performance with optimal integration of solar PV-DG The proposed methodology has been employed on 69-bus RDN for the optimal integration of Solar PV-DG having deterministic parameters. The nmax and swarm size is 130 and 20, respectively. To achieve fast convergence of the optimization technique, the control variables 𝑐1 and 𝑐2 value in (32) and π‘€π‘π‘šπ‘Žπ‘₯ and π‘€π‘π‘šπ‘–π‘› in (34) are chosen as 2, 2, Proposed method [25] RPL (kW) 166.02 189.3761 Reactive power loss (kVAr) 76.79 86.8497 Number of iterations 2 3 84 S. S. PARIHAR, N. MALIK 0.9 and 0.4, respectively [33]. The value of VSI for each branch of 69-bus RDN is shown in Fig. 5. Bus 61 is found out to have the maximum VSI value and is chosen as the best DG placement location at nominal load level. Fig. 5 VSI at each branch in 69-bus RDN The optimal size and site of solar PV-DG is determined at various loading scenarios using the proposed method. The DG size follows a linear relationship with N/w load increment for the 69-bus network, as exemplified in Fig. 6. The N/w performance in terms of Vmin, power losses, RPL reduction and cost of annual energy losses attained at nominal load level after integrating solar PV-DG for CPL and CL model is illustrated in Table 4. Table 4 Results for solar PV-DG at nominal load level Fig. 6 Optimal Solar PV-DG size at various load levels in 69-bus network Base case CPL CL Optimal DG location @ Solar PV-DG size in kW - 61 @ 1888 61 @1888 Vmin in pu @ bus (% voltage improvement) 0.9092 @ 65 0.9684 @ 27 (6.5 %) 0.9700 @ 27 (6.7 %) RPL (kW) 224.80 83.17 70.40 RPL reduction (kW) (in %) - 141.63 (63.00%) 154.40 (68.68) Reactive power loss (kVAr) (in %) 102.09 40.51 (60.31%) 34.88 (65.83%) Annual cost of energy loss ($) 91178.88 33733.75 28554.24 AELS ($) - 57445.13 62624.64 mailto:0.9102@65 Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 85 7.1.1. Impact of solar PV-DG on N/w power loss The integration of solar PV-DG has a considerable effect on N/w losses. To validate, the general mathematical expressions of Ploss and Qloss for any load level in the 69-bus N/w are derived utilizing curve fitting approach as depicted below For CPL model, π‘ƒπ‘™π‘œπ‘ π‘  (π‘˜π‘Š) = 92.40πœ† 2 βˆ’ 12.87πœ† + 3.64 (59) π‘„π‘™π‘œπ‘ π‘  (π‘˜π‘‰π΄π‘Ÿ) = 44.68πœ† 2 βˆ’ 5.8πœ† + 1.64 (60) For CL model, π‘ƒπ‘™π‘œπ‘ π‘  (π‘˜π‘Š) = 55.15πœ† 2 + 20.76πœ† βˆ’ 5.51 (61) π‘„π‘™π‘œπ‘ π‘  (π‘˜π‘‰π΄π‘Ÿ) = 28.21πœ† 2 + 9.09πœ† βˆ’ 2.42 (62) After comparing N/w power losses without DG [(55) - (58)] and with solar PV-DG [(59) – (62)], we can conclude that the integration of renewable DG minimises the N/w losses at all load levels for all types of load models. The relationship between the variation in RPL with solar PV-DG size for 69-bus RDN considering CL model is illustrated in Fig. 7, which seems to follow a parabolic curve. The curve shows the RPL value decreases with the increase in the size of solar PV-DG as presented in the left portion of the curve. The optimum DG size will be attained at the lowest point of the curve when RPL reached to its minimum value after which the RPL losses increase as DG size increases (right part of the curve) due to extra current flow from the DG to the adjacent bus. The RPL in 69-bus RDN without integrating solar PV-DG was found to be 224.80 kW and 166.02 kW for CPL and CL model, respectively. After optimal integration of solar PV-DG, the RPL for CPL and CL model in IEEE 69-bus N/w mitigates to 83.17 kW and 70.40 kW with a decrease of 63.00% and 68.68% percent, respectively, with respect to the base case (From Table 2). The real power demand released is 141.63 kW for the CPL model and 154.40 kW for the CL model after installing SPV-DG, respectively. The RPL magnitude at each branch with and without solar PV-DG is demonstrated in Fig. 8 for 69-bus N/w with CPL model at nominal load. The results validate that the N/w loss minimizes after integration of solar PV-DG. Fig. 