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This chapter focuses on the optimal sizing of BESSs in end-user applications in the frame of time-varying energy pricing structures by adopting a probabilistic approach. More in detail, the proposed procedure focuses on one of the most used time-varying tariff structures (i.e. the ToU tariff) but it can be easily extended to other structures. In this chapter, starting from the procedure proposed, the probabilistic optimal sizing is performed applying the point estimate method (PEM), an algorithm that guarantees accuracy of the results with computational effort significantly lower than that implied by the Monte Carlo procedure.

In this paper, we address the problem of the probabilistic steady-state analysis of an electrical distribution system that includes wind and photovoltaic power plants. We propose a new method that takes into account the uncertainties due to the time variations of power load demands and the random nature of solar and wind energy. The new method extends a probabilistic approach proposed in the relevant literature for balanced electrical distribution systems to unbalanced electrical distribution systems, and it also takes into account the correlation among the random input variables. The method is based on Taguchi's orthogonal arrays. Numerical applications based on an IEEE test system provide evidence of the probabilistic performance of the proposed approach. The results obtained with the proposed method are compared with the results obtained using non-linear Monte Carlo simulation and the Point Estimate Method.

This paper deals with the optimal allocation (siting and sizing) of distributed electrical energy storage systems in unbalanced electrical distribution systems. This problem is formulated as a mixed, non-linear, constrained minimization problem, in which the objective function involves economic factors and constraints address the technical limitations of both network and distributed resources. The problem is cumbersome from the computational point of view due to the presence of both constraints of an intertemporal nature and a great number of state variables. In order to guarantee reasonable accuracy-although limiting the computational efforts-a new approach is proposed in this paper: it is based on a Simultaneous Perturbation Stochastic Approximation (SPSA) method and on an innovative inner algorithm, which allows it to quickly carry out the daily scheduling (charging/discharging) of the electrical energy storage systems. The proposed method is applied to a medium voltage (Institute of Electrical and Electronics Engineers) IEEE unbalanced test network, to demonstrate the effectiveness of the procedure in terms of computational effort while preserving the accuracy of the solution. The obtained results are also compared with the results of a Genetic Algorithm and of an exhaustive procedure.

Electric power systems are increasingly influenced by meteorological conditions due the wide penetration of renewable energy plants. In particular, wind speed plays a key role in transmission and distribution systems, as it determines wind power production and it influences line rating; also, extreme values of wind speed (EWS) affect mechanical reliability of wind towers and blades, causing loss of lifetime and damages. Several physical models provide EWS forecasts on global or local scales; however, such models require accurate initial and boundary conditions, and enormous computational effort. Statistical models are good alternatives to achieve accurate forecasts, but they require assumptions on the statistical distribution of EWS in parametric frameworks. This paper proposes an Inverse Burr - Inverse Weibull finite mixture distribution for EWS characterization and its Expectation-Maximization (EM) procedure for parameter estimation. The proposed mixture model is compared to commonly-used EWS distributions. Actual sites are considered to evaluate the goodness of fitting in different, real conditions; the EM procedure is compared to the maximum likelihood estimation. Sensitivity and error analyses are performed to individuate the main features of the proposed model; numerical results confirmed the suitability of the proposed model and of the EM estimation in the most of considered cases.

In the context of modern distribution systems, power quality is recognized as a crucial concern that requires advanced technical solutions in order to guarantee the correct operation of such complex systems. Among the power-quality disturbances, waveform distortions are one of the most important due to their impact on the equipment, and active filters are considered a useful technical solution to this disturbance even more significantly if they are optimally integrated into the grid. This paper addresses the problem of optimal siting and sizing of active filters in a multiconverter distribution systems by adopting a multiobjective approach. The proposed original formulation includes the control strategies of the active compensators among the decision variables and takes into account the various sources of uncertainties which characterize modern distribution systems. A heuristic simplified approach based on a trade-off/risk analysis is proposed to solve the multiobjective optimization problem under uncertainty with the aim of minimizing several conflicting objectives while meeting proper constraints.

