The distributionally robust day-ahead dispatch is a typical two-stage problem. The day-ahead problem in the first stage is a pre-decision problem, which determines the trading power from the power grid. The intraday operation in the second stage is a regulation problem, which adjusts the intraday trading power and the scheduling plan of the regulating resources to minimize the intraday operating cost.

First stage (day-ahead): in this stage, the RIES trades power from the operator, and the power transaction cost is minimized, i.e., F 1 = ⁡ min ⁡ ∑ t p t P t tr , (1) P _ tr ≤ P t tr ≤ P ¯ tr , (2) where P _t ^tr is the day-ahead trading power, p _t is the power price, and P ¯ tr / P _ tr are the upper/lower restrictions of P _t ^tr, respectively.

The first-stage problem can be refined as follows: min x ∈ X c ⊤ x , (3) where x ∈χ are decision variables in the first stage, i.e., the day-ahead trading power P _t ^tr, and c are day-ahead cost coefficients.

Second stage (intraday operation): in this stage, the RIES adjusts the intraday trading power and the dispatch plan of power storage, transferable power load, biogas generator, biogas storage, heat pump, electric boiler, and heat storage. The intraday operating cost includes the intraday power trading cost, material cost of the biogas generator, and the cost of the transferable power load: F 2 = ⁡ min   ∑ t b t + Δ P t tr , + + b t − Δ P t tr , − + ∑ t b t g G t + ∑ t d t + Δ P t dr , + + d t − Δ P t dr , − , (4) where Δ P t tr , + / Δ P t tr , − are the intraday purchase and sell power from/to the power grid, respectively; G _t is the biogas input of the biogas generator; Δ P t dr , + / Δ P t dr , − are the upward and downward regulated power of transferable power load, respectively; b t + / b t − are the transaction price for the intraday purchase and sell power, respectively; b t g is the material cost coefficient of the biogas generator, and d t + / d t − are the upward and downward cost coefficients of the transferable power load, respectively.

The following security constraints need to be met for the second-stage intraday operation of the RIES.

In the proposed dispatch model of the RIES, the photovoltaic power output fluctuation u _t ^v, the power load u _t ^p, and the thermal load fluctuation u _t ^h are uncertain variables, and they constitute the source and load uncertainties. The day-ahead scheduling model is essentially a two-stage problem, which optimizes the day-ahead operating cost in the first stage and the expectation of the intraday cost in the second stage. In addition, since the possibility for the occurrence of the extreme distribution is relatively low, we further limit the loss of the cost expectation in the historical distribution to reduce the conservativeness of the distributionally robust dispatch. The constrained distributionally robust day-ahead dispatch can be refined as shown below: min x ∈ X max p ∈ P c ⊤ x + E p Q x , u s . t .   c ⊤ x + p k , 0 Q x , u k ≤ F ¯ , (28) where P and P are the probability and the ambiguous set of the source and load uncertainties, respectively; p _k,0 is the baseline probability density of the kth reference sample u _k ; and F ¯ is the allowed maximum for the cost expectation in the historical distribution.

The maximum limit F ¯ can be predefined as shown below: F ¯ = F _ emp + λ F ¯ emp − F _ emp F _ emp : min x ∈ X c ⊤ x + p k , 0 Q x , u k F ¯ emp : min x ∈ X max p ∈ P c ⊤ x + E p Q x , u , (29) where F _ emp is the empirical cost expectation obtained by SO, while F ¯ emp is the empirical cost expectation obtained by DRO without the constraint of the empirical cost expectation, and λ is the equilibrium coefficient. The equilibrium coefficient λ locates within [0, 1], which means that the robustness of CDRO is between DRO and SO. The CDRO is close to DRO and pursues the robustness in extreme distributions when λ moves toward 1, while the CDRO is close to SO and pursues the economical effectiveness in the historical distribution (the common distributions) when λ moves toward 0. Therefore, the equilibrium coefficient λ can be set reasonably to achieve a tradeoff between the robustness and economical effectiveness.