7 Relationship between RPL and solar PV-DG size at optimal DG bus in 69-bus N/w 86 S. S. PARIHAR, N. MALIK Fig. 8 Network RPL with and without renewable DG at nominal load in 69-bus RDN 7.1.2. Impact of solar PV-DG on bus voltage profile The Vmin for 69-bus N/w has been updated from 0.9092 pu at bus 65 to 0.9684 pu and 0.9700 pu at bus 27 for CPL and CL models, respectively, resulting in 6.5 % and 6.7 % increase in bus voltage magnitude (From Table 4). It has been found out that for both the load model the N/w voltage profile is enhanced after solar PV-DG installation satisfying the constraints. Fig. 9 Impact of N/w load variation on voltage profile in 69-bus N/w for CL model The effect of different load levels on bus voltage profile considering realistic loads is analysed in Fig. 9 and found out to have a remarkable enhancement in voltage profile for 69-bus N/w at all the considered load levels after installing solar PV-DG. Fig. 9 also shows that all the bus voltages attained from the proposed method are within allowable voltage limits and hence validates the method consistency. Thus, the obtained integrated solution is very beneficial for DNOs. 7.1.3. Impact of solar PV-DG on AELS The cost of energy loss in 69-bus RDN before integrating solar PV-DG was $91178.88 which is reduced to $33733.75 and $28554.24 with solar PV-DG resulting in AELS of $57445.13 and $62624.64 for CPL and CL model, respectively, at nominal load level with respect to the base case as mention in Table 4. The AELS for CPL and CL model attained considering all the load levels are $85257.73 and $93577.34, respectively compared to base case. Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 87 7.1.4. Comparative analysis To authenticate the efficacy of the method, the test results attained after the penetration of solar PV-DG are compared to other available meta-heuristic methods like WOA [34], GWO [35], PSO [36], SGA [36], CSA [36] and BB-BC [37] and mentioned in Table 5 for the 69-bus N/w. Due to the variable nature of RPL reduction and DG size, it becomes obligatory to compare it on a common platform which is carried out by calculating the ratio of RPL reduction to size of DG. The penetration of solar PV-DG in the N/w yields a ratio of 0.075 superior or comparable to already published literature. The higher value of the ratio compared to the already published results signifies the robustness of the proposed approach used for the optimal integration of DG. To demonstrate its rapid convergence, the computational time required for solar PV allocation at a nominal load for the 69-bus N/w is calculated and compared to the existing literature [38] and [12] (Table 6). Table 5 Comparative analysis of solar PV-DG integration techniques for 69-bus N/w DG allocation method Size of DG/ power factor Optimum Location RPL (kW) % RPL reduction Ratio of RPL reduction to solar PV-DG size Proposed approach 1888/1 61 83.17 63.00 0.075 WOA [34] 1872.82/1 61 83.23 63.01 0.075 GWO [35] 1928.67/1 61 83.24 62.98 0.073 PSO [36] 2000/1 61 83.80 62.75 0.070 CSA [36] 2000/1 61 83.80 62.74 0.070 SGA [36] 2300/1 61 89.40 60.30 0.058 BB-BC [37] 1872.5/1 61 83.22 63.00 NR NR: Not Reported Table 6 Execution time for solar allocation at nominal load in 69-bus N/w Proposed method Analytical method [38] GA [12] CPU time (sec) 0.20 0.70 0.85 7.2. Analysis of N/w uncertainties (line and load) with optimal integration of solar PV-DG For the comparative analysis, the load and line uncertainties for the CPL model are set to 5% and 1%, respectively as described in [19] for IEEE 33-bus RDN. The interval width for line and load uncertainty at Vmin is tabulated and compared with FAA [19] in Table 7. The results clearly illustrate that the interval width determined from the proposed probabilistic-possibilistic approach is narrower. As a result, the solution is less conservative and superior to the probabilistic technique