Accurate probabilistic forecasts of renewable generation are drivers for operational and management excellence in modern power systems and for the sustainable integration of green energy. The combination of forecasts provided by different individual models may allow increasing the accuracy of predictions; however, in contrast to point forecast combination, for which the simple weighted averaging is often a plausible solution, combining probabilistic forecasts is a much more challenging task. This paper aims at developing a new ensemble method for photovoltaic (PV) power forecasting, which combines the outcomes of three underlying probabilistic models (quantile k-nearest neighbors, quantile regression forests, and quantile regression) through a weighted quantile combination. Due to the challenges in combining probabilistic forecasts, the paper presents different combination strategies; the competing strategies are based on unconstrained, constrained, and regularized optimization problems for estimating the weights. The competing strategies are compared to individual forecasts and to several benchmarks, using the data published during the Global Energy Forecasting Competition 2014. Numerical experiments were run in MATLAB and R environments; the results suggest that unconstrained and Least Absolute Shrinkage and Selection Operator (LASSO)-regularized strategies exhibit the best performances for the datasets under study, outperforming the best competitors by 2.5 to 9% of the Pinball Score.

Accurate short-term load forecasting is extremely important for the optimal management of smart grids allowing the increase of energy efficiency and enabling profitable demand response strategies. However, most of industrial and domestic loads are intrinsically affected by uncertainties due to many factors such as devices operational characteristics, time of use, weather conditions and other random effects. Therefore, probabilistic load forecasting is a challenging task and in relevant literature a great interest has been recently developed towards this topic. In this paper, the uncertainties related to the load demand are modeled through different probability density functions and a probabilistic method based on the Bayesian inference and stochastic time series is proposed for the shortterm forecasting of the probability density functions parameters. This paper shows the theoretical aspects of the proposed method and it is a companion paper to a paper of the same title, Part II, in which the numerical applications are reported.

The optimal allocation of capacitors in unbalanced distribution systems can be formulated as a mixed integer, non-linear, constrained optimisation problem. Fuzzy-based approaches, simulated annealing, tabu search and genetic algorithms are some of the techniques used for solving the problem in deterministic scenarios. However, distribution systems are probabilistic in nature, leading to inaccurate deterministic solutions. As a result, a probabilistic optimization model is required to take into account the unavoidable uncertainties affecting the problem input data, primarily the load demands. Of the various techniques for the solution of the problem, one of the most frequently used is the genetic algorithm. However, the application of simple genetic algorithms to solve the probabilistic optimization model involves tremendous computational effort. To reduce the computational effort, this paper proposes a new single-objective probabilistic approach based on the use of a micro-genetic algorithm. Two different techniques, one based on the linearised form of the equality constraints of the probabilistic optimisation model and one based on the point estimate method, were tested and compared. The proposed approaches were tested on the IEEE 34-node unbalanced distribution system to demonstrate the effectiveness of the procedures in generating reduced computational efforts and increased accuracy of the results.

This paper deals with the problem of optimal allocation (siting and sizing) of storage resources in unbalanced three-phase low voltage microgrids. The siting and sizing problem is formulated as a mixed, non-linear, constrained optimization problem whose objective function deals with economic issues and whose constraints involve technical limitations of both network and distributed resources. Emphasis is given to the power quality issue with particular attention to unbalance reduction and voltage profile improvement. Technological issues, such as those related to the preservation of batteries’ lifetime, were also taken into account. The planning problem is solved by means of a genetic algorithm which includes an inner algorithm based on sequential quadratic programming. In order to limit the processing time while maintaining reasonable accuracy, the genetic algorithm search space is significantly reduced identifying a subset of candidate buses for the siting of the storage resources. The Inherent Structure Theory of Networks and the Loading Constraints Criterion were used to identify the candidate buses. The proposed method has been applied to a low voltage test network demonstrating the effectiveness of the procedure in terms of computational burden while also preserving the accuracy of the solution.

