Unified multi‐objective optimization for regional power systems with unequal distribution of renewable energy generation and load

The optimization for large‐scale power systems with unequal renewable energy distribution is an important and urgent task to collaborate operations of the participated sub‐grids. This article proposes a novel method by utilizing the unified multi‐objective optimization (MOO) to integrate diverse strategies to a comprehensive problem. For this aim, individual optimal model is first established to describe the demands of each sub‐grid. The overall objectives are unified in terms of economy costs. This unification integrates evaluate different optimized results without loss of generality. The global objective is the weighted sum of the individual objectives with empirical coefficient. Thus, the internal coupled restrictions and influences among sub‐grids can be solved simultaneously. Finally, by adjusting the corresponding weights according to the preferred requirement, the optimized solution can effectively allocate renewable energy throughout all sub‐grids. Consequently, both individual and global requirements can be met at utmost. The proposed unified MOO is tested on the configured systems based on multiple modified PJM 9‐bus grids. Satisfying the global optimum of the multi‐region joint system, the total system cost increases by 15.1%, the industrial zone cost increases by 21.4%, and the residential load shedding loss cost increases by 27.3%. Although each region has to sacrifice some of its benefits, the compromise operational behavior ensures that the total cost is optimal. Numerical results verify the effectiveness in achieving the promising global optimal solution, and the flexibility in meeting the requirements of different sub‐grids.


INTRODUCTION
Power system has been significantly more complexed with the fast growth of renewable energy integration.7][8] However, in real power system, the situation can be more complex due to the geographically unequal distribution of renewable energy integration and load.Also, a large-scale regional (e.g., province-level) power system can contain several sub-networks that vary obviously in energy composition. 9Shandong Province power network in China is a typical example to illustrate this problem.As can be seen in Figure 1, 10 although there is tremendous wind power natural endowment in the coastal areas, the interior hinterlands still rely on traditional fossil energies due to the fast attenuation of wind speed.Worse still, the major residential and industrial regions are far away from offshore wind farms, leading to a regional unbalance between generation and load.These two problems restrict the consumption of high percentage of wind power generation in large-scale regional power systems with a single dispatch objective. 11raditional studies claimed three major difficulties regarding the above problem of multi-regional operational optimization.Firstly, the cooperative optimization does not satisfy all the requirements from different areas because of the internal competing interests.Reference 12 developed a novel day-ahead scheduling model for prosumers based on a stochastic-robust approach to solve the heterogeneity of the unknown parameters.The study in Reference 13 presented a two-stage stochastic model for customers in terms of an energy hub of electrical and thermal energy storage systems (ESSs).The study in Reference 14 presented a dynamic model to improve the resilience of the distribution network during contingent events.In this model, when an event occurs, the system operator maximizes power supply by changing the network topology as well as utilizing the direct load control (DLC) program.In Reference 15, the authors investigated the impact of DLCs and ESSs on distribution systems based on the technical and economic indicators.In this regard, DLCs include residential, industrial and commercial energy hubs.The results show that the participation of hubs in these programs has led to the release of a lot of flexibility in peak periods.The study in References 16 and 17 proposed complementary strategies to dispatch the optimal energy spatially.However, separate objectives are ignored by the global requirements, leading to the contrary to practical service.Secondly, the criteria of individual objectives are various and cannot be easily merged within the same evaluation system.The study in Reference 18 summarized several mainstream F I G U R E 1 Chinese wind energy resource distribution and wind farms in Shandong province.
assessment criteria of power generation technologies, and proposed a method to reduce curtailments of renewable energy generation.However, these criteria can only be utilized separately.Reference 19 made a comprehensive review on the existing approaches to the optimization of multi-energy systems in mixed-use districts.However, each model only suits the corresponding objective.Thus, the universality and expandability are limited.Last but not the least, the computation efficiency is degraded due to the increasing scale of the integrated objectives, constraints and coupled restrictions.
In 2005, the technique of multi-objective optimization (MOO) was proposed to formulate and solve the power system problems with multiple criterion. 20Since then, MOO has been widely used in power system operation and planning.Reference 21 leveraged MOO with neural network model to improve forecasting stability by considering the internal influences among different electric equipment.Reference 22 proposed a long-term MOO model to integrate solar power system in view of the smoothness of power output and the total annual power generation simultaneously.Despite the expected results, both studies solved MOO with genetic algorithms that ignore the system preferences on each participated sub-grid.In most recent years, the fast-growing deployments of hybrid energy sources, battery storage and power exchanging devices have raised MOO applications and studies to the next level.The study in Reference 23 deeply utilized MOO to obtain the optimal capacity for the complex systems with hybrid energy sources with the promising solution of overall net cost and carbon dioxide emissions.
In addition, a review work of Reference 24 claimed that the core problem of intermittent aspects of renewable energies can be greatly smoothened with MOO in view of various scales, control strategies, and energy managements.Therefore, the problem can be overcomplicated and the computation efficiency is insufficient for short-term dispatch.The representative previous works are summarized in Table 1.
To address the above problems, this article proposes a general MOO model for multi-regional power system with unequal renewable energy integration.The problem is solved by Pareto optimization algorithm.The innovations are summarized as follows.
1.The MOO model is combined with a rolling model to integrate wind farms into a large-scale provincial power system consisting of industrial and residential areas, to establish a multi-timescale active dispatch model for a system containing a large-scale wind power plant, and to evaluate the objective function of each region with an economic indicator.Therefore, it is feasible to accurately consider various objectives from different sub grids.2. Using the weighted-sum MOO method, the three objectives with coupled constraints can be modeled as a new optimization problem, and the global-individual overall optimization is conducted with a unified economic indicator, while the 24 h rolling solution framework is designed to update the decision based on the latest prediction of wind TA B L E 1 A summary of the related literature.

