Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 6379253, 20 pages

http://dx.doi.org/10.1155/2016/6379253

## Energy Optimization for Distributed Energy Resources Scheduling with Enhancements in Voltage Stability Margin

^{1}Automation and Control Group, Department of Electrical Engineering, Denmark Technical University (DTU), Elektrovej, Building 326, 2800 Lyngby, Denmark^{2}GECAD, Knowledge Engineering and Decision Support Research Center, Polytechnic Institute of Porto (IPP), R. Dr. António Bernardino de Almeida 431, 4200-072 Porto, Portugal

Received 30 July 2015; Revised 3 March 2016; Accepted 5 April 2016

Academic Editor: Xavier Delorme

Copyright © 2016 Hugo Morais et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The need for developing new methodologies in order to improve power system stability has increased due to the recent growth of distributed energy resources. In this paper, the inclusion of a voltage stability index in distributed energy resources scheduling is proposed. Two techniques were used to evaluate the resulting multiobjective optimization problem: the sum-weighted Pareto front and an adapted goal programming methodology. With this new methodology, the system operators can consider both the costs and voltage stability. Priority can be assigned to one objective function according to the operating scenario. Additionally, it is possible to evaluate the impact of the distributed generation and the electric vehicles in the management of voltage stability in the future electric networks. One detailed case study considering a distribution network with high penetration of distributed energy resources is presented to analyse the proposed methodology. Additionally, the methodology is tested in a real distribution network.

#### 1. Introduction

The growing use of distributed generation (DG) in different voltage levels has been changing the power systems operation concept. To support the network operation and also to take advantage of the distribution energy resources, it is important to develop new operation and management methodologies. One key aspect to guarantee adequate service levels is the power system stability margin that is ensured by adequate ancillary services. These services are traditionally provided by centralized power plants with high power capacity and coordinated by the system operators in the transmission level. However, in a near future and considering the growing penetration of distributed energy resources in medium and low voltage distribution network, the system stability should also be ensured by the system operator in the distribution level, as well as by the aggregators (e.g., virtual power plants) that manage the distributed energy resources [1, 2].

Behind the DG units, the consumers, storage systems, and electric vehicles (EVs) are also important to support the power system stability [3]. At the distribution level, the distribution system operator and the aggregators can participate in several ancillary services [1, 4] such as primary, secondary, tertiary frequency, and voltage control; fault-ride-through capability; the congestion management; the power losses minimization in distribution networks; the monitoring the waveform quality; and the islanded operation of networks.

Extensive reviews on voltage stability indexes can be found in the literature [5–9], with special focus on online assessment methods for voltage stability. Nevertheless, using this index to enhance scheduling, reconfiguration, and dispatch solutions has shown its potential to improve the solutions regarding the voltage stability limitation [10–12]. These approaches make the -index a possible option to enhance the solutions, which was initially proposed in [13] based on the power flow equations; however, recent attention to its use, application, and possible improvements has been reported in [14]. A thorough comparison with other indexes can be found in [8], as well as a discussion on -index limitations [9].

The line stability indexes like the [15] or the VCPI [16] are more accurate than the -index to predict the voltage collapse proximity in a real time operation. However, in the day-ahead scheduling optimization, the main goal of the voltage stability index is not based on determining the proximity to the collapse, but to influence the distributed resources scheduling for contributing to the system stability. As stated in [16], the evolution of the -index is similar as the other suggested indexes (when the -index increases the and VCPI also increase), meaning if we optimize the -index we are also improving the and VCPI indexes. The -index is used in the present study because it is easier to integrate in the optimization problem than the other suggested indexes. With the -index, the objective function does not depend on the use of and consequently not on due to the load consumption in each bus (also depends on the resources scheduling such as distributed generation and electric vehicles).

The paper proposes an energy resource scheduling problem with a multiobjective function incorporating the operation cost and the voltage stability. This multiobjective optimization problem will be applied for a scenario in a distribution network with a high penetration of distributed energy resources, mainly an intensive EVs penetration. In the operation cost different technologies of DG and the use of EVs with griddable capability are considered, also known as vehicle-to-grid (V2G). The use of -index is proposed to deal with the voltage stability in the joint optimization problem. The -index was initially proposed in [13] based on the power flow equations. Two optimization techniques are proposed in this paper for solving the proposed multiobjective energy resource scheduling problem. These techniques will determine the nondominated solutions of the multiobjective optimization problem, namely, the weighted-sum method and an adapted goal programming methodology. Thus, the nondominated solutions represent the Pareto front that was proposed in [17], yet the application in engineering and science fields only began in the end of the seventies [18]. Furthermore, the goal programming methodology can be very useful in real application due to the complex characteristics of the objective functions. In a regular power system operation, the system operators can establish a predefined range in the operation cost objective function, and, in the critical situations (operation near from boundaries), the system operators can limit the objective function concerning the voltage stability index.