Microgrids include distributed energy resources, controllable loads, and storage devices, and they can be classified into AC and DC types, depending on the characteristics of the supply voltage. In this paper, an optimal control strategy for a DC microgrid is proposed, and the strategy is aimed at minimizing the daily total energy costs. The DC micro grid can include non-dispatchable generation units (such as photovoltaic power generation) and dispatchable generation units, energy storage systems (batteries), and controllable/not controllable loads. The control strategy is based on a two-step procedure, i.e., (1) the implementation of one day-ahead scheduling and (2) a very short-time predictive control. The day-ahead scheduling is formulated using integer linear programming methodology and is aimed at achieving the optimal scheduling of controllable loads. The very short-time predictive control is based on the solution of a non-linear, multi-period, optimization problem and is aimed at achieving the real-time optimal charging/discharging profile of storage powers and the real-time optimal profile of powers of dispatchable generators thereby minimizing the cost of total daily energy. For both procedures, optimization models were formulated and solved, including technical constraints that guaranteed an adequate lifetime of the storage system. Case studies relative to a DC microgrid obtained by a modification of the actual structure of the electrical power plant of an Italian industrial facility were investigated in order to show the feasibility and the effectiveness of the proposed approach.

Voltage stability (VS) has become a fundamental issue in the new, liberalised markets due to the fact that the new power systems are approaching more and more the stability limits. Then, several approaches were proposed in the relevant literature to find the critical conditions and recently the problem was faced also with reference to unbalanced three-phase power systems. The unbalances, in fact, can be responsible of more critical stability conditions than in balanced power systems. Continuation power flow and optimal power flows were applied to analyse such conditions. This study deals with VS analysis in unbalanced power systems and proposes a new optimisation model to determine the critical point based on the use of complementarity constraints. In particular, the maximum stability margin is calculated by a single-stage or a multistage procedure that accounts for the relationship between the actual operating point and the maximum loading point. In addition, the multistage maximum stability margin problem is formulated also in a probabilistic framework to account for the uncertainties affecting the input data (e.g. load powers). An application is presented on a test system highlighting the feasibility and the goodness of the proposed technique.

Modern distribution grids are rapidly evolving toward complex systems (Smart Grids) due to the increasing level of penetration of distributed energy systems and information and communication technologies. In this context, Power Quality (PQ) becomes an important concern and advanced technical solutions to PQ problems are needed. The waveform distortions are one of the most important PQ disturbances and active filters are considered an useful technical solution to this disturbance. This paper deals with multi-converter distribution systems and formulates the problem of siting and sizing of active filters as a multi-objective optimization problem. The problem includes also the control strategies of the active compensators among the problem variables. A heuristic simplified approach based on a trade-off analysis is proposed to solve the multi-objective optimization problem with the aim of minimizing several objectives while meeting proper equality and inequality constraints. The proposed approach is applied to a 17-busbars distribution test system giving evidence of its effectiveness

Distribution networks are undergoing radical changes due to the high level of penetration of dispersed generation and storage systems. This trend is strongly modifying the structure as well as the management of distribution networks, which are progressively approaching the new concept of microgrids (MGs). Also, the level of penetration of storage systems for plug-in electric vehicles (PEVs) is increasing significantly due to the significant potential that PEVs have for reducing both emission levels and transportation costs. The inclusion of these vehicles in MGs leads to a series of challenges in grid operation, especially ensuring the provision of services that can improve the operation of distribution networks. This paper deals with MGs, including renewable generation plants and aggregators of PEV fleets connected to the grid through power electronic devices. A multi-objective optimization model is presented for obtaining optimal, coordinated operation of MGs. A multi-objective model was solved using two different methods, i.e., the exponential weighted criterion method and a compromise programming method. Both of these methods appeared to be particularly suitable when computational time is an important issue, as it is in the case of optimal control. The effectiveness of the multi-objective approach was demonstrated with numerical applications to a low-voltage microgrid; other multi-objective model-solving algorithms also were assessed in order to compare their programming complexity and the computational efforts required.