Sub-grid system preferences System
Single Multi power output.Also, besides the original individual and global objectives, the initial weights can be set in a reasonable range to reduce the redundancy adjustment.3. The computation burden of obtaining the Pareto solutions can be further reduced by relaxing the secondary-priority objective.Thus, from the perspective of the entire system, the penetration of renewable energies into the main grid can be guaranteed, and the operation cost can be reduced correspondingly without causing additional violations.
The remainder of this article is organized as follows.Section 2 describes the theory of MOO and how it can be used in solving electric problems.Section 3 presents the detailed optimization modeling of the three participated areas.Section 4 presents the proposed unified MOO and its computation steps.Section 5 discusses study results that verify the efficiency.Section 6 concludes the paper and suggests future works.

Brief introduction of MOO
The real-world power system involves various criteria to ensure reliable and economic operation.MOO is commonly used to obtain the optimal compromised solution rather than the single-objective solution.The generalized mathematical formulation of MOO problem is given by ( 1) and ( 2). 20n where x represents the vector of decision variables.Equation ( 1) consists of m decomposed objective functions, which may conflict with each other.In other words, the expense reduction of a particular objective function is achieved at the higher expense of other objective functions.In(2), g i (x) andh j (x)represent equality and inequality constrains, respectively.The definition of the multi-objective optimality is introduced as follows.
Definition 1. Pareto Superior.The vectoru Therefore, the superior vector has at least one objective better off and no objective worse off than the inferior vector.
Definition 2. Pareto optimal solution.x ∈ X is the Pareto optimal solution only when there is no x ∈ X, whose objective function value F(x) is superior to F(x).
Definition 3. Pareto front.The Pareto front is defined as the set of non-dominated solutions, where each objective is considered as superior solution.As shown in Figure 2, the optimal solution always aligns with the Pareto front.

Solution algorithms of MOO
The process of MOO involves compromising between different objectives.In general, there are three methodologies for power system MOO problems.The first one is the weighted sum algorithm, which integrates different objective functions into one objective through weighting factors, as shown in ( 3)-( 4).
F I G U R E 2 Pareto optimal solution and front.
where Z is the total objective and w i represents weighting factor of the ith objective.This method is straightforward and easy to implement.However, it requires convex formulation and accurate weight parameters selections.Research work 7 set cost and emission as two objective functions.Different objective functions are converted by different weights to obtain the Pareto optimal solutions.However, the weight is empirically determined and require extensive prior experience.
The second method sets one objective as the major goal and assigns boundary value to the other objectives to convert them as constraints.Therefore, the MOO problem is reformulated to a new single-objective optimization problem.However, this method complicates the formation of Pareto front and the boundary value for each objective requires many trial-and-error steps.In summary, it is feasible and efficient to implement the above two methods.However, the reformulation constantly leads to sub-optimality.
The third method directly solves the MOO problem, instead of reformulating multi-objective function to single one.Many algorithms in this category have emerged since 1990, such as MOGA, NPGA, NSGA, PAES, PESA, NSGA-II, and MOMGA. 23In particular, NSGA is the most popular solution in power system MOO problems.It defines the crowding distance to estimate the density to replace the fitness sharing scheme, which simplifies the computation.The elite population selection scheme ensures the optimality, and thus the NSGA-II is efficient and robust.