To demonstrate the effectiveness of the proposed methodologies, concerning voltage stability, two studies were included: in the first one, the sensitivity analysis is performed considering variations in the power demand, in the voltage angle, and in the voltage magnitude on the slack bus (from the distribution network’s point of view the reference bus is the connection in an upstream network). In the second analysis, the loadability limit is determined for an hour considering three different scheduling objective functions (operation cost, -index, and multiobjective), allowing the determination of the maximum load that can be supplied (voltage stability boundary) considering the voltage control constraints. This approach is equivalent to the bifurcations determined with continuation power flow algorithms that allow to calculate the loadability limit for the power system [19–21]. Both analyses show the improvements in the energy resource scheduling problem through the incorporation of -index as another objective function. In addition, both methodologies are tested in a distribution network with high penetration of distributed energy resources, considering the use of electric vehicles allowing the -index and the operation cost evaluation. The weighted-sum method is also applied to a real distribution network to evaluate its performance in a large network.

After the Introduction, Section 2 presents an overview concerning the energy resource scheduling problem. Section 3 focuses on the mathematical formulation and on the implementation of the proposed methodologies. Section 4 shows the case study considering a 33-bus distribution network, and finally the most important conclusions are presented in Section 5.

#### 2. Energy Resource Scheduling Overview and Contributions

The development of energy resources scheduling methods considering the distributed resources in different voltage levels of power systems is an important research topic. Typically, the energy resource scheduling consists in an optimization problem to determine the best scheduling to minimize the operation cost of the available resources [22]. However, in a smart grid context it is also important to take into account other aspects than just the economic one, such as power quality, voltage stability, environmental aspects, or the load diagram profile. Therefore, all these aspects can be included in the energy resource scheduling providing different solutions to help the system operators in the decision making process.

Several authors have proposed different methodologies to deal with the energy resource scheduling considering distributed energy resources, such as DG and active consumers with demand response programs and the network operation. In [23], it is described a framework for aggregators to determine the energy resource scheduling based on the concept of quality-of-service in power system. A more complex negotiation perspective is presented in [24] considering multilevel negotiation layers between aggregators and electricity market participation. For a microgrid level perspective, [25] proposes a multiagent base platform allowing the scheduling of the distributed energy resources.

Other works deal with the energy resource scheduling to integrate the electric vehicles with V2G capability. A comprehensive and exhaustive review is presented in [26] concerning the impact of EVs in the distribution network. In [27], the authors proved that EVs can improve the management of intermittent renewable resources such as wind farms, and in [28] it is shown that EVs can be used to level the daily load diagram. Wu et al. [29] claim that the charging control in EVs is required for a well accommodation in the power system. To handle the large number of electric vehicles, several artificial intelligence algorithms have been proposed [30–32] to provide the scheduling of charge and discharge energy from EVs batteries. Another innovative perspective is proposed in [33] considering a hierarchical model to coordinate the energy resource scheduling in smart grid with electric vehicles. The integration of plug-in hybrid electric vehicles in microgrids resource scheduling is proposed [34].

The use of multiobjective functions in the energy resources scheduling problems is an important challenge to improve the quality of the obtained solutions. Some approaches are proposed considering the environment aspects [30, 35] or to levelling the load diagram [28, 36] in the energy resource scheduling problem. However, as is possible to see in [26], few work was developed considering the contribution of the distributed energy resources and mainly the electric vehicles to the ancillary services like the voltage stability. The inclusion of a voltage stability index in the energy resource scheduling problem turns into a multiobjective function, because it is a competing objective with the operation cost. The main contributions of this work are as follows:(1)To propose a multiobjective model to deal with the operation cost and voltage stability in the energy resource scheduling problem.(2)To use distributed energy resources, namely, distributed generation and electric vehicles, for contributing to the power system voltage stability.(3)To apply the weighted-sum methodology and to adapt the goal programming methodology to determine the Pareto front of the proposed multiobjective energy resource scheduling problem.(4)Test the proposed multiobjective approach in a real distribution network.

#### 3. Energy Resource Scheduling in Distribution Network

The energy resource scheduling is an important task in the present and the future power systems operation. The growing penetration of distributed generation and other energy resources, such as the electric vehicles, increases significantly the problem complexity [37]. The energy resource scheduling can consider several objective functions, most of them based on the energy costs or on the entities profits. However, technical aspects, such as the system stability, are becoming more important in new operation paradigm of the future distribution networks. In this paper, it is proposed a multiobjective energy resource scheduling for the distributed energy resources, considering two objective functions, namely, the operation cost and the voltage stability, using two different methodologies. The first methodology, called weighted-sum, is one of the most popular methods to solve multiobjectives problems. The second implemented approach is the modified weighted goal programming which is also used in several real applications. These methodologies can be used by an aggregator with the responsibility to control different distributed resources as well as part of the distribution network.

The goal programming approach can result in non-Pareto optimal solutions [38], and the execution time for each simulation should be higher due to the increased number of constraints (one of the objective function is formulated as constraint) [39]. On the other hand, it is possible to obtain an approach of Pareto front with few simulations. Therefore, the use of goal programming approach was selected considering the characteristics of the objective functions. In fact, when the system is operating normally, the system operators can establish a predefined range in the operation cost objective function, and in critical situations (operation near to boundaries) the system operators can define the objective function concerning the voltage stability index.