Capacitors and series voltage regulators are used extensively in distribution systems to reduce power losses and improve the voltage profile along the feeders. This paper deals with the problem of contemporaneously choosing optimal locations and sizes for both capacitors and series voltage regulators in three-phase, unbalanced distribution systems. This is a mixed, non-linear, constrained, multi-objective optimization problem that usually is solved in deterministic scenarios. However, distribution systems are stochastic in nature, which can lead to inaccurate deterministic solutions. To take into account the unavoidable uncertainties that affect the input data related to the problem, in this paper, we have formulated and solved the multi-objective optimization problem in probabilistic scenarios. To address the multi-objective optimization problem, algorithms were used in which all the objective functions were combined to form a single function. These algorithms allow us to transform the original multi-objective optimization problem into an equivalent, single-objective, optimization problem, an approach that appeared to be particularly suitable since computational time was an important issue. To further reduce the computational efforts, a linearized form of the equality constraints of the optimization model was used, and a micro-genetic algorithm-based procedure was applied in the solution method.

Several power system problems require solutions in an uncertain environment. According to the relevant literature, expected utility theory (EUT) has been used extensively to solve such problems. However, the application of Prospect Theory (PT) has demonstrated that people deviate from the expected utility maximization because their effective behaviors reflect loss aversion and risk-seeking, reflection effects. The aim of this paper was to compare and critically analyze EUT and PT with the goal of outlining the different behaviors of a “real” decision maker (DM) and an “ideal” DM, with the real DM operating in the frame of PT and the ideal DM operating as EUT describes. The results of using the two different theories were compared by solving three power-system problems in uncertain scenarios.

Voltage stability has become a fundamental issue in the new, liberalized markets due to the fact that the new power systems are approaching more and more the stability limits. Then, several approaches were proposed in the relevant literature to find the critical conditions and the problem was faced also with reference to unbalanced three phase power systems. The unbalances, in fact, can be responsible of more critical stability conditions than in balanced power systems. Continuation power flow and optimal power flows were applied to analyze such conditions. In this paper a new three-phase optimal power flow is proposed that makes use of complementarity constraints to take into account the effect of reactive generation limits on voltage stability in unbalanced power systems. An application is presented on a test system highlighting the feasibility and the goodness of the proposed technique. Both load and line unbalances are taken into account to capture the dependence of voltage stability on the level of unbalances.

Electrical storage systems are a potential solution for some challenges that will face future Smart Grids due to the large number and variety of services they can perform. These services can be further enlarged if an optimal operation of the storage systems is guaranteed. This paper deals with the energy management of battery storage systems in the frame of real time pricing. An optimal control strategy is proposed in order to minimize the electricity bill costs of an industrial facility. Firstly, the optimization model is formulated considering a multi-battery system. Then, two algorithms are applied to solve the problem: the linear programming and the dynamic programming. Numerical applications to an actual industrial facility demonstrate the effectiveness of the proposed procedure and the differences obtained applying linear and dynamic programming algorithms.

This chapter deals with probabilistic approaches for the steady-state analysis (probabilistic load flow) of distribution systems with wind farms. The probabilistic analysis is performed taking into account the randomness of both the distribution system loads and the wind energy production. Several approaches are presented to obtain the probability functions of state and dependent variables (e.g., voltage amplitudes and line flows). These approaches are mainly concentrated on wind farm probabilistic models, using one of the classical probabilistic techniques (e.g., Monte Carlo simulation, convolution process, and special distribution functions) to perform the probabilistic load flow. Numerical applications on a 17-bus balanced test distribution system and on an IEEE 34-bus unbalanced test distribution system are presented and discussed, considering the various wind farm models.

Microgrids 1 are electrical subsystems which include small generation units, storage devices and loads, which can operate either in grid connected or islanding mode. This paper deals with a DC microgrid that will be realized in an Italian industrial facility of GETRA Group. In particular, in this paper an optimization model is formulated to solve the problem of the day-ahead optimal scheduling of the microgrid. The model aims to optimize storage devices and controllable loads operation minimizing the daily costs of imported energy from AC grid and satisfying operational constraints such as production process constraints. The optimization problem is formulated using the Integer Linear Programming (ILP) theory. Case studies are also investigated in order to show the feasibility and the effectiveness of the proposed approach.

