Abstract
To address the issue of multiobjective control in multienergy systems with diverse operational objectives, a two-stage optimization framework based on expected point tolerance has been proposed in this paper. In the first stage, a single objective function is used for optimization control to obtain the expected point of the multiobjective optimization problem. Then, in the second stage, by defining the allowable deviation between each optimization objective and the expected point, the original multiobjective optimization problem is transformed into a single objective optimization problem solution with tolerance measurement. Finally, in the simulation scene of a multienergy system, it is demonstrated that compared with the optimal results under each single objective method, the proposed method increases power line loss, maximum voltage deviation, new energy consumption, and economy by 2.22, 2.30, 1.02, and 2.45 times, respectively. Compared with the suboptimal results, the proposed method reduces power line loss by 22.26, 1.74, 1.09, and 0.97 times, respectively. Combining the shape of the Pareto frontier, it is demonstrated that the proposed method can comprehensively consider the needs of multiple power optimization objectives for forming a more reasonable and effective system optimization scheduling and also provide a new approach for solving multiobjective optimization problems.
1. Introduction
With the increasing capacity of distributed renewable energy power plants connecting to the grid, the volatility and randomness of renewable energy generation and load consumption pose new challenges to the power balance and stable operation of the grid [1]. Energy storage system (ESS) can effectively accommodate renewable energy generation, improve power quality, and enhance system efficiency. They have been widely applied in modern power systems [2]. However, with the increasing demand for power services and the conflicting optimization objectives of various services, the management and scheduling of renewable energy generation systems and energy storage systems have become increasingly challenging and complex in a multienergy system [3, 4]. The core challenge in further advancing the application of multienergy systems lies in balancing the conflicting demands of various services and establishing a scientific and rational optimization and scheduling strategy. This strategy aims to maximize the fulfillment of diverse optimization requirements and address the multiple objectives effectively.
A significant amount of research has been conducted by domestic and international experts and scholars to address the multiobjective optimization control problem in energy systems, leading to some achievements. For example, in [5], the authors minimize the total generation cost and pollutant emissions of renewable energy systems through two objective functions. This approach enables dynamic economic and emission dispatch control of the system. In [6], a game-theoretic approach is proposed, which utilizes an improved equilibrium coordination algorithm to achieve game optimization for two operational objectives: wind-storage system generation cost and grid voltage support. This study presents an economic dispatch optimization method for energy storage systems.
Based on existing research, there are three main approaches to address multiobjective optimization. The first approach involves transforming multiple objective functions into a single objective for optimization calculations [7, 8]. For example, in [9], the authors add up the operating costs of different energy sources in a microgrid to convert the multiobjective optimization problem into a single-objective optimization problem. The optimized results obtained through this summation approach lie on the Pareto frontier of the multiobjective optimization, thus achieving optimal energy storage scheduling instructions. In [10], a weighted sum method is used to transform the multiobjective optimization model into a single-objective model, and the influence of weights on the operation cost and lifespan of energy storage batteries is analyzed. Similarly, in [11, 12], a weighted sum approach is utilized to unify objectives such as power loss, load fluctuation, and voltage fluctuation in optimal operation strategies for distribution networks with distributed renewable energy sources. In [13], a model predictive control-based scheduling method for user-side energy storage is proposed, optimizing control with the objective of economic efficiency. In [14], a fuzzy control method is employed to dynamically adjust the weight coefficients of multiple objectives, forming an energy management strategy for hybrid oil-electric systems and achieving global optimal control. The advantage of this approach is that it can yield an effective unique solution. However, the transformation method has a significant impact on the results, and it lacks a characterization of the degree to which the global optimal solution for multiple objectives is approximated. In addition, there is insufficient understanding of the interdependencies between different objectives.
