We illustrate this framework in a particular CSP plant. For this case study our results indicate that, for uncertain heat from the sun, the most beneficial schedule is based upon a medium sun heat forecast, while our second case where the pricing of power sold is uncertain, the most beneficial schedule is a low pricing forecast.

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Which authors of this paper are endorsers? Disable MathJax What is MathJax? Optimization and Control math. or arXiv OC] for this version. Related DOI :. The main contributions are: 1 propose a DRO-based microgrid ESCC model, which can balance the ability of withstanding renewable energy fluctuations and investment and operation economics of the microgrid, and establish adjustable distributionally robust joint chance constraints to measure the renewable utilization capability; 2 establish the appropriate affine decision rules for ESS operation equations, present the transformation of constraints containing distributionally robust probability of random variables with adjustable boundaries, and approximately transform the proposed model to a mixed integer programming problem with second-order cone constraints through variable substitution and duality techniques.

The model can be directly solved by commercial optimization software such as Gurobi; and 3 the numeral experiments are carried out based on the IEEE bus system to verify the validity and advantage of the proposed model, and the impact of ESCC on the renewable energy utilization is analyzed.

The rest of this article is organized as follows: Section 2 establishes the mathematical formulation, Section 3 presents the solution methodology, Section 4 performs the case study, and finally, Section 5 concludes this article.

Microgrid can fully promote the large-scale access of distributed power and renewable energy and achieve highly reliable supply of various energy forms to loads, which is an effective way to realize an active distribution network. Figure 1 is a schematic diagram of the structure of the microgrid, which is composed of thermal units, renewable distributed power sources, loads, and energy storage systems.

In the microgrid, photovoltaics PV , wind turbines WT , and thermal units TU are the main sources to supply power to the load. The energy of the energy storage system ESS flows in both directions, which plays an important role in regulation.

The microgrid and power grid are connected by the point of common coupling PCC. The interactive power between the microgrid and power grid, the charging and discharging of ESS, and the output of WT, PV, and TU are controlled by the microgrid through information flow, which are all transmitted through two-way channels.

Based on such microgrid structure, we build the ESCC model as follows. This study considers the impact of three factors on the decision-making of ESCC in the microgrid, namely, energy storage investment cost, system dispatch cost, and the admission on renewable energy fluctuations.

The goal is to reduce the energy storage investment cost and the system dispatch cost as much as possible, while improves the admissible range of renewable energy. Therefore, the objective function consists of three parts as shown in 1 — 3. In this article, the capability of renewable utilization is represented by the probability that the system can withstand the actual fluctuation of renewable energy.

where C sto represents the investment cost of ESS and C disp represents the dispatch cost of the system. The investment cost of ESS is related to P s max and S s max of ESS.

The dispatch cost C disp includes the operating cost and start—stop cost of thermal units, as well as the cost of power transaction between the microgrid and power grid. is the fuel cost function of thermal unit g , which is generally represented by a quadratic function, and can also be approximated by a piecewise linear function.

x g t , u g t , and v g t are both binary variables, respectively, that represent the on—off state 1 is on and 0 is off , starting state 1 is start and 0 is no-start , and shutdown state 1 is stop and 0 is no-stop of thermal unit g during time period t.

δ is a weight coefficient related to renewable utilization, and its numeral represents the additional economic cost of renewable utilization.

The energy and power of the configured energy storage devices should be positive values, and the constraints of the site- and grid-connected power should be considered. The constraints are as follows.

where P M and S M represent the maximum power capacity and maximum energy capacity of the ESS allowed by the conditions of site- and grid-connected power, respectively.

In the microgrid, the power demand should be balanced by the output of the thermal power units, the power of renewable energy including wind and solar power , the power of charging and discharging of ESS, and the power purchased from and sold to the grid.

The power flow on lines should not exceed the limits. Therefore, for any time period t , the balancing and line power flow constraints of the microgrid are as follows:.

where 6 represents the balancing constraint and 7 represents the line capacity restrictions based on the dc approximation of the power flow equations. The on—off state, start-up state, and shutdown state of thermal units are all 0—1 binary variables, and these three variables should satisfy the following constraints:.

where 8 — 9 represent the minimum start-up time constraint and the minimum shutdown time constraint of thermal units; constraints 10 — 11 represent the logical relationship between the three variables. The constraints of upper and lower limit and the ramp-rate of thermal units are as follows:.

where 15 — 16 represent the capacity limits of ESS; 17 represents the relationship between the amount of energy and power of ESS; 18 represents that the ESS has the same amount of energy at the beginning and end of the dispatch cycle, which ensures the sustainable operation of the ESS, which is beneficial to cyclic dispatching and prolong the service life of ESS.

