Abstract
In association with the development of intermittent renewable energy generation (REG), dynamic multiobjective dispatch faces more challenges for power system operation due to significant REG uncertainty. To tackle the problems, a dayahead, optimal dispatch problem incorporating energy storage (ES) is formulated and solved based on a robust multiobjective optimization method. In the proposed model, dynamic multistage ES and generator dispatch patterns are optimized to reduce the cost and emissions. Specifically, strong constraints of the charging/discharging behaviors of the ES in the spacetime domain are considered to prolong its lifetime. Additionally, an adaptive robust model based on minimax multiobjective optimization is formulated to find optimal dispatch solutions adapted to uncertain REG changes. Moreover, an effective optimization algorithm, namely, the hybrid multiobjective Particle Swarm Optimization and Teaching Learning Based Optimization (PSOTLBO), is employed to seek an optimal Pareto front of the proposed dispatch model. This approach has been tested on power system integrated with wind power and ES. Numerical results reveal that the robust multiobjective dispatch model successfully meets the demands of obtaining solutions when wind power uncertainty is considered. Meanwhile, the comparison results demonstrate the competitive performance of the PSOTLBO method in solving the proposed dispatch problems.
1. Introduction
The traditional economic dispatch method aims to determine a generation schedule that minimizes total generation cost while being subjected to generator and system operating limits [1–3]. With increasing concerns of environmental pollution, harmful emissions, such as SO_{2}, NO_{x}, CO, and CO_{2}, have attracted widespread attention. Therefore, simultaneously minimizing total generation cost and emissions has become a crucial research topic. In some studies, the separate economic and environmental dispatch problems are converted to an economic and emission dispatch (EED) problem, formulating a multiobjective optimization issue [3–5]. The EED can provide a set of dispatch solutions for decision marker to choose from with different preferences in economic and emission. Many methods and approaches have been proposed to solve the multiobjective EED problem [6–13]. In the initial studies [6–8], researchers attempted to transform the EED problem into a singleobjective model based on a linear combination of different objectives as a weighted sum. However, this method cannot obtain Pareto front solutions in a single run and does not address how to select weighting factors for the system operators. Moreover, these approaches fail to achieve optimal solutions when the objective functions are not convex or when the objective functions have a discontinuous variable space. To address these problems, several artificial intelligent techniques have shown good performance. An improved Hopfield neural network (NN) in [9] and an improved backpropagation NN in [10] were reported to optimize EED problems. However, these approaches are readily trapped within the local optimum, since the achieved results are not sufficiently accurate. On the other hand, multiobjective evolutionary algorithms have been successfully applied to solve the EED problem [3, 11, 12]. In [3], a novel modified adaptive particle swarm optimization is presented to investigate the multiobjective EED. The fuzzy IF/THEN rule is used to selfadaptively adjust the cognitive and social parameters in the PSO to avoid the evolution stagnation for continuous generation. Reference [11] presents a multiobjective differential evolution algorithm for EED problems. In [12], a robust EED model based on an effective function is built to handle wind power uncertainties. Carbon capture and storage are considered in the model formulation to reduce carbon emission, and a multiobjective bacterial colony chemotaxis method is adopted to solve the proposed robust EED model. However, none of these papers consider energy storage (ES) integration.
Due to the longterm fossil fuel energy crisis and the recent Paris agreement tasks to reduce fossil fuel dependency in traditional power generation, significant renewable energy generation (REG), for example, wind power, photovoltaic energy, and tidal energy, is integrated into multiple levels of the power grid. The uncertainty and variability of the power production provided by REG raise the risk of system instability, particularly in distribution networks. To solve this problem, the integration of ES is an effective solution to alleviate the negative effects caused by the intermittent nature of REG. According to reviews [14–16], ES plays a vital role in smoothing the production of REG, improving the REG penetration level and peak shaving, ensuring system reliability, and increasing the economic and environmental benefits of the power system [17]. Improper dispatch scheduling of generation may lead to undesired cost increases or system reliability deterioration. Typical ES has dynamic power and energy formulations in which the constraints are coupled along the time and space domain. The operation of ES in one time step affects the evolving operation in the other time slots. Therefore, a multistage EED integrated with ES is a strong coupling and dynamic problem in the spacetime domain [18, 19].
