Multi‐time‐scale robust economic dispatching method for the power system with clean energy

Clean energies such as wind energy and solar energy have increased very fast in order to meet the environmental requirements. However, due to the uncertainties of wind and solar energy, the large-scale integration of new energy presents a great challenge to the power system economic dispatch. Thus, a multi-time-scale robust economic dispatch strategy of a multi-source hybrid power system based on the variable confidence level is proposed. The deterministic constraints of each time scale are transformed into robust constraints that take the uncertainty into account. Meanwhile, the robust level whose confidence level increases with the shortening of time scale is set to improve the scheduling conservative degree step by step. The selection principle of the robust level of each time scale is also presented. The proposed approach is applied to an IEEE 9-bus system. The calculation results are compared with those from the traditional multi-time scheduling method and show the effectiveness of the paper, which can reduce the uncertainty impact of wind, solar, and load forecast, as well as achieve a great balance of security, economic, and environmental benefits.


Introduction
Large-scale new energies with uncertainties cause great challenges to the power system economic dispatch. The multi-time-scale dispatching strategy can effectively enhance the clean energy consumption capacity. However, current multi-time-scale scheduling is essentially a deterministic model with weak robustness, which will result in frequent adjustment [1][2][3][4].
Multi-time-scale dispatching approaches with the uncertain factors are being considered [5][6][7][8][9]. Two major methods that are used for uncertainty modelling are stochastic programming [10,11] and robust optimisation [12,13]. The uncertainty analysis methods, which are handled by stochastic programming, are based on the probability theory. They include scene analysis [14], opportunity constraints planning [15], and others. Papavasiliou et al. [16] describe wind power as a series of weighted random scenarios by using a two-stage stochastic programming model to compute the system reserve capacity. Zhang et al. [17] apply the chanceconstrained programming to establish the wind farm economic dispatch model considering the risk of load loss and abandonment. However, the stochastic programming method relies on the new energy probabilistic model. Its computation is complex. Especially, the computational accuracy and security cannot be guaranteed.
The robust optimisation does not rely on the probability distribution of uncertain parameters. It is suitable for large-scale calculation. In recent years, robust optimisation has been widely used to solve the power system operation problem [18,19]. Xu et al. [20] propose a robust economic dispatch model based on the robust trajectory of a conventional unit that is used to deal with all wind power scenarios. To control the conservative degree of a robust model, the concept of robust budget is introduced in [21]. Bertsimas et al. [22] propose an adaptive robust optimisation method. Xuan et al. and Wang et al. [23,24] introduce a robust measure to adjust the robustness of the robust model with multiple uncertainties. However, most of the current publications focus on the regulation of the conservativeness of a day-ahead robust dispatch model without considering the control and coordination of the conservativeness of the multi-time-scale robust dispatch model. This paper proposes a multi-time-scale robust economic dispatch strategy of a multi-source hybrid power system based on the variable confidence level. First, the wind, solar, and hydro power are configured as virtual power (VP). Also, a load-tracking index is defined so that the VP output can track the load curve well. Then, the robust model of the wind, photovoltaic, and load forecasting value of each time scale is established. The deterministic constraints of each time scale are transformed into robust constraints with uncertainties. Meanwhile, the robust level whose confidence level increases with the shortening of time scale is set to improve the scheduling conservative degree step by step. The selection principle of the robust level of each time-scale dispatch is also put forward. The proposed approach is applied to an IEEE 9-bus system. The calculation results show the effectiveness of the paper, which can reduce the uncertainty impact of wind, solar, and load forecast, as well as achieve a great balance of security, economic, and environmental benefits.

Virtual power model
In this paper, both space-time characteristics and regulatory capacity are considered. In this way, renewable energies such as wind, solar, and hydro power are configured as VP. We call it a wind−solar−hydro power substation. We also introduce a loadtracking index N r that is defined to evaluate the ability of the VP output to track the load curve. The small value of N r means good tracking and smooth ability of VP. Through optimisation based on the load-tracking index, the VP output curve can well follow the load curve and stabilise the fluctuations. The optimised load curve P r is acquired when the load curve deducts the VP output curve: where N r is the load-tracking index; D t is the fluctuation rate of the VP output relative to the load curve, which stands for the tracking ability of VP; D s is the standard deviation of the load fluctuation; D c is the changing rate of the load power; D s and D c represent the fluctuation characteristics of P r ; T is the scheduling period; P v.t is the load at time t; P w.t , P p.t , and P h.t are the outputs of wind power, photovoltaic, and hydropower, respectively, at time t; P r.t is the value of the optimised load curve at time t; P L is the average load of period T; and P r.max and P r.min are the maximum and minimum values of P r , respectively. VP and the conventional thermal power station participate in system scheduling together, where VP always keeps the power on.