In recent years, modern distribution networks have rapidly evolved toward complex systems due to the increasing level of penetration of distributed generation units, storage systems, and information and communication technologies. In this framework, power quality disturbances such as waveform distortions should be minimized to guarantee optimal system behavior. This article formulates the planning problem of passive filtering systems in a multi-convertor electrical distribution system as a probabilistic multi-objective optimization problem whose input random variables are characterized with probability density functions. A heuristic simplified approach including trade-off analysis issues is applied to solve the planning problem with the aim of optimizing several objectives and meeting proper probabilistic equality and inequality constraints. The approach is able to quickly find solutions on the Pareto frontier that can help the decision-maker to select the final planning alternative for practical operation. The proposed approach is applied to a 17-busbar distribution test system to evidence its effectiveness.

Distribution system planners have frequently to choose among different solutions either in deterministic framework or under problem data uncertainty. In these conditions, adequate methods are needed to choose the optimal solution. This paper refers to electrical distribution systems with static converters and formulates the planning problem of passive filtering systems as a multi-objective optimization problem. A heuristic simplified approach including tradeoff/risk analysis issues is proposed to solve the problem with the aim of optimizing several objectives and meeting proper equality and inequality constraints. The approach is able to quickly find solutions on the Pareto frontier that can help the Decision Maker in selecting the final planning alternative to be practically operated. Both deterministic and uncertain frameworks are taken into account. This paper is a companion paper to a paper of the same title, Part II [1], in which the proposed approach is applied on a 17-busbars distribution test system. (8 pages)

This is a companion to a paper of the same title, Part I [1], which deals with the theoretical aspects of the optimal allocation of passive filters in distribution electrical networks with static converters. Part I formulates the multiobjective optimization model and proposes a heuristic solution procedure based on trade-off/risk analysis. This Part II presents and discusses numerical applications of the approach proposed in Part I, in order to illustrate the implementation problems and practical applications. The numerical tests refer to a 16-node Medium Voltage distribution system and consider both deterministic and uncertain frameworks. (7 pages)

In this paper, the point estimate method is applied to account for the uncertainties that affect the evaluation of the steady state operating conditions of an unbalanced three-phase power system in the presence of wind farms. The accuracy of the proposed technique is tested on the three-phase unbalanced IEEE 34-bus test system adequately modified to take into account the presence of wind farms; the results obtained by applying the Monte Carlo simulation are assumed as a reference. The main conclusion is that the point estimate method gives good solutions in terms of accuracy and computational efforts.

Genetic algorithms (GAs) are widely used for optimal allocation of capacitors in distribution systems. When dealing with large-scale systems, such as in case of unbalanced multi-converter distribution systems, these algorithms can require significant computational efforts, which reduce their effectiveness. In order to reduce processing time for GAs and simultaneously maintain adequate levels of accuracy, methods based on the reduction of the search space of GAs or based on micro-genetic algorithms have been proposed. These methods generally guarantee good solutions with acceptable levels of computational effort. In this paper, some fast, GA-based methods are compared and applied for solving the problem of optimal sizing and siting of capacitors in unbalanced multi-converter distribution systems. The algorithms have been implemented and tested on the unbalanced IEEE 34-bus test distribution system, and their performances have been compared with the performance of the simple genetic algorithm technique.

The problem of choosing the optimal location and size for shunt capacitors in unbalanced distribution systems can be formulated as a mixed, non-linear, constrained multi-objective optimization problem, and is usually solved in deterministic scenarios. However, distribution systems are stochastic in nature, which leads to inaccurate deterministic solutions.

In this paper, the problem of the optimal allocatio n and sizing of capacitors in unbalanced distribution sys tems is formulated as a multi-objective optimization proble m. This problem is solved by means of the e-constrained technique. Besides, some methods for determining a reduced fea sible region and a micro genetic algorithm are used. The solving method proposed is tested on the IEEE 34-node unbal anced distribution system in order to demonstrate the eff ectiveness of the approaches in terms of reduced computational efforts and accuracy of the results.