The second approach involves decomposing the multiobjective function into single objectives at different stages for optimization calculations [15–17]. Through iterative interactions between these stages, an optimal value is eventually achieved. For instance, in [18, 19], the day-ahead optimal energy scheduling of a multienergy system is determined in the first stage based on deterministic forecasting information. The second stage involves optimizing and adjusting the energy scheduling instructions based on the dispatch instructions from the first stage. In [20], the first stage employs a real-time pricing strategy to determine the optimal ordered power for loads. In the second stage, a stochastic dynamic interaction between users and suppliers is established by minimizing the difference between actual power consumption and ordered power through real-time incentives, creating a two-layer model. However, in multiobjective optimization problems, there is often a complex coupling relationship between multiple objective functions, which poses significant challenges for the multilevel decomposition process. Moreover, this approach generally results in a trade-off among multiple objective functions, lacking an analysis of the importance level between different objective functions [21].
The third approach involves using swarm intelligence optimization algorithms to compute the Pareto solution set, resulting in a series of nondominated suboptimal solutions [22, 23]. Then, the distribution patterns and influences of these nondominated suboptimal solutions are analyzed [24, 25]. Alternatively, the final recommended value can be obtained by evaluating the suboptimal solutions in the solution set. The drawback of this approach is that the solutions in the solution set are often limited and few in number, and the recommended solution is directly influenced by different evaluation methods. The authors of [26–28] utilize the multiple update strategies (MOMUS) and the nondominated sorting genetic algorithm-II (NSGA-II) to perform multiobjective optimization. The technique for order preference by similarity to an ideal solution (TOPSIS) method is then employed to evaluate the quality of nondominated solutions in the Pareto solution set, ultimately obtaining the recommended solution. The authors of [29] employ a potent symphony orchestra search algorithm (SOSA) to solve nonconvex mixed-integer nonlinear master-slave optimization problems. Then, a conservative fuzzy satisfying method is utilized to select the best compromise solution that meets the economic and flexibility requirements of energy storage systems. In [30], a decomposition-based multiobjective evolutionary algorithm (MOEA/D) is proposed to address the balance between investment in energy storage systems and the internal energy autonomy in active microgrid operations. The authors of [31] utilize the NSGA-II to perform multiobjective optimization for a compressed CO2 energy storage (CCES) system, considering both its economic and efficiency aspects. The solution closest to the origin in the objective function space on the Pareto front is selected as the final recommended value. In [32], a modified version of the multiobjective sine-cosine algorithm is proposed. Compared to existing methods such as multiobjective grey wolf optimizer (MOGWO), multiobjective slime mould algorithm (MOSMA), nondominated sorting genetic algorithm-II (NSGA-II), and multiobjective particle swarm optimization (MOPSO), this approach is capable of obtaining a better Pareto front. The comparison of the model and method from some existing literature is shown in Table 1.
To address the limitations of the above-mentioned methods, including the lack of representation for the approximation of global optimal solutions in a multiobjective optimization problem and the local optimal problem of the recommended solution based on a Pareto solution set of finite nondominated solutions, a two-stage optimization framework based on the expected point tolerance has been proposed in this paper. The contributions of this paper are as follows:(1)Unlike the existing literature that uses subjective or objective weighting methods to aggregate multiple objective functions or evaluate the Pareto solution set, the proposed two-stage optimization framework in this paper takes a different approach. In the first stage, the optimal values under each objective function are independently calculated multiple times to obtain the expected points of the multiobjective optimization problem, providing a reference for the measurement of optimal solutions. In the second stage, a measurement function is defined to quantify the permissible deviation between feasible solutions and expected points, enabling the optimal solutions to approach the expected points and achieve superior overall performance.(2)By introducing permissible deviations between feasible solutions and expected points, multiple optimization objective functions are transformed into multiple inequality constraints. Then, the measurement function for permissible deviations is defined to transform the multiobjective optimization problem into a single-objective problem for optimizing the deviations. This approach enables the search for a dominating solution on the Pareto front with the smallest permissible deviation from the expected point, addressing the issue of solution diversity in multiobjective optimization.
The remainder of this paper is organized as follows. In Section 2, a multiobjective optimization problem is analyzed to present the architecture of the proposed method. Then, the mathematical model of a multienergy system with multiple types of objective functions and constraints is established in Section 3. Section 4 analyzes and discusses the optimization performance of the proposed method and the coupling relationships among different objectives. Finally, Section 5 provides the relevant conclusions of this paper and suggests future research directions.
