Optimization of energy storage capacity using distributionally robust optimization for converter-based grids

As the penetration of converters into the grid continues to increase, converter-based power sources are replacing synchronous machine-based power sources, leading to the establishment of a converter-based grid (CBG) for regulating grid frequency and voltage. At this stage, energy storage becomes a necessity to support the operation of the CBG. The grid-interfacing inverters are transitioning from the conventional grid-following (GFL) control to the grid-forming (GFM) control. within the context of this research paper, considering the premise that energy storage capacity must meet CBG stability constraints and accommodate the integration of renewable energy sources, a capacity optimization model is constructed based on distributional robust optimization. The primary objective of this model is to minimize the overall cost with the allocation of energy storage capacity as the central focus. The study includes case analyses conducted within an IEEE-14 node CBG. The results of the case analyses confirm that the determined energy storage capacity can ensure the stable operation of the CBG and meet the demands for integrating renewable energy sources.


Introduction
With the development of the power system, the proportion of power electronic devices in the power system has been continuously increasing, and the 'dual-high' characteristic has become a core feature of the new power systems [1] .In this paper, power sources connected to the grid through converters are referred to as converter-based power sources.When the proportion of converter-based power sources exceeds 70%, or even reaches 100%, it is necessary to construct the grid using converter-based power sources.Such a grid, primarily built and controlled by converter-based power sources, can be termed as a converter-based grid (CBG) [2] .
At this point, CBG requires the integration of energy storage devices (ESD) to support grid operations.Currently, there has been extensive research on ESD configuration.Regarding the allocation of ESD capacity, [3] conducted a spectrum analysis of intermittent renewable energy sources using the discrete Fourier transform method and determined the range of ESD capacity configuration in wind power systems based on the results.[4] studied the ESD configuration method when ESD participates in grid frequency regulation.However, the methods proposed in the above studies did not account for the variability in wind and solar power generation; they are deterministic approaches.Building on stochastic programming (SP), [5] proposed an ESD optimization configuration method for wind farms () ( | ) ( ) ln () In the equation: Ω represents the sample space of uncertain variables; ξ is the uncertain variable; f0(ξ) and f(ξ) represent the probability density functions of the reference distribution and the actual distribution, respectively.To control the degree of deviation between the reference distribution and the actual distribution of renewable energy output, the uncertainty set G is established based on the KLD as follows: In the equation: dKL is a parameter established to ensure the similarity between two distributions; the smaller the dKL value, the closer the distributions contained in G are to the reference distribution P0.According to [9], dKL can be calculated using (3).
In the equation, χ 2 a* (N-1) represents the upper quantile of the chi-square distribution with (N-1) degrees of freedom, and it ensures that the true distribution is enveloped within the uncertainty set G with a probability of no less than α * .The value of α * determines the likelihood of the actual distribution being included in the uncertainty set.

The objective function
The objective function is to minimize the comprehensive cost of configuring ESD in the CBG, which includes the annual average investment cost of ESD and the annual operational cost of the grid.

ESS
res, min ( ) In the equation: D represents the number of typical day categories, Td is the number of days in the dth category of typical days; CESS is the annual average investment cost of configuring the ESD; Md is the operational cost of GFM energy storage for the d-th category of typical days; Cres,d represents the penalty cost for wind and solar curtailment for the d-th category of typical days.Additionally, CESS, Md, and Cres,d can also be expressed as: In the equation: Pmax and Emax represent the maximum charge and discharge power and the maximum capacity of the ESD, respectively; ηp and ηs represent the prices for configuring unit power and unit capacity of the ESD; T is the expected service life of the ESD; Pabs,t and Pdis,t represent the charging and discharging power of the GFM ESD, respectively; ηad is the cost coefficient for unit power of ESD operation and maintenance; Pw,t,max and Pw,t represent the maximum power generation capacity and actual grid-connected power of the wind farm; Ps,t,max and Ps,t represent the maximum power generation capacity and actual grid-connected power of the photovoltaic power station; K1 and K2 represent the penalty prices for unit wind curtailment and unit solar curtailment, respectively.(13) In the equation: soct represents the state of charge(SOC) of the ESD device at time t; ηdis and ηabs represent the discharging efficiencies and charging efficiencies of the ESD, respectively; E0 represents the initial battery capacity of the ESD; vESS,min and vESS,max represent the lower and upper threshold values for the SOC of the ESD; λ is the initial ESD capacity as a percentage of the total capacity.

Power flow constraints:
In order to linearize the constraint, this paper uses the following formula to represent the AC power flow. ] ] Where: n is the number of nodes; θij represents the phase angle difference between nodes i and j; Pi and Qi represent the active and reactive power injected at node i, respectively; vi represents the voltage magnitude of node i; Gij and Bij represent the elements in the nodal admittance matrix.

Wind and solar abandonment rate constraints:
Considering the uncertainty of wind and solar output and the demands for the integration of renewable energy, robust chance constraints are employed to impose constraints on wind curtailment rates and solar curtailment rates.
In the equations: Apv,curt and Aw,curt represent the annual solar abandonment rate and wind abandonment rate, respectively; Apv,curt,lim and Aw,curt,lim represent the upper threshold limits for solar abandonment rate and wind abandonment rate, respectively; G1 and G2 represent the uncertainty sets for wind power output and photovoltaic power output; (18) and ( 19) respectively indicate that even under the worst-case distributions within the uncertainty sets G1 and G2, the probability that the wind curtailment rate and solar curtailment rate meet the restriction requirements is greater than 1-a1 and 1-a2.