OPTIMAL MODELLINGS FOR INDIVIDUAL SUB-GRIDS
In this section, we propose the MOO model of multi-regional power system by using the example of Shandong power system, which consists of renewable energy region, industry region and residential region.The operation of three regions is described as follows.

Characteristics of Shandong power system
Shandong is a large province in economy, population and energy consumption.The coastal area is rich in wind energy potential, as shown in Figure 3. 25 Until the end of 2021, the total renewable energy installation reaches 60.99 GW, which constitutes 30% of the total power generation within the province.The total renewable energy generation is 1.109*10 5 GWh. 26However, the spatial distribution of wind power and load are not equal.The whole province power system can be divided into three areas.Firstly, the residential-domain area is capital city Jinan and its surrounding area, where the power generation is less than the load.The major demand is to minimize the power shortage (load shedding).Secondly, the industry-domain area is a major industrial city, Qingdao and its surrounding area.The major demand is to minimize the energy cost for high-consumption industry sites.Thirdly, the coastal wind power area is located in the north part of Shandong semi-island, in which the renewable power generation is much larger than other areas.The major demand is to export as much renewable energy as possible.

F I G U R E 4
Rolling generation plan model for wind-storage co-generation system.

Rolling optimal scheduling model
Power system operational planning relies on accurate forecast of load demand and reliability of generating unit output.
The stochastic fluctuation characteristics of wind power pose significant difficulties in the development of generation plans.According to the characteristic that wind power forecast improves as the time scale decreases, increasing the forecast resolution and revising the generation plan through rolling optimal dispatch is one of the effective means to mitigate the random fluctuation of wind power and improve the capacity of wind power consumption.In this paper, wind farms are included in the power system active scheduling, and a multi-timescale system active scheduling model with large wind farms is established.The wind power scheduling is formulated based on the short-term forecast reported by wind farms on a rolling basis, and the goal is to improve the wind power consumption capacity and system operation economy.As shown in Figure 4, the decision-maker solves the optimal scheduling model at each time step.Meanwhile, only the decision variable of the first step is executed.Then, the above process repeats before the next time step.Therefore, the impact of forecast error on the optimal decision is minimized.

Objective function
The formulation of MOO is designed as follows.The operation cost minimization is three individual goals, given by ( 5)- (7). min min min C w = a where P G n,t is the output of generator n at time t, a n , b n , and c n are the cost coefficient for generator n; ΔP D m,t represents the load shedding of node m at time t, a m , b m , and c m are the cost coefficient of load m.P W k,t is the generation output of wind farm k at time t; T is the time horizon.Objective functions ( 5) and ( 7) represents the generation cost of traditional power plant, the cost of load shedding and the cost of wind power generation.Without losing generality, three cost functions are modeled as quadratic functions, whose coefficients are determined by the features of generators and loads.
Based on the weighted sum method, the multi-objective function can be reformulated as (8). min where

Constraints
Different areas in the power system have different operating characteristics.

Power flow constraint
The power flow constraints are given by ( 9)-( 16), which are applicable to both industry and generation areas.
The nodal power balance constraint is enforced by ( 9) and (10).The linearized power flow equations are given by ( 11)-( 13), which can approximate the AC power flow with high accuracy. 27,28Constraints ( 14) and ( 15) is the line thermal capacity of line (i, j) in real/reactive power.The nodal voltage limit is given by ( 16).The thermal limits of the distribution lines are given by ( 17).This quadratic constraint is equivalent to a circle in the P- | can be fitted by a circumscribed polygon, as shown in Figure 5. Therefore, the nonlinear constraint ( 17) is replaced by three piecewise linear constraints ( 18)-( 20).

