As an effective carrier of integrated clean energy, the microgrid has attracted wide attention. The randomness of renewable energies such as wind and solar power output brings a significant cost and impact on the economics and reliability of microgrids. This paper proposes an optimization scheme based on the distributionally robust optimization (DRO) model for a microgrid considering solar-wind correlation. Firstly, scenarios of wind and solar power output scenarios are generated based on non-parametric kernel density estimation and the Frank-Copula function; then the generated scenario results are reduced by K-means clustering; finally, the probability confidence interval of scenario distribution is constrained by 1-norm and ∞-norm. The model is solved by a column-and-constraint generation algorithm. Experimental studies are conducted on a microgrid system in Jiangsu, China and the obtained scheduling solution turned out to be superior under wind and solar power uncertainties, which verifies the effectiveness of the proposed DRO model.

In recent years, renewable resources such as wind and solar have been incorporated into the grid on a large scale through the form of the microgrid (MG) to ease the growing energy crisis and environmental pressure. MG is a comprehensive power distribution system [

The deterministic optimization model has a simple form, but the scheduling results are influenced by uncertainty [

Literature [

Given the above defects, some scholars have carried out research on solar-wind correlation. According to the literature [

This paper establishes a distributed robust optimization model for the Microgrid which considers the correlation of wind and solar power output and the probability distribution constraints. The main contributions of this paper are as follows:

(1) A method to generate scenarios of wind and solar power output using non-parametric kernel density estimation and Frank-Copula functions is proposed. The uncertainty and correlativity of wind and solar power output can be better described by kernel density estimation than by parametric estimation.

(2) A two-stage distributed robust optimization model is constructed and combines the 1-norm and ∞-norm while constraining the confidence set of the uncertainty probability distribution. The column and constraint generation algorithm is used to solve the problem.

Other sections of this paper are organized as follows. The framework flowchart of the DRO method is proposed in the

A multi-scenario DRO method is proposed with consideration of the multiple uncertainties of wind and solar power output. The framework flowchart of the proposed method is illustrated in

The distributed robust optimization method based on scenario probability distribution has high requirements for both the number of scenarios generation and its accuracy. In the actual scenario, considering that wind power and photovoltaic are affected by the geographical environment and human factors, there is always some correlation between their output. In this paper, to fully consider the randomness and correlation of output, we use non-parametric kernel density estimation and the Frank-Copula function to generate output scenarios; and use K-means clustering to reduce the many scenarios into four typical daily outputs. Here we choose the data of wind and solar power output for an area in Jiangsu Province, China, for the whole year of 2017.

To describe the correlation between random variables, the Copula function, which connects the joint distribution function of random variables with the respective marginal distribution function, has been widely used. Based on the Copula theory, we can decouple the joint distribution of random variables into the edge distribution and the correlation model, respectively. The edge distribution describes the randomness of variables, while the Copula function describes the correlation [

To select the best Copula function to fit the wind and solar output characteristics, Spearman rank correlation coefficients [

The Normal-Copula, Frank-Copula, and Clayton-Copula functions are used to fit and calculate the Empirical-Copula function, respectively; and obtain the rank correlation coefficient and the Euclidean distance from the Empirical-Copula function are shown in

Copula function | Rank correlation coefficient | Euclidean distance | |
---|---|---|---|

Spearman | Kendall | ||

Normal-Copula | −0.0765 | −0.0342 | 7.1254 |

Frank-Copula | −0.1711 | −0.0786 | 4.7643 |

Clayton-Copula | −0.0245 | −0.0212 | 8.9734 |

Empirical-Copula | −0.1932 | −0.0702 | / |

According to the

First, the Gaussian kernel function based on the non-parametric kernel density estimation is used to generate the probability density function (PDF) of the wind and solar power outputs in each hour per day. Then, the optimized Frank-Copula function is adopted to establish the joint probability distribution function (JPDF) of the wind and solar power outputs in each hour. The JPDF is then sampled and the inverse transformation of Frank-Copula is performed on sampled data of the JPDF to obtain the sampling data of the wind and solar power outputs in each hour. It is difficult to solve the inverse function of its cumulative distribution function (CDF) because the PDF of the non-parametric kernel density estimation is a summation form. Therefore, the cubic spline interpolation method is used to solve the sampling values corresponding to its cumulative probability.

