A load-side resource aggregation method based on Minkowski Sum

In the context of high penetration of renewable energy in new power systems, load-side resources have good regulation potential to improve system flexibility. In order to fully exploit the regulation potential of load-side resources, the power regulation range and regulation cost of load-side resources need to be aggregated first. Based on the establishment of a unified model of the scheduling feasible region of the load-side resources, the computation of the Minkowski sum of the feasible region of the load-side resources is realized through the linear superposition of the maximally inner-approximated feasible region. The decomposition scheme of the instructions of the aggregator and the computation method of the equivalent cost function of the aggregator are proposed. The effectiveness of the proposed aggregation method is verified by arithmetic simulation.


Introduction
Under the dual carbon target, as the penetration of renewable energy in the new power system increases, the regulation capacity of traditional resources on the generation side is decreasing.The system has been continuously experiencing situations such as heavy-load operation and restriction of load power use.At the same time, the problem of wind and light abandonment caused by excess renewable energy output can also bring serious economic losses [1].Flexible load-side regulation resources involved in scheduling can significantly improve the flexible regulation capability of the power grid, ensure the safe and stable operation of the power system, and promote the consumption of renewable energy.However, these resources are small in capacity and large in number, and the operability of participating in grid regulation as individuals is low [2].In order to fully exploit the scheduling potential of load-side resources, it is necessary to first conduct large-scale aggregation, followed by efficient and unified deployment of the aggregates on this basis [3].
The aggregation of load-side resources is a difficult task as the operating parameters of load-side resources are different from each other, and the variables are coupled in time.Commonly used resource aggregation methods [2] mainly include direct aggregation method, geometric calculation method, Monte Carlo simulation method, and optimization method.The direct aggregation method is suitable for aggregating convex response models characterized by the same shape [4], but not for aggregating heterogeneous resources.The Monte Carlo method simulates the flexibility boundary after resource aggregation by generating a large number of samples [5], which is computationally expensive.The optimization method finds the quantitative value of the flexibility boundary by solving optimization problems [6].

Modeling of scheduling feasible regions for load-side resources
The adjustable range of load-side resources is constrained by output, regulation rate, etc. Taking active power as a variable, a convex polyhedron consisting of linear inequality constraints such as output and regulation rate can be obtained, which is the scheduling feasible region of load-side resources.Moreover, due to the existence of constraints such as regulation rate and energy, the upper and lower power limits of the load-side resources at a certain moment are not only related to their own parameters but also to the power state at the previous moment [9].Thus, the scheduling feasible region of the load-side resources is a high-dimensional convex polyhedron that considers multiple time dimensions.

Operation model of load-side resources
This paper considers four types of adjustable loads: electric vehicles, temperature-controlled loads, industrial loads, and residential loads.These loads are small in capacity and large in number, with great potential for regulation.They can provide auxiliary services such as peak shifting and renewable energy consumption.The operation model of load resources can be expressed in geometrical form to obtain the scheduling feasible region.

Electric vehicles.
Electric vehicles consider three states: charging, supplying power to the grid, and driving off-grid.Their operation models consider upper and lower power limit constraints and state of charge constraints [10]: ( ) is the minimum demand value of the state of charge of the battery when the electric vehicle leaves the charging station; EV i  is the charging and discharging coefficients of the electric vehicle; EV,run i P is the power of the electric vehicle when it is traveling.2.1.2.Temperature-controlled load.Temperature-controlled loads consider air conditioning loads with a certain elastic space.The user's perception of temperature has a certain ambiguity, and changing the temperature within a certain range will not affect the user's comfort experience [11].The operation model of temperature-controlled load considers power upper and lower limit constraints, regulation rate constraints, and indoor temperature constraints [12]: ( ) ( ) ( ) where ( )

( )
T i w t is the ambient temperature in time period t ; T i  is the influence factor of ambient temperature.With reference to [13], the indoor temperature can be expressed as: where T R is the thermal resistance of the house; air C is the specific heat of air.
Therefore, based on Equation (8), we have are the upward and downward regulation rate constraints of the load power, respectively.
The regulation compensation cost of the above 4 types of adjustable loads can be unified as in Equation ( 12), considering two cases of load increase and load decrease and taking the absolute value of the change amount of the load. where represents electric vehicles, temperature-controlled loads, industrial loads, and residential loads; ( ) Pt is the power of the load in time period t ; ( ) ( ) i b is the compensation price per unit of regulation paid by the operator to the load; ( ) is the power of the load in time period t when it does not participate in the regulation.