Steam generator constraint
The constraints of traditional steam generator are given by ( 21)- (23).
where P Gmax i and P Gmin i represent the upper and lower limits of the generation output for each unit, r is the ratio between the ramping rate limit.Constraint (21) enforces the lower/upper limits of the generator at node i. Constraints ( 22) and ( 23) enforce the generation ramping up and ramping down limits, respectively.

Load shedding constraint
The constraints for residential area are given by (24).
where ΔP D i,t represents the load curtailment at time t, and  is the ratio of load curtailment to the real load demand and its value is determined by the importance of the load.

Renewable energy constraint
The constraints for wind farm area are given by ( 25)- (26).
where P w,t represents the wind generation output at time t, and P max w,t represents the maximum wind generation.0.1 m represents the wind forecast error following normal distribution.P DW i,t represents the load at wind farm, and 0.2 represents the minimal wind generation (i.e., 200 kW).Constraint (26) restrain the maximum generation received by industrial area and residential area from wind farm cannot exceed the maximum capacity of the wind farm.

MODELING AND COMPUTATION FOR UNIFIED MOO
By utilizing the linear-weighted MOO approach, the comprehensive objective function can be established from Equation (8).To enhance the couplings of individual participant, the weights are adjusted as w i f i ∕f best .In which, f i is the individual best solution with the added constraints from the neighboring grids, f best is the best solution without coupling constraints.The obvious advantage of this adjustment is to reflect both individual importance and the mutual influence with a straightforward result compared with the ideal value of 1.The detailed calculation steps are given as follows: Step 1: Data input and initialization.Based on the raw data and topology of the system, initialize line parameters, bus voltage, node injections, etc.
Step 2: Individual optimization.In view of the separate grid constraints and objective functions, calculate the OPF solutions and results for each participated grid.
Step 3: Global optimization.Connect individual grid by adding the coupling constraints between each other, and then initialize the weights with respect to the real power needs of the whole system.Since this research models the practical needs in a practical power system, the weight priorities are ranked from industrial area, residential area, to wind-farm area.It is also noted that the weight of wind-farm area can be comparatively smaller because of its loose constraints.
Step 4: Final solution.A rolling optimization model with multiple time steps is formulated and updated with forecast information for wind power.Solve the generated MOO problem, achieve the global solution that suits the goals of both province's needs and individual benefit.
For better understanding of the calculation steps, the flow chart is depicted as Figure 6.

CASE STUDY
This section presents case studies on the three-area system, in which two areas are IEEE 9-bus system.The computational tasks are performed on a personal computer with an Intel Core i7 Processor (2.60 GHz) and 8-GB RAM, and the code is implemented via Matlab-based IBM ILOG CPLEX Optimization Studio V12.10.0.

System description
As shown in Figure 7, this research takes a composite system which consists of three various five-bus grids to validate the efficiency of proposed MOO application.The industrial Area 1 and residential Area 2 share the similar structure with attached loads and generators.In addition, the transmitted power from Area 3 (wind-farm) is injected into both areas as the complementary virtual generator.From the perspective of industrial and residential goals, the objective of Area 1 is to minimize the generation cost in economy dispatch, and Area 2 aims to minimize the cost of load shedding.Detailed parameters including generators, loads, and wind turbines are listed in Tables 2-4.

Optimization result
As depicted in Figure 8, the standard load curve of residential area has two high-demand peaks during daytime.The early peak lasts from around 6 a.m.till noon, and it has the main contributions from commercial, industrial, and living From Figure 10, it can be seen that when the wind output is sufficient regardless of the tie-line limit, maximizing the wind injection into the industrial area can greatly help lower the overall cost.In this case, generators of the industrial area cooperate with the wind turbine G 6 to minimize the cost.From Figure 11, when the load curve is at low level, all the load demand can be supplied by local generators without leading to load shedding.However, when there is power shortage F I G U R E 9 Generation power output without considering the system coupling constraints.
F I G U R E 10 Generation power output considering the system coupling constraints.

F I G U R E 11
The comparison of before/after load shedding regardless of the system coupling constraints.from 9 a.m. to 9 p.m., wind power can be an external supplement source to minimize the amount of load shedding.Therefore, the loss costs caused by load shedding can be optimized.
For better comparison, Table 5 lists the compared results achieved with and without the system coupling constraints.They can be classified into four scenarios: • Scenario 1: individual optimization for the industrial area to minimize the generation economy cost.
• Scenario 2: individual optimization for the residential area to minimize the load shedding cost.• Scenario 3: individual optimization for the wind area to maximize the possible wind power output.
• Scenario 4: utilize MOO with the built weighted overall system model to achieve both global and individual optimized solutions.