In summary, the steps of the scenario generation with considerations of the uncertainties and correlativity of wind and solar power outputs are as the following:

(1) After getting the historical

_{t} and _{t} are the wind and solar power outputs, respectively; the _{d,t} and _{dt} are the wind and solar power outputs during the

(2) According to the probability density function of the wind and solar power output in each period, the cumulative distribution function is found, then the joint probability distribution function of the wind and solar power output is established according to the Frank-Copula function, which is specifically expressed as follows:

_{t} and _{t}, are the same value as the _{t} is the correlation parameter. When _{t} ∈ (−1, 1) and _{t} ≠ 0, _{t} _{t} and _{t} have better independence; when _{t} > 0, _{t} is positively correlated with _{t}, when _{t} < 0, _{t} is negatively correlated with _{t}.

(3) The joint distribution function of each period was sampled, and the sampling wind and solar power output of each period with the cumulative probability was solved by the Cubic Spline Interpolation method.

Set a small interval:

Breaking the cumulative probability interval [0, 1] into

_{f} and _{f}, it must fall in an interval _{f} and _{f} into the upper

(4) Considering the large sampling size, in order to balance the calculation speed and accuracy, the K-means [

The MG consists primarily of a wind turbine (WT), a photovoltaic (PV), a combined heating and power (CHP), an electric boiler (EB), battery storage (BT), and heat storage (HS). The inputs of the MG are connected to the power grid. The electricity can be bought or sold to the power distribution system. The direction of the arrows in

In this paper, we use the box decomposition algorithm [_{mp}; the second stage considers the total cost _{ope} in MG operation in different scenarios. As shown in

_{mp} is the start-up and shut-down costs coefficient of the micro gas turbine; the _{s,t} is the linearized state variable of unit start-up and stop-down. The _{ope,s} is the total operating cost of the system under each scenario _{s} is a set of all scenarios, taken as 4; and the _{s} is the probability of scenarios

_{grid,s,t}; micro gas turbine operation cost _{G,s,t}; battery aging cost _{BT,s,t}; wind and solar curtailment cost _{qw,s,t}, _{qv,s,t}; and demand response load compensation _{DR,s,t}.

The above objective function is subject to the following constraints:

_{d} and _{u} respectively indicate the upward and downward climbing rate. The

_{s,t} is the charge state of the battery in the _{max} and _{min} are the upper and lower limits of the charge state of the battery, respectively; the _{bat.max} are the initial and maximum capacity of battery storage.

The deterministic model in multiple scenarios constituted by

_{s} is the probability of occurrence of scenario _{s} is the second stage variable under scenario _{s} is the set of second stage variables under scenario _{s}

The uncertainty and correlation of wind and solar power output, as well as the limitations of historical data, and the scene probability distribution obtained by scene clustering have certain errors. To solve this problem, this paper uses stochastic programming to get the scene set, then uses a robust optimization method to limit the probability distribution of the scene.

By using the method of _{s0} of the reduced

^{p} is the set interval of the probability distribution of the scenarios, representing the confidence set [

The confidence degree of the probability distribution

_{0} is the predicted value of the probability distribution; the _{1} and _{∞} are the allowable deviation values of the probability distribution. For the right side of the above 2 inequalities: the 1 − 2_{s} exp(−2_{θ1}/_{s}) and 1 − 2_{s} exp(−2_{∞}), set them to _{1} and _{∞}, respectively, _{1} and _{∞} denote the confidence levels satisfied by the probability distribution _{1} and _{∞} can be expressed as:

Based on this, the confidence set of the probability distribution can be deduced as:

The model constructed in this paper is a multi-stage optimization problem that cannot be solved directly by commercial solvers. Using the column constraint generation algorithm (C&CG) [

The objective of the master problem is to solve for the optimal solution that satisfies the economy of the system given the known probability distribution

^{∗} and the lower bound _{M} of the model can be obtained by solving the master problem.

The sub problem is a two-layer structure of max-min form, which can be expressed as:

_{s} can be flexibly adjusted according to the scene change adjustment. When the result of solving the master problem is known to be ^{∗}. The lower bound value of the model

1 hown:

_{s} represents the solution to the inner layer problem and can be obtained from the result of solving the master problem. The _{s} is known. Since the absolute value constraint in

The equivalent transformation of the absolute value constraint is obtained as:

_{s} of the scenario _{s0}, respectively; the _{s}, respectively.