In past studies, some economic or EED dispatch approaches are presented. First, some literature has focused on economic ES dispatch or EED focused on singleinterval (1 hour) dispatch [12, 20, 21]. In this setup, the economic or emission objectives in one time interval are optimized and then shifted to the next time interval. Second, in dayahead scheduling problems, current research is focused on investigating the composite economic cost with ES within multiperiods [22–25]. Multiobjective, multistage, dispatch optimization problems with ES are rarely considered. In addition, the prediction error in dayahead REG prediction is inevitable in real cases, which should be carefully taken into consideration in the EED process. At the most time, chance constraint is used to ensure that the loss of load probability is lower than predefined risk level to deal with uncertain REG problems [26, 27]. However, it is difficult to obtain the distribution function of REG.
To the best of the authors’ knowledge, very few researchers emphasized the study on ES scheduling in dynamic, multiperiod, EED while simultaneously dealing with the uncertainty of intermittent REG [17]. Motivated by the above concerns, this paper proposes a dynamic EED approach to schedule the output of ES together with controllable generations over the next 24hour time span. The main contributions of this paper are organized as follows. (i) To address the uncertainties of REG, robust multiobjective optimization is proposed based on minimax optimization approach. (ii) The EED dispatch problem with REG prediction error is converted to a robust multiobjective dispatch model. (iii) Aiming at effectively solving the proposed model, a novel multiobjective PSOTLBO is proposed and employed to seek the Pareto solutions.
The organization of this paper is decomposed into seven sections. Section 2 introduces the energy storage model followed by the dayahead generators and ES dispatch problem formulation presented in Section 3. Further, the robust, multiobjective, dayahead dispatch model is explained in Section 4. Section 5 proposes the multiobjective algorithm to solve the given complex problem. The numerical case study and conclusions are then presented in Sections 6 and 7, respectively.
2. Energy Storage Model
ES is connected to the gird by converters capable of flexibly operating in charge and discharge modes [25, 28]. It is therefore promising for ES to handle the intermittent REG integration problem and to improve the flexibility of energy dispatch. However, ES has significant dynamic and evolving power and energy characteristics, leading to a batch of technical limits with timeevolving decision variables, particularly in multistage scheduling operation [22]. Details of the ES general model and its dynamic constraints are expressed as follows [29].
ES charge/discharge power limits are as follows:where and denote the charge and discharge power of the th ES at the hour , with the power output of ES being positive when discharging, and vice versa. is the power capacity of th ES.
The state of charge (SOC) is a crucial state variable in the process of ES schedule and operation.
When the ES is charging (), When the ES is discharging (),When the ES is idling (),where is the SOC of the th ES at the hour , is the charge or discharge power at the hour , is the charging efficiency, is the discharging efficiency, is the schedule interval, and is the ES energy capacity.
The ES SOC limits are as follows:where is the SOC of the th ES at the hour , and and are the minimum and maximum limits of the SOC, respectively.
The initial SOC of each ES is the same at the beginning of each day, where stands for all the time slots [30].
To prolong the lifetime of the ES, it is assumed that it is only eligible for one chargedischarge cycle per day for optimal operation [31, 32]. Meanwhile, in this paper, ES is used for peak shaving. It stores energy during offpeak and return the power during peakload hours.
3. Problem Formulation
The dayahead dispatch performs the generation dispatch every 24 hours, scheduling all generators and ES units the next day in hourly time slots [33, 34]. The scheme of EED model is illustrated in Figure 1. This is a typical, dynamic, multiperiod decision problem. The decision variables at each hour influence the decisions at the remaining hours. The dynamic multiobjective dispatch problem is described as follows.
3.1. Variables
The input variables of the dispatch problem are the REG power, load demand, and ES parameters over the planning period. The decision (control) variables set is the power outputs of all the controllable units. where stands for the power generation of the th unit at the time slot , is the number of control variables, and presents the total number of time slots.
3.2. Objective Function
The overall objective function of the dayahead dispatch problem includes the total generation cost of the whole system and the pollutant emissions .