We formulate the ambiguous set for the source and load uncertainties according to the confidence set theory. K discrete reference samples can be selected from the M historical observations to characterize the possible value of the source and load uncertainties, and the corresponding baseline probability density is obtained. However, the actual probability distribution values of each reference sample are still uncertain, and we can use the probability density uncertainty to describe the uncertainty of the source–demand distribution. We can construct a set with the baseline density of reference samples as the center and the two sets of 1-norm and infinity-norm as the constraints to restrict the actual probability density of source–demand scenarios. Therefore, the ambiguous set of the source and load uncertainties is formulated as follows: P p k 0 ≤ p k ≤ 1 , ∀ k ∑ k p k = 1 p k − p k , 0 ≤ θ ∞ , ∀ k ∑ k p k − p k , 0 ≤ θ 1 , (30) where p _k is the actual probability of the reference sample k and θ _∞ and θ ₁ are the maximum gap of the probability density under the infinity-norm and 1-norm restrictions, respectively.

The maximum gap of the probability density θ _∞ and θ ₁ can be determined according to the confidence level (Ding et al., 2017). Pr p k − p k , 0 ≤ θ ∞ , ∀ k ≥ 1 − 2 K e − 2 M θ ∞ , (31) Pr ∑ k p k − p k , 0 ≤ θ 1 ≥ 1 − 2 K e − 2 M θ 1 K , (32)

Let α _∞ and α ₁ denote the confidence level associated with the right hand of the above equations, and then: θ ∞ = 1 2 M ln 2 K 1 − α ∞ , θ 1 = K 2 M ln 2 K 1 − α 1 . (33)

Since the scenarios of the source and load uncertainties are denoted with the reference samples, while the probability density of these reference samples is uncertain, the distributionally robust dispatch Equation 28 is reformulated as follows: min x ∈ X c ⊤ x + max p k ∈ P ∑ k p k Q x , u s . t .   c ⊤ x + p k , 0 Q x , u k ≤ F ¯ . (34)

First, we introduce auxiliary variables to linearize the absolute value constraint in the ambiguous set: p k − p k , 0 ≤ Δ p k ↓ , (35) − Δ p k ≤ p k − p k , 0 ≤ Δ p k , Δ p k ≥ 0 , (36) where Δp _k is the introduced auxiliary variable.

Therefore, the ambiguous set is reformulated into P p k 0 ≤ p k ≤ 1 , Δ p k ≥ 0 , ∀ k − Δ p k ≤ p k − p k , 0 ≤ Δ p k , ∀ k ∑ k p k = 1 Δ p k ≤ θ ∞ , ∀ k ∑ k Δ p k ≤ θ 1 . (37)

The constrained distributionally robust dispatch in Equation 28 is a typical two-stage optimization problem, and we develop a C&CG-based decomposition solution for the problem. The principle of the method is to split the two-stage optimization to independently solve the master problem and subproblem. The master problem determines the day-ahead decision variables against all known extreme distributions of the probability density. The subproblem receives the obtained decision variables and determines an extreme distribution of the probability density and adds it to the master problem. These two problems are computed alternately, and hence, the extreme distribution set in the master problem will cover enough extreme distributions to ensure the robustness of the dispatch. The computing framework of the two-stage solution is presented in Figure 2.

1) Master problem: the problem obtains the optimal solution against all known extreme distributions of the probability density.

Second, the expected intraday cost is maximized to determine an extreme distribution of the probability density: f sub = max p k ∈ P ∑ k p k Q x * , u k . (40)

Then, the extreme distribution p of the probability density is added to the extreme distribution set, and the updated extreme distribution set is uploaded to the master problem. c ⊤ x * + f sub is denoted as the upper bound (UB).

3) Convergence criterion check: end the iteration until the following convergence criterion; otherwise, go back to step 1 and modify v = v+1.