The microgrid can purchase energy from the power grid when the system output is insufficient, or sell electricity to the power grid when the system output is surplus. The power purchased or sold should not exceed the transmission power of PCC as the following. This article evaluates the capability of renewable admission of microgrid by the distributionally robust probability that the system can withstand the fluctuation of renewable energy.

Let ξ k t represent the actual output deviation of renewable resource k from its forecasting value during time period t. Let [ ξ k t L , ξ k t U ] represent the admissible fluctuation range of ξ k t. That is to say, if ξ t exceeds the range [ ξ t L , ξ t U ] , the system under a certain operating condition will face risks of load shedding or renewable curtailment.

Based on the DRO method, the following constraints can be established. where a is renewable utilization probability, decision variable, not greater than 1 and a 0 represents a lower bound of a.

In addition, we consider an ambiguity set D consisting of probability distributions P that i match the empirical mean μ k t and empirical variance σ k t of each ξ k t and ii ξ k t is unimodal about μ k t , that is,. where μ k t indicates that the probability density function of ξ k t , if exists, is non-decreasing from 0 to μ k t and is non-increasing afterward.

Furthermore, when renewable generation deviation ξ t is within the range [ ξ t L , ξ t U ] , the system must have the corresponding dispatch scheme which satisfies all operating and safety constraints. Thus, constraints for [ ξ t L , ξ t U ] are as follows:. where 23 and 25 represent the balancing constraint and line capacity constraints during actual operation; 24 and 27 are the capacity and ramp-rate constraint of thermal units during actual operation, respectively; 26 is the ramp-rate constraint of thermal units at response time window; 28 — 30 represent the ESS capacity limits and the transmission capacity constraint of the PCC during actual operation; 31 represents the relationship between the amount of energy and the power of the ESS during actual operation; 32 indicates that the upper and lower bounds of renewable energy utilization need to include its predicted generation, and it is limited by the maximum and minimum output of renewable generation.

In a nutshell, the established microgrid ESCC model is shown in constraints 1 — 32 , which is a two-stage DRO model with adjustable chance constraints. By solving such model, the system operator can obtain the ESCC strategy and the corresponding renewable energy utilization capability. The proposed microgrid ESCC model is a two-stage model with distributionally robust chance constraints, and the boundaries of random variables are adjustable, which cannot be solved directly.

In this section, the model is further transformed. In the second stage of the aforementioned model, P g t ξ k t , P s t dis ξ k t , P s t cha ξ k t , P q t buy ξ k t , P q t sell ξ k t , and S s t ξ k t are recourse variables, which are decided by the random variable ξ t.

In order to facilitate the solution of the problem, we first assume the following affine decision rules:.

where B and b are the coefficients of the decision rule to be optimized, which represent the response of the re-dispatch decisions P g t ξ k t , P s t dis ξ k t , P s t cha ξ k t , P q t buy ξ k t , and P q t sell ξ k t to the forecast deviation ξ t.

It should be noted that the assumption of affine decision rules will reduce the search space of solution and obtain conservative approximation.

However, S s t ξ k t is different from the aforementioned variables. If we assume that S t ξ k t is only determined by the deviation ξ t at time t , the result will lead to a non-negligible error. According to the equality 31 , the expression of the ESS energy can be translated into the following:.

t , and the relationship is linear. This rule conforms to the general law of ESS energy variation. Moreover, it makes constraints 30 — 31 easier to transform and solve.

For adjustable robust constraints 23 — 32 equality constraints and inequalities are handled differently, they are transformed separately. By applying decision rules 33 — 37 , constraint 23 is equivalent to:. where, I k t represents a matrix of all ones.

For constraint 31 , substitute recourse variables by 36 — 39 and obtain:. First, 46 can be transformed into a standard robust optimization form by variable substitution.