The controllable generations are the fuelbased generators. The operational costs of renewable generation and ES within the shortterm dispatch optimization horizon under strong constraints mentioned above are ignored [22]. Therefore, the first objective is the accumulated economic cost of all the conventional generators. These generation costs are normally modeled using a quadratic function of their power outputs [27]. The economic objective is minimized over the scheduling period, for example, 24 hours of the next day, which can be described as follows:
The environmental objective is to minimize the total emission pollutants of all generators as much as possible. These emission costs are also formulated as a quadratic function of their power outputs [27], and the emissions of the ES are taken to be zero. The objective function is expressed as follows:
In (9) and (10), is the total number of time slots for next day, which is 24 hours in this paper, is the number of conventional generators, , , are the cost coefficients of the generators, and are the emission coefficients of the generators.
3.3. Technical Constraints
To maintain the power grid operation, a set of equation and inequation constraints should be considered. The system constraints include the overall system constraints and the ES constraints. The ES constraints are shown in Section 2, and the overall system constraints are expressed as follows.
Power flow balance constraint iswhere , , and are the active power injected by the REG, th controllable generator, and th ES at time , respectively; , , and are the total number of controllable generator, energy storage, and load, respectively; is the power loss coefficient.
Generation output limits are
Generators ramp up and down limits arewhere and represent the minimum and maximum power output of th generator, respectively; and are the ramp up and ramp down limits for th generator, respectively.
4. Robust Multiobjective DayAhead Dispatch
It is well known that renewable energy is intermittent and uncertain. The error in REG prediction is inevitable for any prediction technique. Hence, dealing with the uncertainty of renewable energy is a key issue.
4.1. WorstCase Based Optimization
The robust optimization with the worstcase scenario is one of the most common approaches. The worstcase optimization, in a minimization problem, can be formulated as follows: where is the decision variable over a feasible region and is an uncertain parameter in the uncertainty set .
For environmental uncertainties, the function values of become [35, 36]:where are the decision variables and are the configuration variables, defined by . is the nominal value of the environmental parameter and is the change condition. Once a decision variable is implemented, the configuration variable can be termed as the solution’s adaptability. The value of will be determined according to uncertain environmental parameter. The above robust optimization with the worstcase method is focused on the best decision to worstcase performance and to minimize the risk of severe consequences.
4.2. Robust Multiobjective Optimization
Most robust optimal dispatch problems considering REG uncertainty are singleobjective models or are converted from the multiobjective EED problems to a single objective. They ignore the importance of Pareto optimality, which could provide multiple optimal solutions for operators in the dispatch decisionmaking process. Robust optimization means to search for the solution that is the least sensitive to perturbations of the decision variables in its neighborhood [12]. The REG forecast error is unavoidable, resulting in the possibility that the optimal deterministic solutions may not be practically suitable in reality. On the other hand, stochastic methods based on the probability model have difficulty obtaining accurate probability functions. In this paper, we utilize the minimax multiobjective optimization method based on the worse case in dayahead generator and ES dispatch problems.
We assume a robust, multiobjective optimization problem:where , , and are the decision, configuration, and uncertain input variables, respectively, is the th objective function, denotes the perturbation points, and and are the feasible space of and .
The following robust multiobjective optimization approach is defined in (18), where a robust multiobjective solution, , is defined as the Paretooptimal solution with respect to a neighborhood:Subject to , where is the feasible region of the decision variables, the function is defined as follows:where denotes the sampling point number in the perturbation domain and is the th perturbation sampling point.
4.3. Robust EED Dispatch
In the dayahead predispatch stage, the intermittent renewable power, such as wind, usually deviates from its forecasted value. The forecast error of an REG exists in any prediction models. In view of this, a robust EED dispatch model is developed to adapt to the uncertainty of wind power (WP) and handle its prediction error when spinning reserve is not considered. In order to balance the generated power with the power demand and power loss, one of generators is chosen as the slack generator with output limits. According to (11) and (12), the power output of the slack generator, , can be calculated by solving the following quadratic equation [37]:
According to (20), once the other controlled units is determined, the output of the slack generator is temporarily fixed. is regarded as a configuration variable. Then, the adaptive adjustment of the slack generator output, associated with other fixed control variables, allows the robust dispatch solutions to follow other operational conditions. In other words, only the output of the slack generator will be adjusted with robust optimal dispatch solutions according to the realization of different REG output.