After the above steps, the model is transformed into a mixed linear planning problem, which can be solved directly by the commercial solver to get the _{M} of the model can be obtained. The specific solution process is as follows:

Step1: set the lower bound _{M} = ∞ and the upper bound _{M} = ∞, set the iteration time

Step2: solve master problem, obtain the optimal solution (^{*}, ^{*}), update the lower bound value to _{M} = min(^{T}^{* }^{*});

Step3: fix the first stage variable ^{*}_{sp}(^{*}), update the upper bound to _{M} = min{_{M}, ^{T}^{* }+ _{sp}(^{*})};

Step4: stop the iteration and return the optimal solution ^{*}_{M} − _{M} <

Step5: update the number of iterations,

This paper follows the MG as shown in

Equipment | Parameter | Numerical value |
---|---|---|

Micro gas turbine | 600 | |

_{d}, _{u}/kW |
50 | |

_{mp}($)/kWh |
0.25 | |

_{G}($)/kWh |
1.7 | |

1.2 | ||

Storage battery | 80 | |

0.95 | ||

_{max}(_{min})/kWh |
360(40) | |

_{bat.max})/kWh |
200(400) | |

_{BT}($)/kWh |
0.021 | |

Thermal energy storage | 50 | |

^{ch}(^{dis}) |
0.85(0.90) | |

300 | ||

Distribution network interactive power | 200 | |

Demand response load | _{IL}, _{SL}, _{HCL}($)/kWh |
0.5 |

0.1 | ||

Wind and solar curtailment | _{qw}, _{qv}($)/kWh |
0.62 |

Electric boiler | 0.9 |

The C&CG algorithm was solved iteratively in stages using the algorithm presented in

Based on

As can be seen from

This section mainly analyzes the impact of adding demand response to a microgrid system containing the energy storage, the heat storage, and the micro gas turbines on system optimization. The scheduling results in other scenarios are shown in

According to

During the 24:00 to 8:00 period, the photovoltaic does not work and the wind power output is relatively small. Considering the start-up and shut-down costs of the gas turbine, the heat preparation is not started. At this time, the power shortage of the electric load is obtained by purchasing power from the external power grid, and the heat load is met by the heating of the electric boiler, due to low electricity prices, the battery storage system starts charging.

During the 8:00–14:00 periods, the photovoltaic starts to supply power, the battery storage system discharges; and sells electricity to the grid during the rich periods to make profits, and the micro gas turbine starts to meet the power supply pressure during peak electric load. At the same time, the energy storage and the heat storage system are discharged.

During the 14:00–18:00 period, as the load pressure decreases, photovoltaic and wind power can be used to meet the power supply demand, the micro gas turbine is shut down, and the MG sells electricity to the power grid, and the excess power is sold to the grid as much as possible, and then stored by energy storage, and the heat of the electric boiler is stored by the heat storage system.

From 18:00 to 20:00, with the peak of the electric heat load coming again, the micro gas turbine starts again, and the energy storage and heat storage system respectively discharge the heat. Then from 20:00 to 24:00, the photovoltaic stops working; due to the low electric thermal load supply pressure at this time, the micro gas turbine is stopped by the wind power and electric boiler, charges the energy storage system, and sells electricity to the grid when the power is rich.

To sum up, after the introduction of IDR and the configuration of the energy storage system, the electric thermal load is reduced, and the operation mode of the energy storage device is in line with the strategy of “low charge and high discharge”, which is more conducive to reducing the cost of microgrid energy purchase.

Different confidence intervals were set to calculate the total cost of the model separately, and the results are shown in _{1} and _{∞} are larger, the confidence interval is larger and the range of the corresponding uncertainty probability distribution is larger, which leads to a higher total cost for the system. However, when the value of _{1} is small, the total system cost does not increase as _{1} increases, indicating that the optimization results at this time are mainly affected by the ∞-norm.

_{∞} |
_{1} |
|||
---|---|---|---|---|

0.2 | 0.5 | 0.8 | 0.99 | |

0.5 | 1274.54 | 1274.72 | 1274.72 | 1274.72 |

0.8 | 1274.60 | 1274.82 | 1275.18 | 1275.18 |

0.99 | 1274.75 | 1274.97 | 1275.40 | 1276.68 |

The cost of each scenario obtained from the scenarios analysis method is taken as the known condition. The DRO model is operated under the conditions of the comprehensive norm, only 1-norm, and only ∞-norm respectively, and the corresponding costs are shown in _{1} = 0.5, 0.5 ≤ _{∞} ≤ 0.99, and when considering only ∞-norm, let _{∞} = 0.99, 0.2 ≤ _{1} ≤ 0.99. As shown in