Robust economic operation model
The polyhedron set is used to describe the uncertainties of the renewable energy output and load demand in the system. Let J be the set of uncertainty constraints, I j be the set of uncertain factors on the jth uncertainty constraint and its amount is M, and ũ i j be the parameter of the ith uncertain factors on the constraint j. Then (i ∈ I j ⋅ j ∈ J), where u ij is its nominal value and û i j is its disturbance value (usually positive). The uncertainty set can be expressed as the following constraints: where The uncertain coefficient follows the symmetric distribution. Γ ∈ [0, M] is the robust level, which is used to adjust the worst uncertainty case that can be observed, that is, the conservative level of each uncertainty constraint. Thus, the system robust optimisation model for economic operation can be written as where f(x) is the objective function and x is the decision variable that is the output of all available energy in the system, P min and P max are the upper and lower limits of each energy output, and a j T is the coefficient matrix of x on the jth constraint.
The above model can be transformed into the following form by using the dual cone transformation of the inner-layer programming model in [24]:

Robust constraints for different cases
Robust constraints are different for different scheduling scales. They can be day-ahead 24 h scheduling constraints, intra-day 4 h scheduling constraints, and real-time 15 min scheduling constraints, which are described in detail as follows.

Day-ahead 24 h scheduling constraints based on the robust level
Day-ahead 24 h scheduling constraints include two parts: (i) robust constraints influenced by the uncertainty of wind, PV output, and load forecasting, and (ii) other traditional physical constraints.

Robust constraints:
The robust constraints of the dayahead 24 h scheduling are as follows: i. System power balance constraint: ii. System reserve demand constraints: According to the robust uncertainties set as shown in (11) and (12), the uncertainties of wind, PV output, and the load forecasting are as follows: The robust levels of the uncertainty sets of the uncertainty constraints (20) and (21) can be written as The number of uncertain constraints for the day-ahead 24 h scheduling is 3. That is, Thus, the constraints (22) and (23) can be converted to the following expressions:

Other constraints:
Day-ahead 24 h scheduling constraints also include the unit output limits, ramping rate limits, minimum up/down time constraints, and wind−solar curtailment constraints.
i. Unit output limits: where P w.max is the upper limit of the wind turbines output; P p.
where R u.i and R d.i are the ramping up and ramping down rate for the ith thermal unit, respectively. iii. Minimum up/down time constraints: where T i .
where δ 1 and δ 2 are the maximum permissible wind and solar curtailment rates, respectively; P w . t and P p . t are the maximum output of wind turbines and PV plants, respectively.

Intra-day 4 h scheduling constraints based on the robust level
The intra-day 4 h scheduling constraints also include two parts as follows: (i) robust constraints influenced by the uncertainty of wind, photovoltaic output, and load forecasting, and (ii) other traditional physical constraints.

Robust constraints:
Similar to the day-ahead 24 h scheduling, the number of uncertain constraints for the intra-day 4 h scheduling is 3. That is, System power balance constraint and system reserve demand constraints for intra-day 4 h scheduling are as follows: where T start.i and T stop.i are the start-up and shut-down time of the ith thermal generator, respectively.

Real-time 15 min scheduling constraints based on the robust level
Similar to the day-ahead 24 h scheduling and intra-day 4 h scheduling, the real-time 15 min scheduling constraints also include two parts: (i) robust constraints influenced by the uncertainty of wind, photovoltaic output, and load forecasting, and (ii) other traditional physical constraints. For the robust constraints, the number of uncertain constraints for the real-time 15 min scheduling is also 3. That is, The system power balance constraint and system reserve demand constraints for the real-time 15 min scheduling can be written as follows: The other constraints for the real-time scheduling are the same as those of the day-ahead 24 h scheduling.

Test example
The IEEE 9-bus system is used to verify the proposed multi-time robust dispatch model and approach. For the simulation purpose, the thermal power plant, hydropower station, wind farm, and PV station connect at nodes 1,2,3, and 5, respectively. The thermal power plant contains 10 units. The parameters of those units refer to [25].  Table 1.
In Table 1, the day-ahead 24 h scheduling has fully considered the fluctuations of uncertainties and had enough reserve capacity. In that case, in order to meet the power balance under the robustlevel Г2, only 23 MW power shortage needs to be filled. Since the hydropower output has reached its upper limit at this time, it is necessary to regulate the most economic unit no. 6 to its regulation rate limit so that 12.5 MW power shortage can be filled. The remaining 10.5 MW power shortage will be filled by increasing the output of unit no. 7. While in the real-time 15 min scheduling, the hydropower station can reduce the output by 29 MW to meet the power balance under the robust level Г3, the output of the thermal power unit does not need to change.
For the purpose of comparison, the thermal power unit output in the traditional multi-time-scale dispatch at 34th time point is analysed as shown in Table 2. In the traditional multi-time-scale dispatch, the thermal power unit output is adjusted frequently, and there is a large risk of load curtailment when the forecast error is relatively great.

Conclusion
This paper presents a multi-time-scale robust economic dispatch strategy for a power system with renewable energy. The proposed method is based on the variable confidence level. The robust uncertainty model of the wind, solar, and load forecast in different time-scale scheduling is established, where the deterministic constraints of each time scale are transformed into robust constraints that consider the uncertainty. The robust level whose confidence level increases with the shortening of time scale is set to improve the scheduling conservative degree step by step. This strategy can effectively reduce the impact of wind, solar, and load forecast uncertainty, relieve adjustment pressure as well as decrease the wind and load curtailment amount, which achieves a great balance of security, economic, and environmental benefits. The proposed approach is applied to the IEEE 9-bus system. The calculation results are compared with those from the traditional multi-time scheduling method and show the effectiveness of the paper.    1  455  455  455  130  2  455  455  455  130  3  130  130  130  60  4  130  130  130  60  5  162  162  162  90  6 51.