2. Multiobjective Optimization Problems and Solutions
2.1. Multiobjective Optimization Problems
In multiobjective optimization problems, a mathematical model with S objectives can be described as equation (1). In the above equation, X = [, , …, ] represents a vector of optimization variables , where i = 1, 2, …, m, represents the s-th objective function to be minimized and represents the k-th inequality constraint, while represents the l-th equality constraint and and denote the lower and upper bounds of the optimization variable , respectively.
Pareto optimization is a conventional approach to solving the aforementioned multiobjective optimization problems. It involves analyzing the superiority relationships between different solutions to obtain the Pareto-optimal solutions, which are considered nondominated. The relevant definitions regarding Pareto optimality are shown in [32]. However, using Pareto optimization often results in a set of nondominated solutions rather than a single global optimal solution, which can make it challenging to select the best solution. To better illustrate the proposed solution, this paper analyzes the results of Pareto optimality using two optimization objectives as an example, as shown in Figure 1. It can be observed that using Pareto optimization can yield a set of nondominated solutions, forming a convex surface in the multiobjective function space known as the Pareto front. It is worth noting that if this paper independently optimizes each objective, a global optimal point can be obtained in the function space. This global optimal point can be defined as the expected point of the multiobjective optimization problem, as illustrated by point A in the figure. Clearly, in most multiobjective optimization problems, it is not possible to reach this expected point. However, from the figure, it can be seen that the distance between the nondominated solutions on the Pareto front and the expected point is different. This is illustrated by the distances and between points B and C on the Pareto front and the expected point A. Therefore, the problem of multiobjective optimization can be transformed into finding the solution in the function space that has the minimum distance to the expected point. This solves the problem of selecting nondominated solutions on the Pareto front. It is worth mentioning that the term “distance” here is just a measure function that describes the proximity between a point in the function space and the expected point. It is not limited to the Euclidean distance as shown in Figure 1.
2.2. A Two-Stage Optimization Process Based on Expected Point Tolerance
Based on the analysis of Figure 1, this paper proposes a two-stage optimization framework based on the expected point tolerance. The objective of this framework is to minimize the deviation between the recommended solution and the expected point by using a measurement function. The specific implementation process is as follows: Step 1: Under the constraints shown in equation (2), we minimize each objective function to calculate the expected point (min (), min (), …, min ()), as shown in the following equation: Step 2: Based on the single-objective optimization model derived from equation (3), the expected point (min (), min (), …, min ()) is obtained. This paper defines the allowable deviation for a feasible solution on the s-th optimization objective. This allows us to transform multiple optimization objectives into inequality constraints involving the allowable deviation, as shown in the following equation:
Based on equation (4), this paper can transform the multiobjective optimization problem described in equation (2) into an optimization problem involving multiple allowable deviations . The advantages of this transformation include the following: (1) eliminating the biases in optimization results caused by different scales among objective functions, (2) defining a simple and clear measurement function for the allowable deviation , such as a norm, and (3) tailoring the specific value of the allowable deviation according to the specific requirements of practical applications and enabling optimization of the system’s optimal results within a specific allowable deviation range for the s-th optimization objective. Taking two norm as an example for the measurement function of the allowable deviation , the above optimization problem can be defined as shown in the following equation:
3. Mathematical Model of Multienergy System
This paper applies the aforementioned methods to a multienergy system consisting of distributed generation, distributed renewable energy, and energy storage system. From the perspectives of grid performance, renewable energy integration, and system economy, four optimization objective functions have been established, including grid line loss, maximum voltage deviation, renewable energy integration, and system generation cost. The calculation formulas for each objective function are shown in Table 2. It is worth noting that in this paper, when establishing the objective function for renewable energy integration, in order to maintain consistency with the other objectives in terms of seeking the minimum value, the negative of the active power output of renewable energy is defined as the optimization objective . Among them, represents the current flowing from node j to node i at time t, A. represents the voltage at node i at time t, and represents the base voltage of the power network, kV. and represent