The original EED dispatch objective functions that minimize cost and emissions in (9) and (10) are transformed in the robust model, which can be expressed as follows: where is the power output of controllable power units and is the uncertain set of WP.
The robust dayahead EED dispatch model is expressed as follows:The constraints are formulated as (1)–(6) and (11)–(14).
4.4. Uncertainty Wind Power
To further analyze the uncertainty of WP, the actual WP output at time () limits can be described as follows:Commonly, the minimum output of WP is 0, and the maximum is defined as the installed capacity.
In the robust optimization, the uncertainty has a direct impact on its performance. WP generations follow the Gaussian distribution, with the predicted value as the mean and prediction error as the standard variance. A 95% percentage of samples fall in the range . To generate typical scenarios, Latin hypercube sampling (LHS) has been utilized to generate WP uncertain dates each time slot [38]. To efficiently capture and with uncertainty in 24hour time slots in (21), two extreme scenarios, as well as several stochastic scenarios, are selected using LHS. To ensure robust optimal solutions, the method to obtain 24hour WP scenarios is shown as Algorithm 1.

5. Multiobjective PSOTLBO to Solve the Problem
The robust dispatch problem is formulated as a mathematic model, and the task in this section is to rapidly and effectively seek a balance between the total economic and emission benefits over all intervals during the next day. We adopt a novel, multiobjective, metaheuristic based method called the PSOTLBO to solve the problem.
5.1. Multiobjective PSOTLBO
To find the optimal solutions in the aforementioned multistage, dayahead dispatch problem, an effective and efficient multiobjective optimization method is required. It is noted that PSO is a popular technique but lacks sufficient exchange within different particles. Its premature convergence limits the performance of the MOPSO [39]. The multiobjective Teaching Learning Based Optimization (TLBO) can achieve the satisfactory performance on some benchmark functions with respect to convergence rate and calculation time but sometimes weak in diversity and distribution, especially in nonconvex Pareto functions and multimodal function [40]. Meanwhile, EED dispatch problem has lots of equation and inequation constraints. When energy storages with strong coupling constraints incorporated into EED, EED problem becomes more complex. The current multiobjective PSO or TLBO cannot simultaneously capture satisfactory optimal solutions with respect to good convergence and wellspread goals for such a complex EED problem. To capture better dynamic and robust EED schemes, the effective PSO update is employed for the global search and gives a good direction to the optimal region. In addition, the TLBO algorithm is activated periodically to improve the exploration and exploitation ability of the PSO. Meanwhile, a circular crowded sorting (CCS) strategy is proposed to truncate the nondominated solutions archive. The multiobjective PSOTLBO is implemented in [41].
5.2. Constraints Handling
To satisfy the power balance in (11), one generator is assumed to be in slack generator operation, and the limit constraints will be handled using the static penalty method [37]. The constraints of other generators and ES outputs will be handed using their boundaries. If the dimension of the particle moves outside of its boundaries, it will be assigned the value of the corresponding boundary. The constraint of the ES charge/discharge power will be handled by the following equations:
The slack generator limits, generator ramp limits and SOC level in the end constraint will be handled using the static penalty method with the penalty function below. The penalty function will be expressed as (25) when the constraints are violated:where is the total number of constraints handled using the static penalty method, and is the penalty coefficient ( in subsequent simulation of this paper). if there are no constraint violations in the given variable and if a constraint is violated for a given variable. The penalty function will be added to the objective function, and the infeasible particles have a lower chance to be selected.
5.3. Solution Framework of the Dispatch Problem
The outline of the multiobjective PSOTLBO to solve the robust dayahead EED problem is presented as follows.
Step 1. Choose the appropriate parameters of the multiobjective PSOTLBO, such as the population size, the maximum iteration numbers, inertia weight, and social coefficient.
Step 2. Initialize all particles in the swarm randomly and ensure that the positions of the particles are in the problem space.
Step 3. Obtain the REG prediction data during the 24 hours of the next day. Obtain the robust dayahead EED model from Section 4. Update the particle positions according to the multiobjective PSOTLBO from Section 5.1. Compute the fitness () (objective functions) of all particles in the swarm.
Step 4. Judge the constraints of the control and state variables. If constraints are violated, handle the constraints with the references from Section 5.2.
Step 5. Iterative process continues if the maximum iteration is not reached. Otherwise, output the optimal Pareto solutions.