alone. It can be concluded that increasing N/w load uncertainty creates a bigger voltage drop than increasing N/w line uncertainty. Table 7 Interval width of Vmin for CPL model in 33-bus N/w Output variable Type of uncertainty Interval width (p.u) Interval width reduction in % Probabilistic-possibilistic approach FAA [19] Vmin (p.u) Line 0.0019 0.0021 9.5 Load 0.0094 0.0142 33.8 88 S. S. PARIHAR, N. MALIK In this case, the analysis of uncertainties in input parameter is presented with solar PV- DG for 69-bus N/w. The fixed variation of Β±3% in N/w line data has been considered. The solar PV-DG is positioned at bus 61 with DG size of 1888 kW as determined in case 1 from the proposed method. The simulated results for Vmin, total real and reactive N/w losses in solar PV-DG integrated IEEE 69-bus N/w for the deterministic case and when uncertainties occur in N/w line and load parameter at various degree of belongingness at different load models are tabulated in Table 8. At Ξ± = 1, the interval widths for CPL, IL, RL, CML, and CL are 0.002 pu, 0.0017 pu, 0.0017 pu, 0.0016 pu, and 0.0018 pu, respectively, based on the upper and lower bounds of the Vmin. For all practical load models, the interval of voltage magnitude at bus 65 is narrower than that obtained from the CPL load model. As a result, the CL model, as opposed to the CPL model, produces more realistic results. Fig. 10 shows the effect of adding uncertainties on the voltage profile of 69-bus N/w for the CL model and three degrees of belongingness (Ξ± = 0.2, 0.6, 1). As expected, with deterministic input values, the voltage magnitude at every bus fall within the range of potential N/w states obtained by varying input parameters. Table 8 Results for IEEE 69-bus N/w with load and line uncertainty and DG penetration at nominal load Degree of belongingness Ξ±pl, Ξ±ql =1 Ξ±pl, Ξ±ql =0.6 Ξ±pl, Ξ±ql =0.2 Load model Deterministic result Lower Upper Lower Upper Lower Upper CPL Vmin 0.9684 0.9674 0.9694 0.9535 0.9820 0.9425 0.9915 Ploss 83.1722 80.5589 85.7935 28.1596 172.5547 6.3011 262.4480 Qloss 40.5177 39.2487 41.7903 13.7375 83.9240 3.0769 127.4850 IL Vmin 0.9709 0.9700 0.9717 0.9586 0.9828 0.9499 0.9916 Ploss 60.4966 59.1220 61.8500 23.3002 110.3419 6.0157 153.6309 Qloss 30.499 29.7787 31.2103 11.5943 56.3890 2.9513 79.266 RL Vmin 0.9710 0.9702 0.9719 0.9590 0.9829 0.9507 0.9916 Ploss 65.6427 64.0243 67.2424 24.5247 122.8959 6.0953 173.7163 Qloss 32.7774 31.9483 33.5982 12.1344 61.9703 2.9863 88.2260 CML Vmin 0.9714 0.9706 0.9722 0.9599 0.9830 0.9519 0.9916 Ploss 67.1456 65.4526 68.8209 24.8706 126.6893 6.1169 179.9339 Qloss 33.4364 32.5746 34.2904 12.2859 63.6350 2.9957 90.556 CL Vmin 0.9700 0.9691 0.9709 0.9569 0.9825 0.9476 0.9916 Ploss 70.40 68.5249 72.2729 25.5387 136.0021 6.0119 196.6515 Qloss 34.88 33.9405 35.8256 12.5824 67.7841 2.9496 98.4141 Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 89 Fig. 10 Voltage profile with solar PV-DG with fixed and varying line and load parameter at various degree of belongingness considering CL model and nominal load Fig. 11 shows the variation of total reactive and real power losses at different degree of belongingness without and with solar PV-DG for the CL model in IEEE 69-bus N/w. It was obvious that when solar PV-DG was integrated into an IEEE 69-bus N/w, power losses were dramatically decreased. As can be seen in Fig. 11, the interval between power losses reduces as the degree of belongingness increases. Fig. 11 Variation of total N/w losses at different degree of belongingness with CL model at nominal load level The generalized equations for determining lower and upper real and reactive losses in 69-bus RDN considering the CL model with line and load uncertainty at Ξ±=0.6 using curve fitting technique are given as π‘ƒπ‘™π‘œπ‘ π‘  π‘™π‘œ (π‘˜π‘Š) = 22.02πœ† 2 + 5.15πœ† βˆ’ 1.3 (63) π‘„π‘™π‘œπ‘ π‘  π‘™π‘œ (π‘˜π‘‰π΄π‘Ÿ) = 11.05πœ† 2 + 2.12πœ† βˆ’ 0.577 (64) π‘ƒπ‘™π‘œπ‘ π‘  𝑒𝑝(π‘˜π‘Š) = 97.50πœ† 2 + 52.05πœ† βˆ’ 13.55 (65) π‘„π‘™π‘œπ‘ π‘  𝑒𝑝(π‘˜π‘‰π΄π‘Ÿ) = 50.88πœ† 2 + 22.86πœ† βˆ’ 5.96 (66) The coefficient of variation (CV) in RPL decreases with the penetration of solar PV- DG with CL model for 69-bus N/w, and is greatest for the base case, as tabulated in Table 9. This implies that the integration of DG decreases the power loss variation in the feeders of the distribution N/w around its mean value and thereby provide better security against overheating of feeders and instability. It is found that the CL model provide better results consistently as specified by their better voltage profile, lowest power losses and minimum 90 S. S. PARIHAR, N. MALIK CV. The minimum, maximum, mean, standard deviation (Std) and CV of power loss lies within their lower and upper limits for all values of Ξ± but has been illustrated for Ξ± =1 only, in Table 9. It has been observed that higher the DG penetration, higher will be the CV value due to its higher degree of uncertainty. Table 9 Statistical analysis for RPL without and with DG in IEEE 69-bus N/w with CPL and CL model and uncertainty at nominal load level Load model π‘ƒπ‘™π‘œπ‘ π‘  (kW) Deterministic Lower Upper Without DG CPL Min 1.2562e-05 1.2184e-05 1.2940e-05 Max 49.6749 47.8832 51.4904 Mean 3.3059 3.1890 3.4242 Std 8.3880 8.0892 8.6906 CV 2.5373 2.5366 2.5380 With solar PV-DG CPL Min 1.2561e-05 1.2183e-05 1.2939e-05 Max 15.0325 14.5640 15.5022 Mean 1.2231 1.1847 1.2617 Std 2.7308 2.6450 2.8169 CV 2.2327 2.2326 2.2328 CL Min 1.2468e-05 1.2096e-05 1.2841e-05 Max 12.4737 12.1515 12.7931 Mean 1.0354 1.0077 1.0628 Std 2.2805 2.2205 2.3401 CV 2.2026 2.2018 2.2035 8. CONCLUSIONS This paper proposes a probabilistic and possibilistic strategy to solve the power flow problem with optimally integrated solar PV-DG to investigate the impact of line and load uncertainties in the RDN. The N/w line and load vary in fixed and as function of Gaussian distribution, respectively. A new VSI is proposed to search the optimal site strategically for solar PV-DG to reduce power losses and enhance bus voltages. PSO method is further applied to determine the optimum solar PV-DG size. The independent method for finding the optimal site and size of the renewable DG provide openness and flexibility to the method. Two test cases have been designed and solved for varying levels of complexity in the PF problem. The bus voltage characteristic for various degree of belongingness is found to be affected by various realistic loads. The solution obtained from the proposed approach comprises all possible states of the N/w and converges faster than the existing results. The robustness of the method has been demonstrated on 33- and 69-bus N/w. It has been statistically approved from the analysis that the voltage profile and reduction in N/w power losses are under-represented for CPL model when compared to the CL model for all N/w loading conditions. The results imply that the proposed technique is more feasible and effective for the design of the large-scale N/w with a high degree of uncertainty. The narrower interval width signifies less conservative solution and numerical stability when compared to the FAA method. The findings revealed that uncertainties have a major impact on the RDN and so cannot be overlooked. A generalized set of equations for calculating N/w power losses with and without solar PV-DG considering uncertainties has been developed under various loading conditions which will help the DNOs in N/w planning and expansion of the RDN. Uncertainty Assesment in RDN with Optimally Integrated Solar PV-DG Considering Realistic Loads 91 REFERENCES [1] Noorkami et al., "Effect of temperature uncertainty on polymer electrolyte fuel cell performance", International Journal of Hydrogen Energy, vol. 39, no. 3, pp. 1439–1448, 2014. [2] Z. Wang and F.L. 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