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Currently the renewable energies including wind power and photovoltaic power have been increasingly deployed in power system to achieve contamination free and environmental-friendly power production. However, due to the natural characteristics of wind and solar, both wind power and photovoltaic power contain uncertainty and randomness which may significantly impact the stability, security, and economic efficiency of the conventional power system mainly consisted by hydropower and thermal power. To deal with the issue, this paper presents a two-stage robust model which is able to achieve the optimal day-ahead dispatch strategy in the worst-case scenario of wind and photovoltaic outputs. Because of the strong interactions between the two stages, the original optimization has been decomposed into the day-ahead dispatch master problem and the additional adjustment subproblem considering the uncertainty and randomness of the wind and the photovoltaic outputs. Also, the piecewise linearization technique is employed to convert the original problem into a MILP problem. Afterward, the dualization of the additional adjustment subproblem can be obtained by using linear programming strong duality theory. Additionally, the Big-

To achieve the contamination free and environmental-friendly power generation, clean energies such as wind power generation and photovoltaic power generation have been increasingly deployed in modern power systems [

To study the integration of the clean energies, researches [

Due to the uncertainty in the renewable energy sources, the adjustment between the base generation and the actual generation should be considered. Researches [

Motivated by the previous works, this paper presents a two-stage robust model considering the generation of wind power, photovoltaic power and hydropower in enabling the day-ahead economical dispatch strategy for power system. In our model the fluctuations of wind output and photovoltaic output caused by the randomness and uncertainty are evened out using hydropower and thermal power. Especially due to the strong adjusting ability of hydropower, the model deals with the fluctuations prioritizing the hydropower and then the thermal power because of its feeblish adjusting ability. Additionally, to improve the wind and photovoltaic power accommodation capability of the power system, this paper regards the wind power and photovoltaic power curtailment costs as penalty function in the presented model. This paper also linearizes the original two-stage robust problem and then employs the strong duality theory to handle the max-min subproblems. At last the ultimate optimal solution can be achieved using the model decomposition and iterations. The effectiveness of the presented approach has been evaluated using the IEEE-30 bus system.

The rest of the paper is organized as: Section

Modern power systems contain not only thermal power and hydropower but also wind power and photovoltaic power. In this section, the optimal dispatch model is presented for a power system with thermal power and renewable energies including wind, photovoltaic, and hydropower. At the very beginning of the modeling, without considering the uncertainty and randomness of the wind and photovoltaic power, the day-ahead dispatch strategy is an optimization problem based on the identification of the minimum operating cost, which is represented by

In (

Combining the day-ahead operation and the additional adjustment a model is represented by (

In (

To clarify the model represented by (

The uncertain variable

Additionally, this paper also employs the adjustable robust parameters shown by

To achieve the optimal economical day-ahead dispatch scheme

Day-ahead dispatch constraints in the model represented by (

(_{t} is the reservoir storage at time

_{b} is the total number of the nodes;

Additional adjustment constraints in the model represented by (

Equation (

The day-ahead economical dispatch is a mixed integer quadratic programming (MIQP) problem. Using robust optimization directly may be time consuming to achieve the optimal solution. Therefore, this paper employs piecewise linearization approximation [

In our model, the nonlinear factor (the first equation in (_{j,t} and the state variable_{j,t}, linear approximation can been done. As a result, the linearized cost of the thermal units is presented by _{j} is the slope of _{j} is the intercept of the _{j,t} is the output of the _{j,t} is the status of the _{j} is the

Piecewise linearizing of generation cost curve of thermal units.

Based on the linearization, the original two-stage model can be represented using the matrix shown by

In our model, the worst scenario and its corresponding additional adjustment scheme in the second stage are optimized based on the day-ahead dispatch scheme in the first-stage optimization. Moreover, the optimization in the second stage also impacts the optimized results of the first stage. The interactions between the two stages result in the employment of C&CG algorithm [

This paper employs strong duality theory to convert the SP1 into an equivalent maximization problem. Big-M approach is also adopted to handle the nonlinear terms in the dual problem. The converted SP2 is represented in^{-} are the upper and lower limit of uncertainty variable; ^{+}, ^{-} are the positive and negative values of

Our model is finally converted into the MP and the SP2. The optimizations of OP (

Initialize the worst-case scenario

According to the worst scenario _{k}, _{1},_{2}, …,_{k}), and then set

After given the first-stage solutions_{k}, the solution (

If_{k}. Otherwise add the new variable

To evaluate the effectiveness of the presented two-stage robust optimal model in enabling the economical day-ahead dispatch considering the uncertainties, this paper implements the model using MATLAB. The solution of the model is by employing CPLEX. A modified IEEE-30 bus system is employed as the power system model. The load fluctuations are shown in Figure

The values of

Convergence of C&CG approach.