In the proposed dispatch model of the RIES, the decision variable x in the day-ahead stage is the trading power, while the intraday decision variables y is constituted by the dispatch plan of power storage, transferable power load, biogas generator, biogas storage, heat pump, electric boiler, and heat storage. Note that the following analysis for the intraday operation is presented with the expected value of the reference samples.

Figure 5 shows the day-ahead trading power and power export of the biogas generator. Due to the low power generation cost of the biogas unit, the power output of the biogas generator is always maintained at a high level. At the dispatch periods t = 10:00–12:00, the power load is at the valley, while the available photovoltaic power output is at a relatively high level, which means that the net power load is at the lowest level during these periods, and hence, the power export of the biogas generator does not reach the maximum. Similarly, the RIES purchases very little electric power at the dispatch periods t = 8:00–16:00, with relatively lower net power load and heat load. During the dispatch periods t = 00:00–7:00 and 17:00–24:00, the power load and heat load are both at the peak while the photovoltaic power output is at the valley, and hence, the RIES purchases much electric power from the power market.

Figure 6 illustrates the distribution of transferable power load whose transferable periods are t = 13:00–17:00. The transferable power load is mostly transferred to t = 13:00–14:00 with lower power price, and thus, the operating cost of the RES is decreased.

Figure 7 shows the charge–discharge state of the power and heat storage, respectively. The power price is relatively low during the dispatch periods t = 1:00–6:00 and 23:00–24:00, and so, the power storage charges power. While the power price is relatively high during the dispatch periods t = 7:00–9:00 and 19:00–21:00, the power storage discharges power to supply the demand. Finally, during the dispatch periods t = 10:00–12:00, the power load is at the valley while the photovoltaic power output is at a relatively high level, and hence, the power storage charges power to accommodate photovoltaic power. As indicated from the above analysis, the energy storage flexibly charges and discharges power according to the power price, and hence can decrease the operating cost of the RIES. While for the heat storage, its charging and discharging status is similar to that of the power storage since the heat load is supported by the waste heat of the biogas generator and the electric boiler.

To verify the economy and robustness of the CDRO, this subsection compares the proposed method and the commonly used uncertainty methods, including SO, RO, and DRO. SO minimizes the operating cost expectation corresponding to the historical distribution of the source and load uncertainties, RO minimizes the operating cost corresponding to extreme scenarios, and DRO minimizes the operating cost corresponding to extreme distributions. The proposed CDRO minimizes the operating cost expectation corresponding to extreme distributions while restricting the sacrifice of the operating cost expectation in the historical distribution. The comparison results related to these methods are shown in Table 2. As displayed in the table, SO has the lowest operating cost expectation in the historical distribution, while it is significantly less robust than DRO and CDRO under some extreme distributions. RO minimizes the operating cost corresponding to extreme scenarios, which is very conservative and relatively uneconomical since the probability of extreme scenarios is extremely low. Although the historical distribution cost of the CDRO is slightly higher than that of SO, its economical effectiveness under extreme distributions is significantly better than that of SO. Compared to SO, the historical distribution cost of CDRO is slightly sacrificed by 6.18E-04, while the cost efficiency under extreme distributions is significantly improved by 2.7E-03. In view of that the practical distribution of the source and load uncertainties does not strictly obey the empirical value, it is essential to address the distributional deviation and enhance the robustness of the dispatch plan.

To further verify the performance of the CDRO, we change the fluctuation degrees of source and load uncertainties, i.e., change to 0.75–1.25 times of the original. Since RO is very conservative, we compare the optimization results of SO, DRO, and CDRO. As shown in Table 3, no matter how the uncertainty degree changes, CDRO always achieves a tradeoff between the economical effectiveness and robustness, i.e., it sacrifices the historical distribution cost slightly to significantly improve the robustness. In addition, as the uncertainty degree increases, the advantage of CDRO in balancing the economical effectiveness and robustness becomes more obvious.