Discussion
From the comparisons of Figures 9-11, it is observed that when the load demand is at low level in the residential area, the coupling constraints will not affect the overall optimization since power balance can be guaranteed by local generators.However, from the time period of 9 a.m. to 9 p.m., the power shortage becomes the major concern since the power supply for residents is the first priority.Meanwhile, upon the potential minimization of load shedding, the lowering generation cost for the industrial area has the second priority.It should be noted that the first and second priorities have internal conflict of interest.Since the wind power output is limited, the generating units in the industrial area increase their generation by 12.13% to reduce the demand for wind power.Consequently, the wind power district supplies 68.3% of its external output to residential areas, which will help minimize load shedding in residential areas.In addition, since there is no direct tie-line between the industrial and residential areas in a practical system, wind-turbine area has to supply both areas simultaneously with the dynamic balance without violating the line limits.
From the given comparisons, the individual optimization for each area can only guarantee each best solution with the consideration of several single objectives.The simulation results suggest that in the case of satisfying the global optimum, the total cost of the system increases by 15.1%, the cost of the industrial area increases by 21.4%, and the cost of the loss of residential load-shedding increases by 27.3%.However, for such a multi-region joint system, although each area should sacrifice partial interests, behavior of compromising operations can ensure the optimal total cost.

CONCLUSION
Aiming at the urgent problem of global optimized operation for the large system consisting of sub areas, this paper proposes an innovative MOO model to ensure that the overall cost can be minimized with the agreements of all participants.The model splits the system of Shandong Province into three typical areas: industrial area, residential area, and wind-farm area.In view of different operating features and objectives, the first contribution is to unify them with standard costing objectives.In this way, each area can be compared and merged in the same judgment by solving all individual optimal problems.Subsequently, by utilizing MOO to build the problem for the overall system, the previous solutions can be used to get the weights.Together with the internal and coupled constraints of power balance, flow thermal limits, and security limits, the final solution can give both the overall minimized cost and power distribution.In addition, the unified equation provides a straightforward result.Compared with traditional methods, the proposed approach in this study can guarantee the interests of both the overall system and each participant.
Based on the validation and achievement of this work, the improved MOO can be seen as a very promising research direction to solve the real-world large system optimization problem.Future work will improve the MOO model with more detailed coupling constraints, and the selection of weights can be better adjusted to include more areas.

F I G U R E 3
Spatial distribution of yearly average wind speed in Shandong, China.

w 1 ,
w 2 , and 1 − w 1 − w 2 are the weighting factors of objective function C gen , C shed , and C w , respectively.C min gen represents the minimal generation cost of objective function C gen .C min shed represents the minimal load shedding cost of objective function C shed .The wind cost is already reflected by C gen , which makes C w C min w = 1

F I G U R E 5
Piece-wise linearization of line thermal constraint.

F I G U R E 6 TA B L E 2 TA B L E 3 4 TA B L E 4 8
MOO Computation flow chart.F I G U R E 7 Simplified system model constructed with three sub-grids.Generator cost coefficients of the industrial area.Load shedding coefficients of the residential area.Parameters of wind turbines.Typical load curve of 24-hour for residential area and industrial area.consumptions.Another peak occurs from 6 p.m. and lasts to 10 p.m., with the major contribution of living consumption.The load curve of industrial area has a different feature compared to the curve of residential area.Despite the same two high demands, the peaks occur in very different periods.It is because industrial consumers prefer utilizing the lower electricity price.Thus, the night load demand remains at a high level.Figures9 and 11give the optimal results for the industrial and residential area without the consideration of the system coupling constraints.In other words, the individual OPF solution.In Figure11the dashed and solid line represent original loads and load shedding, respectively.The simulation results are organized into two parts.The first one solves individual optimal problems separately in view of internal constraints to achieve the solution.And the second one adds system coupling constraints into the comprehensive model to study the influence of wind power on industrial generation economy, load shedding, and the overall cost.
Comparisons of results with/without system coupling constraints.