The output power of thermal units and hydrounits.

The worst-case scenario of WTG and PVG.

Figure

Under the robust optimized dispatch scheme, the day-ahead operation cost is 2359.9 USD while the adjustment cost under the worst-case scenario is 134.8 USD. To evaluate the robustness based on the adjustable robust parameter constraints, this paper also employs Monte Carlo approach to generate a number of 500 random scenarios. The adjustment cost of each scenario is calculated based on the robust optimized day-ahead dispatch scheme. Figure

Robustness analysis.

The economy of the optimized dispatch scheme is evaluated by the number of 500 scenarios generated by Monte Carlo approach. The adjustable robust parameter in the static robust planning is as the same as the two-stage robust planning. The comparisons are listed in Table

Comparison for operation costs.

Category | Two-stage robust planning | Static robust planning | Deterministic planning | ||||
---|---|---|---|---|---|---|---|

Avg. | Max | Avg. | Max | Avg. | Max | ||

Day-ahead operation cost /$ | 2359.9 | 2374.2 | 2264.1 | ||||

| |||||||

Adjustment cost /$ | Thermal units | 2.7 | 30.8 | 85.0 | 251.5 | 375.9 | 1153.7 |

Hydro units | 123.2 | 169.4 | 153.9 | 221.8 | 51.1 | 145.3 | |

Wind power curtailment | 0 | 0 | 73.0 | 500.9 | 314.9 | 1053.1 | |

PV power curtailment | 0 | 0 | 53.1 | 473.2 | 225.2 | 813.9 | |

| |||||||

Total cost /$ | 2485.8 | 2541.4 | 2739.2 | 3348.3 | 3231.2 | 4098.5 |

Three points can be clearly seen in Table

As mentioned in the introduction section, the hydropower has strong adjusting ability. And it should be also pointed out that robust model considers the worst-case scenario, in which the output of both wind and photovoltaic power may be lower than the predicted values. Thus, in the robust scheme, hydrounits retain more capacity of upregulation than that in the deterministic scheme to ensure the economic benefits. As a result, though the adjustment cost of hydropower increases, the total adjustment cost is significantly decreased.

In addition, the two-stage robust model formulates a better base generation for thermal units and hydrounits than that of the static robust model based on the iterations between the main problem and the subproblem. Consequently, wind power curtailment cost and PV power curtailment cost are reduced to 0. The renewable energy resources accommodation capability of the power system is greatly enhanced.

To reveal the importance of the hydropower in the dispatch scheme, extra dispatch scheme optimizations with different hydropower penetrations have been carried out and evaluated. Four different cases are studied as listed below.

Not considering the hydropower.

The capacity of the hydropower is 9MW.

The capacity of the hydropower is 18MW.

The capacity of the hydropower is 30MW.

The results are shown in Figures

The output power of thermal units in different cases.

Day-ahead operation cost in different cases.

It can be observed that, along with the hydropower penetration increases, the output of the thermal power decreases. As a result, the day-ahead operation cost can be reduced. In addition, the cost of the dispatch scheme also has been significantly impacted as shown in Table

The effect of hydropower.

Category | | | | | ||||
---|---|---|---|---|---|---|---|---|

Avg. | Max | Avg. | Max | Avg. | Max | Avg. | Max | |

Adjustment cost of thermal units /$ | 404.1 | 1109.5 | 21.2 | 107.9 | 2.7 | 30.8 | 0.5 | 7.2 |

| ||||||||

Adjustment cost of hydro units /$ | — | — | 110.3 | 143.35 | 123.2 | 169.4 | 124.3 | 174.0 |

| ||||||||

Wind power curtailment cost /$ | 154.2 | 1089.0 | 0 | 0 | 0 | 0 | 0 | 0 |

| ||||||||

PV power curtailment cost /$ | 193.3 | 763.0 | 4.9 | 37.3 | 0 | 0 | 0 | 0 |

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Total cost /$ | 751.6 | 1852.0 | 136.34 | 245.4 | 125.9 | 181.5 | 124.8 | 176.7 |

Table

The previous experiments are based on the values of the adjustable robust parameter

Optimization results and economic efficiency with different adjustable robust parameter