the resistance and reactance between node i and node j, respectively, Ω. and represent the actual output active and reactive power of the distributed power source at node I at time t, MW and MVar. and represent the actual discharge active power and reactive power output of the battery energy storage system at node i at time t, MW and MVar. and represent the rated capacity of the distributed power source and rated power of the battery energy storage system at node i, MW和MW. and represent the maximum active power output and actual active power output of the new energy generation station at the i-th power network node at time t, MW. The new energy generation station primarily refers to photovoltaic (PV) power plants and wind power plants, which are widely used forms of renewable energy with large installed capacities. , , and represent the generation cost coefficients of the distributed power source, ¥/kWh. and represent the cost coefficients of the battery energy storage system and the renewable energy station, ¥/kWh. The cost per kilowatt-hour of the battery energy storage system is calculated based on its initial investment cost, maintenance cost, electricity cost, and recovery cost, as well as the amount of discharge throughout its lifecycle. This cost calculation is dependent on the parameters specific to the type of battery energy storage system being considered. In this paper, the distflow optimal power flow (OPF) method [33] has been utilized. Also, the energy storage system is applied to lithium iron phosphate batteries. represents the number of nodes in the power network, T represents the duration of a typical day, and max(·) and abs(·) represent functions that calculate the maximum value and absolute value, respectively.
From the definition formulas of multiple objective functions shown in Table 2, it can be observed that (1) the physical dimensions of the multiple objectives established in this paper are different, making it difficult to directly compare and analyze the importance of each objective. (2) Each objective function is only used as one indicator to evaluate the performance of the multienergy system. This means that in the mathematical modeling of the multienergy system, all objective functions are considered equivalent. Furthermore, it should be noted that the four objective functions mentioned above are only listed to illustrate the methods proposed in this paper. It does not mean that the methods presented in this paper are limited to optimizing only these four objective functions.
In addition to the multiple service objectives mentioned earlier, the optimization and operation control mathematical model of a multienergy system also includes several fundamental requirements for the interaction of energy among different energy sources, the power grid, and loads. These requirements are referred to as constraints. This paper establishes the constraints from the perspective of energy balance and basic physical laws. The constraints include the range of the output power of distributed power sources, the power output and state of charge requirements of energy storage batteries, the actual power output capabilities of renewable energy stations, as well as constraints on active power, reactive power, node voltage, and branch current in the power grid. The definition formulas for all the constraints are shown in Table 3. It can be seen that the operational constraints of a multienergy system are numerous and include many equality constraints, inequality constraints, and second-order cone constraints [34]. The optimization variables consist of continuous and discrete 0-1 variables. Therefore, the optimization problem of a multienergy system is a type of mixed-integer multiobjective programming problem with complex constraints.
Among them, , , , , and represent the rated capacity, lower and upper limits of active power output, and lower and upper limits of reactive power output of the distributed power source at node i, respectively. In this paper, = 0, = , = −, and = . (t + Δt) and represent the energy state of the battery energy storage system at node i at time t + Δt and t, respectively, %. Δt represents the time interval in hours, h. , , and represent the discharge power, charge power, and actual active power of the battery energy storage system at node i at time t, respectively, MW. , , and represent the charging efficiency, discharging efficiency, %, and rated capacity of the battery energy storage system at node i, MWh, respectively. and represent the lower and upper limits of the energy state of the battery energy storage system at node i, respectively, %. and represent binary variables indicating the charging and discharging status of the battery energy storage system at node i at time t, with values of 0 or 1. and represent the active power and reactive power demand of the load at node i at time t, MW. and represent the injected active power and reactive power at node i at time t, respectively, MW. and represent the active power and reactive power flowing from node j to node i at time t, respectively, MW. represents the magnitude of the current flow from node j to node i at time t, . and represent the square of the voltage at node i and node j at time t, respectively, kV2. and represent the square of the upper and lower limits of node voltage, respectively. represents the maximum current flow in the branch. and represent the resistance and reactance between node j and node i, respectively, Ω.
