1. Introduction
The ever-increasing integration of variable renewable energy has brought great challenges to the economic dispatch (ED) of power systems. Current studies on the ED coping with the uncertain renewable energy mainly are divided into two parts: day-ahead ED and intra-day real-time ED. Day-ahead ED mainly focuses on a day-ahead energy plan or power reserve [
1,
2,
3]. It has enough time to model the entire day’s dispatch in detail based on the predicted information. For example, a chance-constrained day-ahead ED which co-optimizes the power reserve and the curtailment strategies of renewable energy is proposed in [
4]. Ref [
5] simulates day-ahead ED as a multi-stage process in order to ensure a sufficient day-ahead flexible reserve. In addition, to coordinate the integrated transmission and distribution systems, Ref. [
6] conducts a two-stage robust day-ahead ED with a distributed framework. However, the same technique of day-ahead ED cannot be directly applied to intra-day real-time dispatch due to the complexity of computation.
In the literature, several methods have been reported that aim to reconcile the gap between solution quality and efficiency. The well-known and industry-practiced model is look-ahead economic dispatch (LAED). The key idea of LAED is to use a moving horizon to reduce the computational burden, which can be traced back to [
7,
8]. To solve the problem caused by the penetration of uncertain renewable energy, researchers combined LAED with the stochastic methodology. The two-stage stochastic LAED [
9], chance-constrained LAEDs [
10,
11] and stochastic LAEDs combined with risk-constraints [
12] have achieved good effects under different respective applied scenarios. Nevertheless, as described in [
13], the probability distribution of renewable energy generation data in the stochastic methodology is difficult to acquire in practice. Robust optimization is another alternative method which ensures both the safety and economy by simulating the system operating under the worst-case scenario of renewable energy. In [
14], the moving horizon framework of LAED is embedded to the second stage of the two-stage robust optimization, also called the adaptive robust optimization (ARO), which shows better results of classic LAED and stochastic LAED. In addition, robust LAEDs based on ARO or the data driven ARO are proposed in [
15,
16,
17] to achieve more effective and robust solutions of real-time dispatch than LAED based on stochastic methodology.
However, ARO’s second-stage dispatch decisions are made with full knowledge of uncertain future parameters, making it anticipative and overestimating the unit’s adjustment capability [
18,
19]. Affine rules which convert the unit output to a linear function of the real wind power are extensively used to alleviate the anticipation issue of ARO [
16,
20,
21]. However, the linear approximation of affine rules greatly decreases the quality of the dispatch solution [
22]. Piecewise linearization decision procedures have been proposed by [
23,
24] to increase the optimality, but they are rarely used in the field of power system operation. The absence of a general and efficient algorithm that determines the number of segmentation for each decision variable prevents the procedure from being widely adopted [
23].
The temporal decomposition, as proposed in [
18,
19,
25], is another approach to overcome the anticipation issue of ARO. Box variables are incorporated into ARO in [
19] to eliminate dynamic constraints. The resulting problem consists of a set of single-stage economic dispatch problems and thus avoids the anticipativity issue. Since the box variables only decompose the coupling between adjacent dispatch periods, the flexibility from equipments with global coupling, e.g., ESS, is overestimated. Accordingly, Lorca et al. [
18,
25] proposed to use the multi-stage equivalence of ARO and proved that non-anticipative dispatch decisons can be guaranteed. Nonetheless, they do not propose an appropriate mechanism for obtaining the global optimum, where the multi-stage problem is still solved using the approximation method of affine rules.
On the other hand, to find a way that can improve the flexibility of power systems, the multi-stage dynamic programming is introduced into ED [
26]. Papavasiliou et al. [
27] proposed to model the equipments with long-time coupling by using the stochastic dual dynamic programming method. The corresponding multi-stage stochastic dynamic economic dispatch model (MSED) incorporates ESS in a natural way. They enhance the flexibility of ESS-integrated power systems in the field of stochastic optimization by a multi-stage dynamic framework. Nevertheless, in the field of robust optimization, the largest obstacle in applying this multi-stage dynamic programming framework to ED is to find an efficient approach to solve the strongly NP-hard problem, not as the affine rules in [
18,
25] mentioned before.