The ambiguous set of the source and load uncertainties is related to the amount of historical scenarios (M), the amount of the reference sample (K), and the confidence degree (α _∞ and α ₁). The default setting of M, K, θ _∞, and θ ₁ in this paper is 200, 50, 0.99, and 0.95, respectively. This subsection will study the impact of these parameters on the optimization performance.

First, we study the influence of the amount of historical scenarios M on the optimization results. The optimization results with different M from 50 to 10,000 are displayed in Table 4. The results in the table reveal that the objective decreases with the increase of the scale of historical observation. The reason is that the increase of historical observation limits the allowable gap of the probability distribution, and hence decreases the conservativeness of the proposed CDRO. It should be noted that the optimization results of CDRO actually approach that of SO with the increase of the scale of the historical observation, which further validates the flexibility of the proposed CDRO.

Second, we study the impact of the amount of reference samples K on the optimization results. The optimization results with different K from 10 to 200 are shown in Table 5. As K increases, the possible scenarios in the probability distribution of the source and load uncertainties are more dispersed, and hence, the maximum possible range of the probability distribution widens. Therefore, the optimized objective increases with the increase in the number of reference samples.

Finally, we study the impact of the confidence degree on the optimization performance, and the simulation results are displayed in Table 6. With the increase in the confidence degree α _∞ and α ₁, the allowable gap of the probability density θ _∞ and θ ₁ widens, and hence, the maximum possible range of the probability distribution widens. Therefore, the optimized objective increases with the increase in the confidence degree. It is worth noting that when α _∞ = 0.9, the optimized objectives are the same for α ₁ = 0.9 and α ₁ = 0.99. As shown in Equations 30–33, only the infinity-norm works if the infinity-norm and 1-norm share the same confidence degree. At this time, even though the confidence degree for the 1-norm increases, the ambiguous set of the source and load uncertainties remains the same, and so, the optimized objective will not change.

In the proposed CDRO, the equilibrium coefficient λ plays a crucial part in coordinating the economical effectiveness and the robustness of the dispatch plan. Figure 8 presents the operating cost expectation in the historical distribution and extreme distribution associated with the equilibrium coefficient within [0, 1]. As shown in the figure, if the equilibrium coefficient approaches 1, the operating cost expectation in the historical distribution increases, while the expected cost in extreme distribution decreases, which means that the CDRO pays more attention to the robustness than to economical effectiveness. When the equilibrium coefficient λ approaches 0, the operating cost expectation in the historical distribution increases and the expected cost in extreme distribution decreases, which means that the CDRO pays more attention to economical effectiveness than to robustness. In addition, when the equilibrium coefficient locates between 0 and 0.2, the expected cost in the extreme distribution significantly decreases with a very small loss of the operating cost expectation in the historical distribution, which means that the robustness can be ensured with a very slight sacrifice of economical effectiveness in common distributions. The above numerical results reveal that the CDRO can acquire a tradeoff between the robustness and economical effectiveness through a reasonable setting of the equilibrium coefficient. In practical applications, considering that historical distributions are more likely to occur while extreme distributions are less likely to occur, the proposed CDRO shall give priority to economical effectiveness in the historical distribution. Therefore, the equilibrium coefficient can be set within [0.1, 0.3] in practical applications.

The tailored two-stage solution procedure splits the dispatch problem to independently solvable master and subproblems. The master problem determines the day-ahead decision variables, and the subproblem determines an extreme distribution of the probability density. The iteration steps and computing time with different number of historical scenarios (M) and amount of reference samples (K) are displayed in Table 7. As indicated in the table, the iteration times of the proposed solution are few, and the calculation time is only tens of seconds, which validates the computational efficiency of the solution. In addition, the convergence of the two-stage solution is related to the convergence gap ε. We change the convergence gap and compare the optimization results. As shown in Table 8, the optimized objective descends when the convergence condition becomes strict. Although the iteration times and calculation time increase correspondingly, the computational efficiency of the proposed algorithm is still very high.