Solving the Model
The optimization model that have constructed obtained the text contains robust chance constraints, which are challenging to directly solve.By constructing uncertainty sets using KLD, the robust chance constraints can be transformed equivalently into traditional chance constraints [10] .Taking Equation (18) as an example, its equivalent form is as follows: In the equation: w a represents the probability correction value; φ is the introduced auxiliary variable.However, (20) remains a non-convex constraint, which can be handled using the following approach: Extract q scenarios of random variables: ξ1, ξ2, ..., ξq, corresponding to probabilities π1, π2, ..., πq, and define 0-1 auxiliary variables zw (1), zw(2), ..., zw(q).For scenario ξk, setting the values of zw(k) as 1 and 0 represents wind curtailment rate meeting the specified standard and exceeding the given limit, respectively.In this case, the probability of meeting the wind curtailment rate in this scenario is πkzw(k).Therefore, equation (20) can be equivalently represented as: In the equation: ϕ is a positive constant.The equivalent transformation process for (19) is similar.As a result, the ESD capacity optimization model has been converted into a linear model, which can be established and solved using Yalmip in MATLAB along with the commercial solver Cplex.

Case study
This paper modifies the IEEE-14 node system to create a CBG for the purpose of case validation ，as shown in Figure .1.In this setup, Node 1 is equipped with a GFM converter connected to the VSC-HVDC, transmitting a fixed power of 34MW.Nodes 2 and 3 are transformed into an 80MW wind farm and a 40MW photovoltaic power station, respectively, integrated with the grid using GFL converters.Since the load is primarily concentrated near Node 9, this paper selects Node 9 for the installation of the ESD.Utilizing the k-means clustering method [11] , output curves for wind and photovoltaic power during four typical days in spring, summer, autumn, and winter are generated based on historical data, as shown in Figure .2 These 96 data points represent the annual output data.Typical day load data and renewable energy output.The uncertainty considered in this study is generated from the output of typical days, assuming that the errors in wind and photovoltaic power output relative to the typical day's output follow a normal distribution with a mean of 0 [12] .It is assumed that the standard deviation of the errors in wind and photovoltaic power output is 20% of the reference value.Using the Monte Carlo method, a total of 2000 scenarios are generated, and the probability distance scenario reduction method is applied to reduce the 2000 scenarios to 100 scenarios.In the robust chance constraints, both 1-α1 and 1-α2 are set to 95%.A confidence level of α * =0.95 is chosen, and dKL=0.0124,which yields w a =0.0228.Other parameter settings for this study are provided in Table 1.

Analysis of case study validation results
The case study validation was conducted using the reduced set of 100 scenarios as mentioned earlier.
The optimization results were compared with those obtained from the random optimization model and the robust optimization model, as shown in the table.The results indicate that the RO yields the most conservative optimization results, leading to higher costs.On the other hand, SP provides the most economical results but with slightly reduced reliability.DRO offers optimization results that are more economical compared to RO while also achieving improved reliability compared to SP.Based on Table 2, the energy storage capacity is set to 151.294 MW. 2000 scenarios were generated and reduced to 100 scenarios.In these 100 scenarios, the optimization results of the energy storage capacity optimization model constructed in this study were tested to determine whether they meet the requirements for accommodating renewable energy.The results, as shown in Figure .3, indicate that in the majority of the 100 scenarios, wind and solar curtailment rates are within the 5% threshold.There are three scenarios where the wind curtailment rate exceeds 5%, but the combined probability of these three scenarios is 0.019, which is less than 0.0228.

Conclusions
This paper is aimed at CBGs primarily dominated and controlled by converter-based power sources.Based on distributionally robust optimization, an energy storage capacity optimization model for CBGs was constructed.The results of the case validation indicate that the optimization results of the energy storage capacity model built in this paper can meet the stability requirements of CBGs and accommodate the energy needs of the majority of scenarios involving renewable energy integration.

Figure 2 .
Figure 2. Typical day load data and renewable energy output.The uncertainty considered in this study is generated from the output of typical days, assuming that the errors in wind and photovoltaic power output relative to the typical day's output follow a normal distribution with a mean of 0[12] .It is assumed that the standard deviation of the errors in wind and photovoltaic power output is 20% of the reference value.Using the Monte Carlo method, a total of 2000 scenarios are generated, and the probability distance scenario reduction method is applied to reduce the 2000 scenarios to 100 scenarios.In the robust chance constraints, both 1-α1 and 1-α2 are set to 95%.A confidence level of α * =0.95 is chosen, and dKL=0.0124,which yields w a =0.0228.Other parameter settings for this study are provided in Table1.

Figure 3 .
Figure 3. Wind curtailment rate and solar curtailment rate.
Pload,t represents the load power; Ploss,t represents line losses; PGFM,t represents the power output through the GFM inverter for all time period t 3.2.2Energy storage charge and discharge rate constraints: Pdis,t and Pabs,t represent the discharge and charge power of the ESD, respectively; Udis,t and Uabs,t are binary variables, representing the discharge and charge states of the ESD, respectively.

Table 1 .
Configuration of other parameters.

Table 2 .
Optimization results of energy storage capacity for different optimization models.