4. Case Analysis and Discussion
4.1. Example Parameters
In order to analyze and validate the proposed methods, this paper has considered an IEEE33 distribution network [34] with the integration of two photovoltaic stations (at nodes 6 and 13), two wind power plants (at nodes 22 and 32), one distributed power source (at node 1), and one lithium-ion battery energy storage station (at node 6), as shown in Figure 2. Figure 3 illustrates the power profiles of the load, PV, and wind power generation during a typical day. Also, Figure 4 represents the distribution of active and reactive power demand at each nod in a typical day [34]. The time interval Δt is set to 1 hour, and the duration of the typical day T is 24 hours. The parameters and simulation conditions of the power grid, distributed power sources, PV stations, wind power plants, and lithium-ion battery energy storage station are presented in Table 4. The simulation environment for this case study is MATLAB 2020a running on a Windows 10 system with 32 GB RAM, an Intel Core i7 CPU with a frequency of 2.3 GHz, and 16 cores. To solve the above optimization problem, a mathematical programming solver CPLEX that was developed by IBM has been adopted.
From Figure 2 and Table 4, it can be observed that in the simulation, the PV and wind power generation are dispersedly integrated at four different nodes. The total installed capacity of the renewable energy sources is 11 MW, slightly larger than the rated power of the power sources and the apparent power of the loads. This highlights the challenge of accommodating high penetration levels of renewable energy in the power grid. Particularly, from Figure 3, it can be seen that the load demand is relatively high. During the period from 5 AM to 11 AM and from 4 PM to 12 AM, the output power of the renewable energy sources is insufficient to meet the load demand, and the generators supply the active and reactive power. During the period from 1 AM to 5 AM and from 12 PM to 3 PM, the output power of the renewable energy sources exceeds the load demand, resulting in curtailment of wind and solar power when no energy storage system is installed. In addition, as shown in Figure 4, the load power exhibits two peak periods in the morning and evening, following the users’ consumption pattern. The load at each node has significant differences in active power and reactive power demand, and nodes with high power demand are located far from the generator node, resulting in significant network losses.
4.2. Analysis of System Optimization Control Results
Under the simulation parameters mentioned above, this section compares the proposed methods with the results obtained from single-objective optimization. Table 5 presents the variations in the objective function value under the single-objective optimization in the first stage of the proposed methods. Figures 3 and 4 have illustrated the performance comparison of the multienergy system in terms of power sources, renewable energy, energy storage batteries, and the power grid under different optimization approaches. In Table 5, the reason for being negative is calculated based on Table 2. The larger the absolute value of the objective function value , the better the performance of that objective.
From Table 5, it can be observed that when optimizing with a single objective, only that specific objective is guaranteed to be optimal, while other objectives may perform poorly and fail to achieve a dominant solution across all objectives. For example, when optimizing with the objective function as the single objective, the objective function value is 0.21, at least 10 times larger than the other optimization results. Similarly, when optimizing with the objective function as the single objective, the objective function value is −2.9924, more than twice as large as the other optimization results. Therefore, in the control of multienergy systems, it is necessary to consider the needs of various objectives and develop comprehensive optimization and scheduling strategies. Compared to the results of single-objective optimization, the proposed methods in this paper perform suboptimally in terms of all objective function values except for . By comparing the variations of objective function and , it can be observed that as the objective function value decreases, the objective function value increases. This indicates a significant trade-off between reducing grid voltage fluctuations and increasing system generation costs.
Figure 5 illustrates the differences in the generator system, renewable energy system, energy storage system, and grid voltage of the multienergy system under different optimization objectives. In Figure 5(a), which shows the comparison of active and reactive power output of the generator, it can be observed that under the single optimization objective of , the active and reactive power output of the generator is significantly higher than the results of other optimization objectives. This is mainly because the objective function is designed to reduce grid voltage fluctuations and meet the active and reactive power demand of the grid load, as shown in Figure 5(b). On the other hand, under the single optimization objective of , the generator’s active power output is minimized, and it only provides reactive power to support the grid voltage. This is because represents the system’s generation cost, as shown in formula (9), which is a quadratic function of the actual active power output of the generator. However, in the proposed optimization methods in this paper, the actual power output of the generator lies between these two scenarios, balancing the optimization requirements of objectives and .