The problem of sizing and location of capacitor banks in unbalanced distribution systems with nonlinear loads is addressed. This problem can be solved by means of genetic algorithm based procedures that have been proven to find good solutions. However, when dealing with multiconverter unbalanced large scale systems, these algorithms may require significant computational effort which will reduce their effectiveness. In this paper a hybrid method, which significantly reduces the computational effort without affecting the level of accuracy is proposed. The hybrid method involves a two step procedure: first a small number of nodes are selected as candidate buses for the capacitor allocation through the application of the inherent structure theory of networks; then, a heuristic procedure based on sequential placement of discrete capacitor units is applied to the selected buses. Numerical examples are presented to demonstrate the efficiency of the proposed procedure.

In this paper, the problem of the optimal allocation and sizing of capacitors in unbalanced distribution systems is formulated as a multi-objective optimization problem. Financial factors as well as technical issues are considered among the objectives. The costs include, in the most general case, at least initial investment cost and Joule losses costs; the other objective functions link the capacitor placement to power quality indices, such as fundamental voltage profile, harmonic distortion and unbalances. The multi-objective problem is solved by means of trade-off methods, that are valuable tools for distribution engineers that have to select the best-compromise solution. A case study referred to the IEEE 13-node test feeder is presented and discussed

Distribution system engineers have frequently to choose among different solutions under problem data uncertainty; in most cases, among the uncertainties the ones deriving from the presence of static converters must be included. In these conditions, the decision theory can offer efficient and valuable tools to choose the optimal solution. This paper refers to distribution electrical systems with static converters and considers three different approaches based on the decision theory for the economical sizing of medium voltage cables and passive filters under uncertainty (separately and together), properly accounting harmonics due to nonlinear load presence. In the last part of the paper the implementation and practical application of the sizing procedures are discussed and compared in some case studies.

In this paper the economical damage due to the harmonic losses is quantified by means of the expected values of the operating costs and of the aging costs. To evaluate the above quantities a simplified approach based on closed form relations is proposed. It can be applied when only one group of nonlinear loads is the main cause of the harmonic pollution. The evaluation is easy and quick, since it requires the knowledge of only the mean value, the standard deviation and the covariance of the random variables on which the expected values depend. A numerical application to a test system is presented and discussed; it is shown that the results of the simplified method are not far from those obtained using an exact method.

This paper describes a new probabilistic method for harmonic analysis of multiconverter power systems which allows the evaluation of the percentiles of the voltage and current harmonics, that are the probabilistic quantities of interest in the most recent international recommendations or standards. In order to reduce the computational efforts, the proposed method firstly linearizes the well known nonlinear equation system describing the steady state behavior of multiconverter power systems and then calculates the mean and other moments of the voltage and current harmonic probability density functions: finally, the percentiles of interest are obtained applying the Pearson distributions, which require the knowledge of only the first four moments of the probability density functions. Numerical applications to a test system are also presented and discussed to demonstrate accuracy and speed of the proposed method.

In designing a distribution system, engineers have to choose optimal solutions under environmental uncertainty; in most cases, among the uncertainties the ones deriving from the presence of nonlinear loads must be included. Under these conditions, the Decision Theory appears to be a valuable tool for solving distribution design problems. This paper considers three approaches based on the Decision Theory for the economical sizing of medium voltage distribution electrical lines under uncertainty, paying particular attention to harmonics due to nonlinear load presence. In the last part of the paper the implementation and practical application of the sizing procedures are discussed and compared in some case studies.

The most adequate way to take into account the uncertainties which affect the modelling of harmonic penetration is to express the input data by random variables and to apply probabilistic techniques of analysis. In this paper a probabilistic method is proposed for assessing harmonic distortion levels in power systems. This method transfers to the field of probabilistics a deterministic iterative procedure which is extremely useful for analysing multiconvertor systems. The proposed method, called the 'probabilistic iterative harmonic method', employs a Monte Carlo simulation technique. A numerical application to a test system is also discussed.