Scheduling feasible regions for load-side resources
The operation models of the above types of resources can be unified into the following universal multi-cycle operation model: Based on the above operation model, the scheduling feasible region for load-side resources can be obtained.Taking the active power as a variable, the feasible region is a convex polyhedron characterized by a set of linear inequality constraints, denoted by i Ω :

Aggregation model for load-side resources
In order to reduce the complexity of the operator's decision-making when scheduling massive heterogeneous load-side resources, it is necessary to first aggregate the scheduling feasible regions and scheduling costs of load-side resources.

Aggregation of scheduling feasible regions for load-side resources
The Minkowski sum refers to solving the expansion set of multiple point sets in Euclidean spaces [14], as defined in Equation ( 15) [15].Convex polyhedra are closed under Minkowski addition [16]; that is, the Minkowski sum of a convex polyhedron remains convex.The Minkowski sum of two polyhedra can be geometrically described as the union of the region swept by polyhedron 1 S along the edge of polyhedron 2 S and polyhedron 2 S .In other words, each point within the aggregated polyhedron can be obtained by taking a point from each individual polyhedron to be aggregated and summing them.The power-feasible region of a single resource has been represented in the form of a convex polyhedron in the previous section.The aggregation of the feasible regions of multiple load-side resources is the Minkowski summation of the polyhedra.The corresponding power value of each point in the resulting aggregated feasible region can be obtained by taking a certain power in the feasible region of each of the individual resources to be aggregated and summing them.In other words, the mathematical essence of the feasible region aggregation problem can be viewed as solving the Minkowski sum of the feasible region, and the resulting aggregated feasible region AGG k Ω is:

 
where AGG k Ω is the scheduling feasible region of aggregate k ;  is defined as the computational symbol of Minkowski sum; AGG Θ k is the set consisting of massive heterogeneous resources.Tiwary [17] proved that when solving the Minkowski sum of two polyhedra (solving for all vertices or all edges), the problem is an NP-hard problem if both polyhedra are described by halfspace; or if one is a polyhedron described by vertices and the other is a polyhedral cone described by half-space.The feasible region established in this paper is the convex polyhedron described by halfspace.It needs to consider the aggregation of the feasible regions of the massive load-side resources in 24 time periods of a day.Thus, the Minkowski sum of massive high-dimensional polyhedra is difficult to be solved accurately.
In order to reduce the computational complexity, this paper adopts the approach in [12], where the feasible region of a single user is inscribed as the maximal inner-approximation feasible region generated by the homogeneous polyhedra through the translation and scaling of the basic homogeneous polyhedra.This in turn realizes the minkowski sum of the scheduling feasible regions of multiple types of load-side resources.The main improvement made in this paper is based on [12].Based on the regulation characteristics of the adjustable loads, the upward and downward potentials of the adjustable loads are defined in the following text.In addtion, the decomposition scheme of the aggregator instructions and the calculation of the equivalent cost function is proposed.The basic isomorphic polyhedron is shown in Equation (17), and the translated and scaled polyhedron is shown in Equation (18): where 0 k Ω is the feasible region of the basic isomorphic polyhedron; i Ω is the polyhedron formed by the basic isomorphic polyhedron after translation and scaling; N are the coefficient matrices of the isomorphic polyhedron, which are taken to be the median of the coefficient matrices of all the resources in the set, respectively.
The maximum inner approximation feasible region problem for a single resource is shown in Equation (19).The optimization objective is to maximize the scaling coefficients of the isomorphic polyhedra to obtain the maximum feasible region volume.
The above equation can be transformed into a linear programming problem, as shown in Equation (20).It can be solved using a commercial solver to obtain the translation coefficients and scaling coefficients.The specific derivation process can be referred to in [12]. where , and dim L is the dimension of matrix i N .Since the obtained approximate feasible regions exhibit similar morphology, the solution of their Minkowski sums only requires arithmetic addition.This greatly simplifies the computation.The feasible region of the aggregator is shown in Equation ( 21).The translation and scaling coefficients are the sum of the coefficients of the maximal inner approximation feasible regions of all load-side resources in the aggregator:

Aggregation of regulation costs for load-side resources 3.2.1. Upward and downward regulation potential of adjustable loads.
The difference between load scheduling and generator scheduling is that the generator only needs to ensure that the power is within the feasible region; however, the regulation capability of the adjustable load is not only related to the feasible region of the power regulation but also related to the original load value in the case of nonparticipation in the scheduling.Therefore, this section will define the upward and downward regulation potential of the adjustable load based on the feasible region and the original load value.
Based on the scheduling feasible region of an individual resource, the regulation range of an individual resource at each moment can be calculated separately: where ( )

D
is the set of power values of a single load-side resource in time period t constrained by the scheduling feasible region, which is the regulation range of resource i in time period t .Based on the regulation range and the original power value of the load before participating in regulation, the upward and downward potential of a single resource at each moment can be obtained:

Disaggregation of scheduling instructions.
Before aggregating the regulation cost, the overall regulation instruction of the aggregator is first decomposed to determine the output plan of each user.Considering that the operating parameters, such as load regulation range, are reported by users, the size of regulation potential, to a certain extent, reflects users' willingness to regulate.Therefore, this paper chooses the following decomposition idea.Based on the calculation result of load regulation potential, the regulation instruction is allocated within the aggregator according to the size of regulation potential of each load resource in equal proportion.The allocation equation of regulation instructions is as follows: where ( ) ( ) are the power upward and downward orders assigned to resource i in time period t , respectively; ( ) is the operator's overall scheduling order for aggregate k in time period t , which is the day-ahead power plan of aggregate k ; ( ) is the overall power of aggregate k in time period t in the case of non-participation in regulation.

Equivalent cost function of aggregator.
Based on the decomposition results of the scheduling commands, the day-ahead scheduled power of a single load resource in the case of participation in regulation can be obtained in Equation ( 27), which is categorized into two cases of upward and downward adjustment.Combining the scheduling costs of individual resources shown in Equation ( 12), the equivalent regulation cost function of aggregator k for both power up-regulation and down-regulation is: ( ) is the equivalent regulation cost of aggregator k in time period t ; ( ) bt are the cost coefficients of load aggregator k in time period t for power upward and downward adjustments, respectively, which are calculated as:

Analysis of aggregation results for feasible regions
In order to visualize the process of feasible region aggregation, the superposition of scheduling feasible regions of 2 industrial load users in 2 time periods is first taken as an example.The feasible regions are two-dimensional polygons.Table 1 gives the values of the relevant operating parameters of the 2 industrial loads. = , respectively.Figure 1(d) shows the aggregated feasible region of 2 industrial loads.As shown in Figure 1, the feasible region polygons have irregular shapes due to the different parameters of the 2 industrial loads.By using the basic isomorphic polygon to approximate them, the approximate feasible regions can be made to show similar morphology.Then, the Minkowski sum of the feasible regions can be realized by adding the translation and scaling coefficients respectively.Although the aggregation results in sacrificing part of the feasible regions, it can make the computation much less difficult.To illustrate the feasible region aggregation effect of the method adopted in this paper in the case of multiple time periods and massive heterogeneous resources, the following scenario is considered.It is assumed that the aggregated objects are 500 electric vehicle loads, 500 temperature-controlled loads, 100 industrial loads, and 1, 000 residential loads, respectively.The system scheduling period is 24 h, and the unit scheduling duration is 1 h.
The Monte Carlo method is used to randomly generate the relevant parameters of each type of load-side resource.The probability distribution of the parameter values is shown in Table 2.In addition, the upper and lower limits of the state of charge of EV loads and the minimum demand value of the state of charge when leaving the charging station are referred to [10].The settings of travel and charging/discharging periods are referred to [18].The outdoor temperature corresponding to the temperature-controlled loads is referred to [19].The upper limit of the ideal indoor temperature is 26°C, and the lower limit is 22°C.In the case of 4 types of adjustable loads not participating in scheduling, the original values of each load in each time slot are randomly generated using the Monte Carlo method.The generated values need to be within the feasible region.( ) The above four types of adjustable loads are aggregated.The average scaling factor Φ k for each type of resource after feasible region aggregation is calculated as an evaluation metric.The calculation is as follows: It is calculated that the average time consumed for 24-point data aggregation of every 2 load-side resources is 11.13 s.The average scaling factor of each type of load-side resource is shown in Figure 2. The feasible regions of industrial loads and residential loads have fewer constraints, and the average scaling factor is greater than 90%, indicating a good approximation effect.The average scaling factor of electric vehicles is about 86.7%.The constraints of the feasible region for temperature-controlled loads are more complicated.The loss of the feasible region after approximation is relatively high, with an average scaling factor of about 74.3%.

Analysis of aggregation results for regulating costs
Based on the aggregation results for the load resources corresponding to Table 2, the upward and downward regulation potentials of the 4 types of adjustable load aggregators can be calculated.Figure 3 gives the overall regulation potential of the 4 types of load aggregators from 1:00 to 6:00, where positive numbers represent upward, and negative numbers represent downward regulation potential.It is assumed that the compensation price for residential load participation in regulation is uniformly distributed in the interval of   0 2 0 5 kW ., .￥ . The regulation cost coefficients of 1000 residential loads are randomly generated.The regulation cost coefficients for each time period of the residential load aggregator are calculated as shown in Figure 4. aggregators at different time periods.

Conclusion
Based on the background that the penetration rate of renewable energy in new power systems is increasing and the regulation capability of traditional generation-side resources is decreasing, this paper efficiently aggregates the scheduling feasible regions of high-dimensional and massive load-side resources.It proposes a decomposition scheme for the aggregator instructions and a calculation method for the equivalent cost function.It enables the load-side resources to participate in the scheduling in the form of aggregation.The feasibility of the aggregation method is verified through example simulation, which can fully exploit the regulation potential of load-side resources and effectively expand the regulation resources of the grid.The identification and shaping correction of dynamic time-varying feasible regions considering the influence of network topology and the uncertainty of load-side resources is a subsequent research direction.

Ω
where i is the scheduling feasible region of load-side resource i ; is the power vector of resource i ; T is the optimization period; i M and i N are the coefficient matrices.

Figure 1 (
Figure 1(a) shows the feasible region of the basic isomorphic polyhedron.The blue and green polygons in Figure 1(b) and Figure 1(c) are the exact feasible regions of the 2 industrial loads.The gray polygons are the maximal inner approximation feasible regions obtained by translating and scaling the basic isomorphic polyhedron.The translation and scaling coefficients of load 1 are   T

Figure 1 .
Figure 1.Superposition process of feasible regions of 2 industrial loads.

Figure 2 .
Figure 2. Average scaling factor of each type of load-side resource.

Figure 3 .Figure 4 .
Figure 3. Regulatory potential of 4 types Figure 4. Regulation costs of residential load of adjustable load aggregators.aggregators at different time periods.
the power of the temperature-controlled load in time period t ; is the day-ahead scheduled output of load i in time period t .

Table 1 .
Relevant operating parameters of 2 industrial loads.

Table 2 .
Probability distribution relevant parameters of load-side resources.