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On the other hand, from Figure 5(b), which shows the comparison of actual active power output of the PV and wind power generation systems, it can be observed that under the single optimization objectives of , , and , both PV and wind power generation experience significant and high curtailment, especially for wind power generation. However, under the proposed optimization methods in this paper, the curtailment of renewable energy is significantly reduced. Combined with Figure 5(c), which shows the active power output and energy state of the energy storage battery, it can be seen that between 0 AM and 6 AM, the energy storage system actively charges to absorb the excess wind power generation that exceeds the load demand. Between 6 AM and 11 AM, when the load demand for active power exceeds the maximum output power of renewable energy, the energy storage system discharges to support the load demand. Between 11 AM and 4 PM, due to the higher contribution from PV generation, the renewable energy generation exceeds the load demand, and the energy storage system charges again to prepare for the shortfall in load demand and renewable energy supply resulting from the decrease in PV generation. Figure 5(d) compares the voltage distribution at different nodes of the grid under different optimization objectives. It can be observed that when minimizing grid losses, () is the optimization objective and the grid voltage approaches 1.21, which is the square of the reference voltage of 110%. Under other methods, the grid voltage remains close to the reference value, indicating better control effectiveness.
4.3. Analysis and Discussion
To analyze the impact of different deviation metrics on the optimization results, Table 6 presents the optimization results of the multienergy system under different norm forms. In this table, the infinity norm measures the maximum allowable deviation, the two norm measures the sum of squared deviations, and the one norm measures the sum of absolute deviations. From the comparison results presented in Table 6, it can be observed that under different norms, the variation in objective function values is small, and there is no dominance among the optimization results for different norms. This indicates that the optimization results under different norms are all nondominant solutions of multiobjective optimization, and the optimization results are not sensitive to the norm definition.
In the multienergy system, to reflect the bias towards different optimization objectives in the actual application process, this paper introduces weight coefficients for different optimization objectives in equation (5), as shown in equation (6), where represents the weight coefficient for the allowable deviation of the s-th optimization objective. To further discuss the impact of weight coefficient variations on the optimization results, Table 7 presents the optimization results of the multienergy system under different weight combinations for the four optimization objectives.
From Table 7, it can be observed that under the condition of weight variation for to , as the weight value decreases, the allowable deviation of the objective function decreases. This means that the optimization result is closer to the desired value on the objective. By reducing the weight coefficient of a specific optimization objective, the allowable deviation for that objective can be controlled, thereby achieving a preference for a specific optimization objective. At the same time, there are some differences in the variation patterns of the objective values under the condition of equal weights for the other three objectives. The objective shows the greatest improvement when changes from 7 to 1, while and show the greatest improvement when changes from 0.1 to 0.01. This indicates that different optimization objectives have different sensitivities to the weights, and therefore, it is necessary to conduct further discussions. Figure 6 shows the variation curves of the four objective values under different weight variations for the optimization objectives. It should be noted that the weights of the other three objectives change in equal increments along the x-axis, representing the weight of the optimization objective.
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In Figure 6(a), it can be observed that under the condition of equal values for to , as the weight increases, the objective value shows an exponential growth trend. Before = 0.7, the growth is relatively slow, but after = 0.7, it increases rapidly. On the other hand, the objective value experiences a sudden drop and then remains unchanged, indicating that is not sensitive to the variation of . Objective values and have a plateau period, and after this period, increases while decreases. The increase in implies an increase in the abandonment of renewable energy, which is due to the decrease in weight caused by the increase in , indicating that the optimization objective is given higher priority, thus affecting the integration of renewable energy. Similarly, from Figure 6(b), it can be observed that under the condition of equal values for , , and , as the weight increases, the objective value shows an exponential growth trend. Before = 0.9, the growth is relatively slow, but after = 0.9, it increases rapidly. On the other hand, the objective values , , and gradually decrease and enter a plateau phase. This indicates that , , and are only influenced by within a certain range. In Figure 6(c), under the condition of equal values for , , and , as the weight increases, all the objective values to show rapid changes and then tend to stabilize. Simultaneously, as the abandonment of renewable energy increases, the objective value , which represents voltage fluctuations in the grid, decreases. However, the system network losses and operation costs increase. Nevertheless, the changes in the objective values , , and are very small. This implies that the impact of allowable deviation of the optimization objective on the optimization results can be almost negligible. In Figure 6(d), under the condition of equal values for to , as the weight increases, the objective values and gradually increase. This means that as the system operating costs are relaxed, the abandonment of renewable energy will increase continuously. At the same time, the objective , which represents network losses, decreases rapidly, and the voltage fluctuations in the grid decrease slowly, but the numerical changes are small. This indicates that the system operating costs have a significant impact on network losses within a small range, while their effect on grid voltage is minimal. From the above analysis, it can be concluded that there is a complex coupling relationship among the optimization objectives in the multienergy system. Simply modifying the weight of a single optimization objective may have a minimal impact on the overall optimization results or even cause step changes. Therefore, the coupling relationships and sensitivities among the optimization variables will be one of the important research topics in subsequent multiobjective optimization calculations.