##### 3.1. Operation Cost Objective Function ()

The operation cost function is composed by several terms concerning different distributed energy resources use/operation costs that are given by

For the DG units, a quadratic function is used, which is commonly employed for fossil fuel units [40]. In DG units based on renewable sources (e.g., wind or solar), the linear term of the quadratic function is the only one considered. The cost with energy acquisition to external suppliers is also considered that allows the balance between the DG, EVs, and demand in the distribution network. In this formulation the cost with EVs discharge is considered and also the benefit to the aggregator from charging EVs . In addition, the battery degradation cost is considered during the EVs discharging process [41, 42]. Finally, two penalization costs are considered. The first one, , penalizes the aggregator when nonsupplied demand situations occur. The second one, , refers to “take-or-pay” contracts violation. These contracts are considered mainly for wind and solar units, and the penalization occurs when generation curtailment is necessary. The penalization terms are important to make a robust mathematical formulation in order to handle with critical situations from high consumer demands or high power generation from DG units.

##### 3.2. Voltage Stability Objective Function ()

In the proposed mathematical formulation, the voltage stability is achieved considering the load index (-index) minimization. In [13] the following expression is proposed that determines the -index (), considering bus as a generation bus and bus as the load bus:

In [43] and most recently in [44], a new expression is proposed to determine the -index using measurements of voltage phasors at the bus and is defined as

The -index value is between 0 and 1, and the optimal value is close to 0. If the maximum -index in the system is less than 1, the system is stable in terms of voltage level. The system is unstable if the -index value is above 1 [13]. From the optimization point of view, the goal is to minimize the maximum value of -index in all buses. Basically, the -index minimization involves taking into account the bus far from the stressed condition boundaries.

The evaluation of -index implies the use of expression (3) in all consumption buses. However, in the future distribution networks there will be generation connected in several buses, changing the consumption buses to the generation buses in some periods of the day depending on the distributed energy resources installed and on the generation and load forecast in each one. Therefore, all buses are compared with the bus connected to the high voltage level in order to determine the -index, where the minimization of the function , which is the maximum -index in each period , is formulated:

Function is a nonconvex function, which requires more time to find the optimal solution. The epigraph variable is used to turn the function into a convex one:where the epigraph variable removes the nonconvexity of function (the maximum -index) turning the optimization problem simpler to be solved. The use of the epigraph variables is detailed, explained, and illustrated in [45], turning a nonlinear optimization problem into a linear optimization problem.

##### 3.3. Multiobjective Function: Weighted-Sum Approach ()

The weighted-sum method [46] transforms the multiobjective function into a single one by summing all functions ( and ), where each function is multiplied by a different weight ( and ), as it is formulated:where the weight factors are between 0 and 1 for giving more or less relevance to each objective function. Additionally, the sum of the two weight factors must be equal to 1. To uniform the objective functions the voltage stability price factor () is included. The voltage stability can be quantified as a price signal meaning that the multiobjective function can be treated as a single objective function to optimize the cost. In the present paper, the value of is equal to the energy cost of the most expensive distributed resource, as given bywhere the contains the prices of all resources scheduled (DG, external suppliers, and EVs) solving the optimization problem with just the operation cost function . For the DG units that use a quadratic function, it is considered an average price determined by the multiplication of the DG maximum generation power and the coefficients of the quadratic function and then divided by the same maximum generation power. Typically, the price selected will be the most expensive resource scheduled in the peak periods, because in those periods it has the highest consumption power. However different expression can be also used depending on the aggregator’s strategies and on the normal network operation conditions. The weighted-sum method is the most traditional and popular method that parametrically changes the weights among objective functions to obtain the Pareto front [47].

##### 3.4. Multiobjective Function: Goal Programming Considering the Utopia Point Approach ()

The goal programming was firstly proposed in [48, 49] and it is used in a large range of problems in different areas [50]. Several variations of the original method have been proposed, such as the reference goal programming [51], or the Archimedean goal programming (also known as weighted goal programming) [52]. The goal programming consists in the definition of a goal for the objective function, converting the original objective function into a constraint, as it is described

In order to cope with variations in the initial goal, positive and negative deviation variables for each objective function should be added to the new constraint. The objective in (8) is to minimize the positive and negative deviation variables [39]. Additionally, a weight factor can be multiplied in each deviation variable turning the method into a weighted goal programming. The Pareto front can be also obtained by this method through changing the weights of the positive and negative deviation in each simulation [39, 53].

The proposed methodology is based on the goal programming method with additional changes in order to adapt this approach to the envisaged problem. The proposed methodology uses some principles of the normal-boundary intersection proposed in [54]. The main idea is to establish only one objective function as constraint, trying to optimize the other objective function. The main advantage is the ability to execute simulations only in the predefined ranges for each parameter. Figure 1 presents the flowchart of the proposed goal programming methodology.