6. Case Study
The six generators bus system is used for the case study. The algorithm parameters are listed in Table 1. The detailed generator limit and the economic and emission coefficients [27] are shown in Table 2. Generator G1 is assumed to be a slack generator. A wind farm is connected and its prediction power production for the 24 hours next day is shown in Figure 2. The prediction load information in the next 24 hours is illustrated in Figure 3. It is assumed that load prediction is accurate. The storage is incorporated with a capacity of 200 MWh and maximum charge/discharge power is 80 MW. The charge and discharge efficiencies of ESS are 92%. It is assumed that the initial SOC is 0.5 before dayahead dispatch, and the limit of the SOC is .
To verify the efficiency of the robust EED optimization in the dispatch problem with ES, three cases are tested.
Case 1. The deterministic EED with ES test without WP prediction error is considered.
Case 2. The robust EED with ES test with different WP prediction errors is considered.
Case 3. Repeat Cases 1 and 2 without ES, respectively.
6.1. Simulation Results of Cases 1 and 2
For comparison, the Pareto fronts obtained from the deterministic EED without WP prediction error in Case 1 and with WP prediction errors of 10%, 15%, and 20% in Case 2, both of which use the PSOTLBO, are shown in Figure 4. We can see from Figure 4 that there is a shift in the Pareto front in Case 2 from the Pareto front in Case 1 without the WP prediction error. This means that the acquisition of the robustness is at the expense of a greater generation cost and more emissions. It is clear that as the WP prediction error increases, the gap between the robust Pareto fronts in Case 2 and original front in Case 1 enlarges. This reveals that more accurate WP prediction techniques result in lower costs and emissions.
The aforementioned results can be understood as follows. In Case 1, the output of the generator and ES is controlled to satisfy the power consumption and to maintain the system reliability while ensuring that all equations and inequation constraints are satisfied. In the robust EED with WP prediction in Case 2, the methods attempt to find the best Pareto fronts under the worstcase problem, for example, to minimize the maximum objective values in (21). In this situation, the generators and ES dispatch have to remarkably change their output when the WP shows dramatic changes. Compared with the deterministic EED in Case 1, the robust EED has to schedule with more costs and emissions when dealing with the uncertainty of the WP.
6.2. Robustness Analysis
To obtain the extreme and stochastic scenarios in uncertainty sets, we use the LHS method to generate 1,000 variables in each time slot. Figures 5(a), 5(b), and 5(c) represent the boxplots of the WP values at each hour with errors of 10%, 15%, and 20%, respectively. Fifty deterministic solutions (DS) in Case 1 and fifty robust solutions (RS) in Case 2 (WP error of 10%) shown in Figure 4 are used for the robustness analysis. As generator G1 is assumed to be a slack generator, its output is determined by other decision variables and WP output. The adaptive adjustment of the slack generator allows the dispatch solutions to adapt to other WP scenarios. Twenty new stochastic scenarios from the boxplots with a 10% WP error in Figure 5(a) are selected for the case study. Figure 6 shows the Boxplot for function values of DS and RS under worse case on twenty 24hour WP stochastic scenarios. It is clearly that most and values of DS are larger than the penalty coefficient (15). However, the and values of the Pareto front in Figure 4 with a 10% WP error are smaller than the penalty coefficient (15). This means that most DS are not all feasible simultaneously for twenty 24hour WP stochastic scenarios and punished by the penalty when dealing with the dispatch problem with a 10% WP error. Most of RS in Figure 6 are feasible and less sensitive to WP changes. It is proved that RS have a better robustness to uncertain WP.
(a)
(b)
(c)
(a)
(b)
6.3. Generators and ES Dispatch Analysis
The generators and ES dispatch of the compromise solutions [11] in Case 1 and in the scenario of Case 2 (10% WP error) are given as the examples in Tables 3 and 4, respectively. All generator outputs and ES at each time period properly satisfy the output limits. The functional values are smaller than the penalty coefficient (15), which reveals that no constraints are violated. This means that the constraints have been properly handled.