Then, the random variable ξ bounded by [ ξ L , ξ U ] can be replaced by the random variable v bounded by [ 0 , e ] , as follows:. There exist quadratic terms in the aforementioned formula. Furthermore, using standard technique in robust optimization, 50 is equivalent to:.

Therefore, using the aforementioned method, Eqs 24 — 29 can be transformed into linear inequalities that are easy to solve. Take an example, 24 can be transformed to:. where λ g k t R and γ g k t R are auxiliary variables. The transformations of constraints 25 — 29 are detailed in Supplementary Appendix A.

For the inequality constraint 30 , substitute S t ξ k t by 39 and the subsequent transformation process is similar. However, S t ξ k t is jointly affected by the fluctuation deviations of multiple time intervals.

As a result, the dimensions of the random variables contained in constraint 30 are different at different time intervals. Thus, it is necessary to transform at different times one by one, and the equations can be further simplified by making a difference between the formulas at adjacent times.

The final transformation is as follows:. where λ s k t S and γ s k t S are auxiliary variables. where s k t , r k t , z k t represent auxiliary variables [see, e.

In this study, we conducted simulation analysis based on the IEEE bus system to verify the effectiveness of the established model. The platform used for the test is Matlab b, the model is solved based on Gurobi, and the processor of the test computer is Intel Core iU CPU, running at 2.

A microgrid system is constructed based on the IEEE bus system for simulation, and 2 WT and 1 PV are added to nodes 5, 28, and 14 of the system, respectively. According to Xiao et al. The typical daily forecast load curve of the microgrid and the forecast output of renewable energy wind plus solar power are shown in Figure 2.

ESS parameters are set as Table 1 ; the transaction price between microgrid and power grid is shown in Figure 3. The upper limit of the PCC power between the microgrid and the power grid is set to 60 MW, and PCC is set at node 1.

According to the assumed probability distribution, a large amount of historical data is generated and divided into two groups for sample training and out-of-sample testing separately.

The corresponding day-ahead dispatch results output of thermal units, the power transaction between microgrid and power grid, the charging and discharging results of the ESS are shown in Figure 4 , where the net load is equal to load minus renewable energy predictive power generation.

Figure 4 shows that a large portion of the load in this microgrid can be met by the renewable energy generation, and the remaining load net load is matched by thermal units, power grids, or ESS, according to the principle of economy.

For example, during time period 18—21 h, the electricity price of power grid is higher than the thermal power generation, so the output of the thermal unit is higher, while the system preferentially purchases electricity from the power grid when the electricity price is low. The charging and discharging of ESS comprehensively considers the net load demand and the electricity price.

For example, during time period 4—5 h, net load and electricity prices are low so that ESS is charged, and ESS discharges when the net load and electricity price are high.

It is verified that the proposed model can effectively derive the energy storage configuration scheme, which adapts to the regulation needs of the microgrid.

Configuration of energy storage can improve the renewable utilization capability of microgrid. In this section, we set different weight coefficients δ and analyze their influences.

By gradually increasing δ and solving the corresponding model, we can obtain different ESCC schemes and the corresponding renewable admission capability. By increasing δ sequentially and solving the corresponding models, we obtain 28 groups of simulation results, including ESCC schemes, investment costs, renewable admissible ranges, and renewable utilization probabilities.

Based on this, Figure 5 show the variation of renewable utilization capability with ESCC schemes. In Figure 5D , the upper and lower bands of renewable utilization and the renewable output all are the average values within a day.

From Figure 5 , we can easily see that with the increase of energy storage investment cost and its capacity, the admissible range of renewable energy gradually widens, and renewable utilization probability also increases.

This means that increasing the investment cost of ESS can improve the flexibility of the system, thereby increase the capability of renewable admission.

FIGURE 5. Relationship between ESCC strategy and renewable utilization. A Relationship between maximum power of ESS and renewable utilization. B Relationship between maximum energy of ESS and renewable utilization. C Relationship between Energy Storage investment and renewable utilization. D Relationship between energy storage investment and admissible range.

System operators can adjust δ , according to the required renewable utilization probability and the cost they are willing to pay.

In order to analyze the advantages of the proposed distributionally robust ESCC model considering renewable utilization in this article, we compare the proposed ESCC model with an ESCC model without considering the renewable utilization.