Category | | | | |||
---|---|---|---|---|---|---|

Avg. | Max | Avg. | Max | Avg. | Max | |

Day-ahead operation cost /$ | 2264.1 | 2359.9 | 2430.0 | |||

| ||||||

Adjustment cost /$ | 969.6 | 1831.8 | 125.9 | 181.5 | 123.4 | 169.4 |

| ||||||

Total cost /$ | 3233.7 | 4095.9 | 2485.8 | 2541.4 | 2553.4 | 2599.4 |

Table

In order to reveal the influence of the different prediction errors of the renewable energy sources in the two-stage robust model, the deviation indicator

where

In this section, five deviation indicators (0, 0.25, 0.5, 0.75, and 1) are employed to the presented model. The day-ahead operation cost under the different prediction error is shown in Figure

Adjustment cost under different adjustable prediction error.

Category | | | | | | |||||
---|---|---|---|---|---|---|---|---|---|---|

Avg. | Max | Avg. | Max | Avg. | Max | Avg. | Max | Avg. | Max | |

Adjustment cost of thermal units /$ | 0 | 0 | 0 | 0 | 0.6 | 10.6 | 2.6 | 28.1 | 2.7 | 30.8 |

| ||||||||||

Adjustment cost of hydro units /$ | 0 | 0 | 31.3 | 42.3 | 61.2 | 83.6 | 91.2 | 122.4 | 123.2 | 169.4 |

| ||||||||||

Wind power curtailment cost /$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

| ||||||||||

PV power curtailment cost /$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

| ||||||||||

total adjustment cost /$ | 0 | 0 | 31.3 | 42.3 | 61.9 | 83.6 | 93.8 | 148.0 | 125.9 | 181.5 |

The parameters of thermal generator.

Category | Node | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 40 | 10 | 15 | 10 | 0.02 | 2 | 0 | 8.152 |

2 | 2 | 40 | 10 | 20 | 15 | 0.0175 | 1.75 | 0 | 12.170 |

3 | 22 | 25 | 5 | 10 | 10 | 0.0625 | 1 | 0 | 7.771 |

4 | 27 | 22.5 | 5 | 10 | 8 | 0.00834 | 2.25 | 0 | 6.899 |

5 | 23 | 15 | 5 | 8 | 5 | 0.025 | 2 | 0 | 5.024 |

6 | 13 | 20 | 5 | 8 | 5 | 0.025 | 2 | 0 | 6.020 |

The parameters of hydrogenerator.

^{3}/(Mwh)) | ^{3}/(Mwh)) | ^{4}m^{3}/h) | ^{4}m^{3}/h) | ^{4}m^{3}/h) | ^{4}m^{3}/h) |
---|---|---|---|---|---|

2500 | 0 | 4.86 | 0 | 7.00 | 3.25 |

| |||||

^{4}m^{3}/h) | ^{8}m^{3}) | ^{8}m^{3}) | ^{8}m^{3}) | ^{8}m^{3}) | |

| |||||

4.10 | 1.3586 | 1.1623 | 1.3366 | 1.3366 | 3.5 |

Comparison for the planning results under different prediction error.

Load demand.

Wind power fluctuation interval.

PV fluctuation interval.

As can be seen from Figure

This paper presents a two-stage robust model in enabling economical day-ahead dispatch considering wind power, photovoltaic power, hydropower, and thermal power. Based on the model decomposition and linearization, the original model can be converted into MP and SP, which are finally handled by C&CG approach. Based on the experimental results, the contributions of the presented model are as follows:

The two-stage robust model is able to achieve the optimal day-ahead dispatch scheme under the worst wind and photovoltaic output scenarios. The scheme is proved to be economy and has great potential to improve the capability of the power system to accommodate wind and photovoltaic power.

Hydropower can effectively handle the uncertainties caused by wind power and photovoltaic power. In our model by considering the adjusting ability of the hydropower, the adjustment demand of the thermal unit can be significantly reduced. Additionally, the wind power curtailment and PV power curtailment can also be potentially avoided.

The adjustable robust parameter enables the performance adjustment of the optimized dispatch scheme by tuning their values. As a result, the compromise between economy and robust could be achieved under different dispatching requirements.

Taking more factors (demand side management, battery storage system, electric vehicle [

See Figures

The IEEE bus model used to support the findings of this study has been deposited in the Department of Electrical Engineering at the University of Washington. The details of the modifications of the models and the customized parameters are stated in the paper. IEEE bus model site is available at

The authors declare that there are no conflicts of interest regarding the publication of this article.