In order to present the optimization results of the proposed method more intuitively, we take the example of a two-objective optimization consisting of objective functions and and analyze the position of the obtained optimization results on the Pareto frontier as shown in Figure 7.
In Figure 7, it can be seen that in the optimization model with objective functions and , the coordinates of the expected point A are (0.0294, 1.158 × 104), and the coordinates of the optimal point B obtained by the proposed method are (0.04528, 2.752 × 104). The optimal point B is located on the Pareto frontier of the two-objective optimization model, indicating that the optimization results are effective. Moreover, from the shape of the Pareto frontier, when the optimization objective increases, the optimization objective first decreases rapidly, then decreases slowly, and finally approaches the corresponding ordinate value of the expected point. This means that under the condition of sacrificing the optimization objective , the optimization objective gains rapidly and then levels off. The optimal point obtained by the proposed method is exactly at the point where the optimization objective starts to decrease rapidly, indicating that the proposed method balances the overall benefits of the two optimization objectives and is superior. Furthermore, the tolerances on and are = 0.5586 and = 1.3756, respectively, and it is evident that is greater than . This means that the multienergy system requires a higher tolerance level for the optimization objective . More specifically, if the tolerance for the optimization objective is reduced and the allowable deviation is reduced, the cost required for the multienergy system to achieve the optimization objective will be higher.
5. Conclusion
To address the problem of multiservice objective optimization in multienergy systems, a two-stage optimization framework and method based on the expected point tolerance are proposed in this paper. Based on the expected point derived from the optimization results of each single objective function, the characteristic of the proposed method is to transform a multiobjective optimization problem into a single optimization objective solution by describing the distance between the nondominated solutions on the Pareto frontier and the expected point. In the IEEE 33-bus simulation example including a multienergy system, the comparative results and analysis conclusions have been as follows. Compared with the optimal results under each single objective method, the proposed method increases power line loss, maximum voltage deviation, new energy consumption, and economy by 2.22, 2.30, 1.02, and 2.45 times, respectively. Compared with the suboptimal results, the proposed method reduces power line loss by 22.26, 1.74, 1.09, and 0.97 times, respectively. The proposed method can effectively balance the contradictions between different optimization objectives and ensure that the optimization results are more reasonable. Under different deviation metrics of the proposed method, the maximum relative errors on each objective function are 9.0%, 1.59%, 1.88%, and 2.93%, respectively. This indicates that the optimization results under different norms are all nondominant solutions of multiobjective optimization, and the optimization results are not sensitive to the norm definition. It means that the impact of different deviation metrics on optimization results is relatively small. Based on the comparison results of tolerance weights for different objective functions, there is a complex coupling relationship between the optimization objectives, and the impact of a tolerance weight on the optimization results is significant. Therefore, the coupling relationship and sensitivity analysis among optimization objectives will be the important topics in future research.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request. Also, Figures 2 and 3 used to support the findings of this study are reproduced from the below literature, [34].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was sponsored by Beijing Nova Program (Z211100002121081) Research start-up fund of North China University of Technology (110051360023XN224-22).