_{1} |
Comprehensive norm | 1-norm |
---|---|---|

0.2 | 1274.75 | 1279.38 |

0.5 | 1274.97 | 1279.38 |

0.8 | 1275.40 | 1279.38 |

0.99 | 1276.68 | 1279.38 |

_{∞} |
Comprehensive norm | ∞-norm |
---|---|---|

0.5 | 1274.54 | 1278.64 |

0.8 | 1274.60 | 1278.64 |

0.99 | 1274.75 | 1278.64 |

The decision results of the DRO model are analyzed in comparison with those of the traditional stochastic optimization and robust models. The optimization results obtained by the two methods are compared by generating 1000 scenes in random simulation. where the values of _{1} and _{∞} are both taken as 0.99 in the DRO method. A stochastic programming approach for optimal scheduling with a probability of 0.25 for each scenario. The robust optimization methods are performed according to the [

The results are shown in

Optimization results | Average value | ||
---|---|---|---|

DRO | Stochastic programming | Robust optimization | |

Electricity purchasing costs | 2293.02 | 2293.02 | 2298.42 |

Energy storage costs | 208.62 | 208.62 | 208.52 |

CHP power generation costs | 838.82 | 842.32 | 834.32 |

Daily profit | 1276.68 | 1284.45 | 1272.21 |

Photovoltaic curtailment ratio | 12.61% | 12.54% | 13,73% |

Wind curtailment rate | 23.42% | 23.42% | 24.25% |

According to the results, it can be seen that the robust optimization model corresponds to the worst clean energy consumption rate and profit; and the stochastic programming corresponds to the optimization results with the best economy and the best clean energy consumption rate, but conservation is not guaranteed. Compared with robust optimization and stochastic programming, the DRO method can achieve a better balance of economy and conservativeness, maximize the profitability of the distribution network while improving the consumption rate of clean energy, and has more advantages in dealing with uncertainty optimization.

Firstly, based on the stochastic optimization model, the output correlation between wind power and photovoltaic is analyzed. Secondly, based on the nonparametric nuclear density estimation and Frank-Copula function, a typical sample of 1000 sets of wind and solar power output with correlation was generated, and the reduced scenario is obtained through K-means clustering. Finally, we constructed the distributed robust optimization model that considers the confidence interval of a probability distribution. After simulation verification and comparison, indicates that our model can well balance system economy and robustness. The following conclusions can be drawn:

(1) The method of scenario generation has an important impact on the overall optimization results. The Frank-Copula function is used for scenario generation, which can better reflect the wind power and PV output correlation. Combined with the K-means scenario clustering algorithm for scenario reduction, the final generated scenarios can be more representative.

(2) The simulation results show that the participation of demand response reduces the peak-to-valley difference of the overall load. Rather than considering only 1-norm and ∞-norm, the combined norm reduces costs.

(3) By considering the confidence interval of the probability distribution based on the stochastic optimization model, the proposed DRO model in this paper can achieve a better balance of economy and conservativeness. Compared to stochastic programming and robust optimization, this method has more advantages in dealing with uncertainty optimization.

(4) C&CG algorithm can effectively and quickly solve the proposed distribution robust optimization model.

The wind turbine.

The photovoltaic.

The combined heating and power.

The electric boiler.

The battery storage.

The heat storage.

The micro gas turbine.

Demand response load.

Interruptible load.

Shiftable load.

Cuttable heat load.

Time intervals (in h), scenarios of uncertain parameters.

System equipment (WT, PV, CHP, EB, BT, and HS).

System equipment that counts start-up and stop-down (CHP, BT, and HS).

Renewable energy equipment (WT and PV).

Energy storage equipment (BT and HS).

Demand response load (IL, SL, and CHL).

Time horizon of the problem (in 24 h).

A collection of scenarios.

_{s}

Probability of scenario

Purchased and Sold prices at period

Heat efficiency of CHP and EB.

The charging and the discharging efficiency of equipment

Maximum purchased and sold electricity power (in kW).

_{s}

Scenario numbers of the uncertain power output of WT and PV.

_{i}

Operation cost of the system equipment under scenario

_{b}

Operation cost of the renewable energy equipment under scenario

_{l}

Compensation factor of the demand response load under scenario

Electricity purchased and sold power under scenario

Power forecast output of WT and PV under scenario

Power actual output of WT and PV under scenario

Power output of demand response load

Power output of equipment

Charging and discharging power of equipment

State of charging and discharging of equipment

We would like to thank the editorial and reviewer panel for their input and comments.

This work was supported in part by the National Natural Science Foundation of China (51977127); in part by the Shanghai Municipal Science and in part by the Technology Commission (19020500800), “Shuguang Program” (20SG52) Shanghai Education Development Foundation and Shanghai Municipal Education Commission.

The authors declare that they have no conflicts of interest to report regarding the present study.

The specific mathematical expressions of the 5 common types of Copula functions are shown below:

Multidimensional Normal-Copula:

The expressions of the probability density functions of the 5 common types of Copula functions are shown in