The robust dual dynamic programming (RDDP) method [
28], which is a robust counterpart of SDDP, is proposed in 2019 to obtain the global optimum of multi-stage robust optimization problems. However, because the approach utilized in the upper approximation of RDDP has exponential time complexity, it cannot be used to solve the short-term ED with more than 10 units [
28,
29]. In the authors’ previous work [
30,
31], the relaxed inner approximation (RIA) method is integrated to accelerate RDDP. The proposed fast robust dual dynamic programming (FRDDP) algorithm has been tested on the multi-stage ED problems. Test results indicate that FRDDP has strong scalability for large-scale systems and can efficiently administer robust solutions that are better than MSED [
30]. However, in [
26,
27,
30], the framework of multi-stage dynamic programming isn’t implemented to simulate a real-time ED, but a day-ahead off-line training process, which only utilizes the predicted information. Among them, the intra-day real-time ED runs based on the static operation strategy or policy. It can be reasonably assumed that if the multi-stage dynamic programming framework is directly applied to the real-time ED, it will achieve better results than the above strategy, which is due to the fact that the real-time information observed can directly participate in the process of dispatch decision making.
In the existing work of robust real-time ED, the uncertainty set are most conducted by the nearest predicted wind power [
16,
17], which does not make full use of the information of observed wind power. In [
14], a dynamic uncertainty set is proposed to capture the highly dynamical and time-coupled variable renewable energy in ARO. This simple explicit formulation is less accurate and efficient than deep learning [
32].
Fascinated by the rolling horizon structure in LAED which can reduce the computational burden and reserve the efficiency of dispatch decision in real-time ED, this paper proposes a novel model that introduces the multi-stage dynamic programming framework in robust form to the intra-day real-time ED with rolling horizon. The main contributions of this paper are fourfold:
We propose a multi-stage robust real-time ED (MRRTD) model in this paper. It uses the rolling horizon to lessen the computational load. Compared to the ARO, it is non-anticipative and maximizes the flexibility of timing coupled equipment such as ESS during real-time dispatch.
A policy guided real-time dispatch mode based on MRRTD with expanded time-slot scale is designed for large-scale systems to improve the scalability and industrial applicability of the proposed model.
A dynamic uncertainty set is built using a long short-term memory network (DUS-LSTM), which is real-time updated by refining the most-recent predicted available wind power during the process of rolling dispatch.
We employ a fast robust dual dynamic programming method to efficiently solve the MRRTD, where the forward pass and backward pass procedure are effectively embedded in the look-ahead scheme to realize the fast application of MRRTD in real-time dispatch.
7. Conlusions
In this paper, the MRRTD is proposed to cope with the intra-day real-time ED problem. MRRTD utilizes the framework of rolling horizon to alleviate the calculation burden. In each period, a multi-stage dynamic robust optimization problem is solved in MRRTD, which overcomes the non-anticipative problem of ARO and maximizes the flexibility of time-sequence coupled equipment such as ESS. To enhance the scalability of MRRTD, a PGRTD mode is proposed, which shows great effectiveness in large-scale systems via testing. In addition, an embedded DUS based on deep learning is proposed to update the uncertainty set in real-time, showing better performance than the existing uncertainty set. A MTD-FRDDP algorithm is designed to tackle the strongly NP-hard problem caused by MRRTD, where two accelerating mechanisms are integrated to improve the applicability. Case studies confirm the enhanced scalability of the proposed model and solution methodology and indicate the potential for real-world application.
Tight, yet accurate, DUS contributes to both the computational efficiency and the solution quality of the proposed MRRTD model. As suggested by a recent work [
39], type-3 fuzzy logic system (T3-FLS) is a promising generic mathematical tool for modeling uncertain phenomena. MRRTD with T3-FLS enhanced DUS is interesting for future work.