38 PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 88 NR 7a/2012 Francisco Javier RUIZ-RODRÍGUEZ, Francisco JURADO, Salah KAMEL University of Jaén (Jaén, Spain) Application of probabilistic three-phase load flow for electrical distribution systems with photovoltaic generators Abstract. This paper shows how to solve a probabilistic three phase load flow in radial distribution networks with photovoltaic distributed generation (PDG). Voltage regulation is one of the principal problems to be addressed. This research study applies a three phase power flow combined with the Monte Carlo method to solve this problem. Load and PDG are modeled as random variables. A case study is presented. The results obtained show the decrease of the unbalance factor due to the presence of PDG. Streszczenie. W artykule pokazano jak rozwiązać problem trójfazowego przypadkowego obciążenia w sieci dystrybucyjnej ze źródłami fotowoltaicznymi. Do rozwiązania problemu zastosowano metodę Monte Carlo. (Zastosowanie metod probabilistycznych do przypadku trójfazowej sieci dystrybucyjnej ze źródłami fotowoltaicznymi) Keywords: Monte Carlo method; Probabilistic load flow; Three-phase load flow; Photovoltaic systems. Słowa kluczowe: metoda Monte Carlo, sieci zasilające, ogniwa fotowoltaiczne Introduction The electric distribution systems have unbalanced lines supplying three-phase loads. These systems present voltage unbalances. The unbalance is a state of a three- phase system in which the RMS values of the line voltages, and/or the phase angles between successive line voltages, are not all equal and/or 120º displaced. Distributed Generation (DG) is electricity generation sited close to the load it serves, typically in the same building or complex. DG creates a variety of well documented impacts on distribution network operation and implies significant changes to planning and design practices. Research has suggested that the benefits of distributed resources could be substantial. However, these distributed advantages are site specific [2,3]. A Photovoltaic Grid-Connected System (PVGCS) is chosen for DG. To assess the unbalanced voltages of the electric distribution systems, the unavoidable uncertainties that are relevant to the information of entry of the model must be considered. The uncertainties normally are due to variations in the time of the demands of phase load, generation and topology of the system. In the last years, the new and significant causes of the uncertainties in the distribution systems are due to the DG. For example, the production of energy by renewable sources - as the photovoltaic one - is powerfully associated to the uncertainties that concern the available energy of the sun. Hence, the right valuation of the impact of renewable sources on current systems of distribution must be faced by means of a probabilistic approach [3] that also considers the unbalance of the electrical systems. In addition, probabilistic approaches seem to be particularly useful to study in depth the influence of voltage unbalance on the performance of induction machines in stationary condition, with the objective to set up recommendations for their functioning [4]. This work studies the unbalanced three-phase systems with PVGCS. These PVGCS will be considered as negative charges, because they have a very low performance and it is advisable to work normally at a power factor close to the unit [5]. In this work, the model of PVGCS is incorporated into the three-phase load flow equations and the Monte-Carlo simulation method is used to consider the random inputs, not only the active and reactive loads, but also the distributed generation. Using deterministic load flow analysis, it is not possible to measure with objectivity how often and where overvoltages or undervoltages happen in the network during a period of time. This can be accomplished by employing probabilistic techniques like the probabilistic load flow or the Monte Carlo simulation. Probabilistic load flow demands modeling of loads and power productions as probability density functions and supplies the complete range of all probable values of the node voltages and load flows in the study with their respective probabilities assuming generation and load uncertainties and correlations and topological changes. The probabilistic load flow was enunciated in [6, 7] and further developed at a greater extent in [8, 9]. In [10] the probabilistic power flow was extended to the three-phase field to evaluate the uncertainties which affect the steady-state operating conditions of an unbalanced power system. Both Monte Carlo procedure and a linearized form were proposed. Various distribution system load flow algorithms, based on the forward/backward sweeps, were reviewed, and their convergence ability was quantitatively evaluated for different loading conditions in [11, 12]. In [13] was presented a three-phase power flow solution method for real-time analysis of primary distribution systems. This method is a direct extension of the compensation-based power flow method for weakly meshed distribution systems from single phase to three-phase. For asymmetrical three-phase load-flow study, two methods based on symmetrical component theory, the bus admittance method and the decoupling compensation method were proposed in [14]. Probabilistic PV system model Solar irradiation on a horizontal surface inside the atmosphere cannot be predicted exactly; it depends on the irregular presence of clouds. The randomness introduced by clouds on terrestrial radiation is characterized with two random variables [15, 16]: the daily clearness index K T and the hourly diffuse fraction k d . The statistical properties of the components of solar radiation allow constructing a probability model, using probability density functions (PDFs) and cumulative distribution functions (CDFs). These properties provide the probability for K T [15] and k d [16]. If the random variables K T and k d are known, then it is possible to determine the total irradiance on a surface sloped, G t,  , as a linear combination of K T and k d . At this point, the Cumulant Method [17] allows a statistical information mapping of predefined random variables K T and k d with the new random variable G t,  . If the variable G t,  is