The results of the ES dispatch analyses are reported in this part. The compromise solutions from the PSOTLBO in Tables 3 and 4 are listed for the ES SOC analysis. Figure 7 reports the daily SOC profiles during the entire dispatch process. Under the SOC profiles with relevant ES dispatch patterns in Figure 7, ES helps to successfully ensure that all system operation constraints are satisfied during all intervals and to improve the economic and environmental benefits. As foreseeable in Figure 7, the SOC varies within the admissible range between 0.2 and 1. The ES works in charging mode followed by discharging mode and is then charged to the same SOC level as the initial SOC. The ES nearly follows this same pattern each day. The ES has an average of one complete chargedischarge cycle every day. Therefore, the ES constraints in each period, and over the entire dispatch period, are properly satisfied.
(a) Without WP error
(b) With 10% WP error
6.4. ES Benefits Analysis
ES in this paper is used for simultaneously improving flexibility of energy dispatch and peak shaving. To test the benefits of ES, the deterministic dispatch without WP error and the robust dispatch with a 15% WP error but without ES in Case 3 are shown in this part. The Pareto solutions of EED problem with ES and without ES are compared in Figures 8(a) and 8(b), respectively. It is observed that the Pareto solutions without ES are dominated by the Pareto solutions with ES. The results reveal the test without ES performance worse in minimizing and compared with that with ES in both cases. This is because the dispatch problem becomes more flexible and generators with lower cost and emission can be better utilized when ES is incorporated. Also, peak shaving characteristics of compromise dispatch solutions in Cases 1 and 2 are shown in Table 5 with respect to the distance ratio between peak to valley in (26). It is clear in Table 5 that the demands at the peak hours decrease and shift to the valley periods when ES is incorporated.
(a) Without WP error
(b) With 15% WP error
6.5. Comparison of PSOTLBO with TVMOPSO
To investigate the performance of the PSOTLBO algorithm in the multiobjective, dayahead dispatch problem, comparison simulations for Cases 1 and 2 are studied. One of the most popular and effective multiobjective PSO algorithms, the TVMOPSO [42], is used for the comparison. The typical Pareto fronts using two algorithms on the aforementioned cases are shown in Figure 9. We observe from the subfigures in Figure 9 that most of the solutions obtained with the TVMOPSO are dominated by those achieved with the PSOTLBO. It is observed that the TVMOPSO performs worse when finding the whole Pareto front. The multiobjective PSOTLBO has been found to have better convergence and better spread solutions compared with the TVMOPSO in both cases. This is because that hybrid PSOTLBO search strategy and CCS strategy in multiobjective PSOTLBO algorithm promote the population diversity. They are helpful in improving the search ability. The performance of multiobjective PSOTLBO has been improved by comparing with single PSO evolution. Hence, the PSOTLBO performs well in solving dayahead EED dispatch problems.
(a) Without WP error in Case 1 and 10% WP error in Case 2
(b) With 15% WP error and with 20% WP error in Case 2
7. Conclusion
The increasing development and penetration of REG are a major challenge for transmission and distribution networks. ES presents the potential to supportively optimize system operations with REG and reduce other excessive participations to grid operation, such as voltage and VAR regulation. In this paper, based on conventional EED experiences, we combine a robust, multiobjective optimization method and an EED problem formulation to solve the dayahead ES dispatch problem with REG uncertainty. The robust optimal dispatch solutions are adaptive to uncertain REG and only one generator (slack generator) production needs to be adjusted. The numerical study explores the different WP prediction errors on the impact of the robust objective function values. The results show that a higher WP prediction error will lead to more economic cost and greater emissions. Moreover, the ES dispatch patterns provide ancillary services, such as peak shaving, to provide economic, emission, and technical benefits. The simulation case results also reveal that the multiobjective PSOTLBO performs well in solving robust EED dispatch problems compared with TVMOPSO.
In future research, the complex dispatch tasks require more objective functions that could be added, such as the REG penetration rate, and carbon emission, especially technical objective functions. The proposed robust generators and ES dispatch method can be verified in real power system with a high penetration of REG, and REG curtailment can be considered. Moreover, the increasing number of ES deployments to active distribution networks will be an increasingly complex problem. These will be dealt with in the future improvements.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 51177177, National “111” Project of China under Grant no. B08036, and the Science and Technology Project of State Grid Corporation of China under Grant no. 5220001600V6 (Key Technology of the Optimal Allocation and Control of Distributed Energy Storage Systems in Energy Internet).