Distributionally Robust Collaborative Dispatch of Integrated Energy Systems with DNE Limits Considering Renewable and Contingency Uncertainties

Abstract


I. INTRODUCTION
With the advancement of renewable energy generation and green energy consumption, the increasing penetration of renewable energy has become an emerging trend for future power systems [1]- [3]. However, the inherent uncertain nature of renewable generation can pose an immense challenge to the safe and stable operation of integrated energy systems (IES). The combined heat and power (CHP) Manuscript  unit has the characteristics of quick start & stop, which can effectively mitigate the fluctuation of renewable output. Therefore, the CHP units, considered as the coupling elements to facilitate the renewable energy consumption in the IES, have raised increasing attention.
On this background, many researchers have proposed approaches to handle the uncertainty of renewable energy, such as stochastic programming (SP) [4], [5] and robust optimisation (RO) [6]- [8]. The authors in [9] proposed a stochastic programming method using probabilistic scenarios to deal with the uncertainty of wind power. This method relies on obtaining the probability distributions of the wind power forecast error in a priori, but the accuracy and rationality are difficult to verify. The authors in [10] established a scheduling model based on the interval optimisation method, but interval representation of dispatch decisions is not easy to use for practical implementations. Despite the merits of SP and RO methods, the following three problems remain unsettled: 1. The decision-making results obtained by RO can be too conservative, while the probability distribution of renewable energy is difficult to accurately estimate; 2. The uncertainty of contingency events, such as equipment and line outages, is not considered in the dispatch model, which may affect the stability of system operations; 3. Considering the renewable energy as non-dispatchable resources, it is too costly to balance the power-load mismatches by regulating other dispatchable resources, especially in the case of high proportion of renewable integration.
To address the first challenge, the distributionally robust optimisation (DRO) is applied because it can combine the probabilistic distribution information of uncertain parameters and the technical bounds of uncertainty. DRO quantifies an Distributionally Robust Collaborative Dispatch of Integrated Energy Systems with DNE Limits Considering Renewable and Contingency Uncertainties ambiguity set to represent distribution information, which is particularly useful given limited datasets and could provide less conservative results than traditional RO [11]. The authors in [12] established an ambiguity set for wind power constrained by combined conditions of the l 1 -and l ∞ -norm, where dispatch decisions of electric-gas coupled systems are made in the worst-case scenario. In [13], the wind power prediction error is assumed to follow the normal distribution, and its mean and variance are characterised by the uncertainty set in the form of a polyhedron. In [14], statistical values of the mean and covariance matrix of the wind power forecast error were obtained to solve the scheduling results under the worst probability distribution, which satisfies the given moment information obtained from the sampling set.
Instead of restricting to a specific distribution type, the authors in [15] utilised the moment information of the wind power output, which is incorporated into the piecewise linear model, but the spatial-temporal correlation of neighbouring wind farms is not considered. The authors in [16] presented a two-stage DRO model, which quantifies the renewable consumption rate by an affinely adjustable mechanism, to jointly optimise power dispatch outputs and spinning reserves.
In terms of the second challenge, in [17], the N-k criterion was used to quantify the operating reserves of the system for possible contingencies, such as equipment failure and line outages. The authors in [18] adopted a heuristic rule that reserve requirements are set as a percentage of the expected renewable output and load. However, a critical concern is whether the committed reserve is deliverable or not when congested issues occur. For this reason, a stochastic programming method was presented to optimise the spinning reserves within an acceptable risk level. In [19], the operational flexibility of the system was enhanced by employing reserve margins through an offline training algorithm. This offline approach eliminated the real-time computational challenge of stochastic programming. In [20], a distributionally robust framework was constructed by capturing the distribution of fault events in an interval ambiguity set. However, in-depth analysis and research is yet to be done on how to fully account for the combined impact of both contingency and renewable uncertainty.
The concept of do-not-exceed (DNE) limit can effectively tackle the third challenge and has been widely adopted and incorporated in the optimal dispatch model of renewable integrated systems. The DNE limit is defined to allow an acceptable range of renewable energy output power to ensure the safety of system operations. This provides a clear guide on the use of renewable energy and measures the maximum capacity of the system to accommodate renewable power. In [21], the rate of consumption of renewable energy is obtained based on the dispatch results given, but this is not the best strategy to coordinate the renewable power and the generation units, which can underestimate the ability of the system to adapt to renewable energy. The DNE limit is presented in [22] without accounting for the renewable power curtailment and load reduction, and a stringent renewable energy control is adopted. In recent years, many studies have advocated for optimising the power dispatch output and the DNE limit simultaneously. In [23], an adjustable RO model is established to jointly optimise the DNE limit and the power scheduling strategy. In [24], an affine policy rule is proposed for an RO dispatch model, in which the specific coefficients are optimised to quantify the DNE limit, as well as the power dispatch level of the generation units.
To address the research gap mentioned above, we have implemented a robust cooperative optimal dispatch approach for IES. The primary contributions of this study can be outlined as follows.
1. A two-stage distributionally robust framework is proposed as an alternative to the traditional robust or stochastic methods, which can result in dispatch outcomes that are either excessively radical or overly conservative. In our approach, the reserves of IES and the DNE limit are co-optimised, leading to improved utilisation of renewable energy generation.
2. An adjustable ambiguity set is established to effectively address the uncertainty associated with both the continuous output of renewable energy and the discrete probability of contingencies.
3. Various techniques are employed including absolute value linearisation, duality theory, and Karush-Kuhn-Tucker (KKT) conditions to transform the dispatch model into a manageable mixed integer linear programming (MILP) problem that can be solved directly by commercial solvers. The following sections of this paper are arranged as follows. In Section II, we present the mathematical formulation of the deterministic optimal dispatch model of IES. Section III outlines the distributionally robust framework that considers uncertainties arising from renewable energy and contingencies. Section IV presents the numerical results of the proposed method in the test case using different scenarios. Lastly, in Section V, we conclude this paper.

II. DETERMINISTIC OPTIMAL DISPATCH MODEL OF IES
The goal of the optimal dispatch model of IES is to minimise the operational cost by considering the fuel cost, start-up cost, no-load cost, and spinning reserve cost of the unit. It is noted that the CHP unit is kept ON during the period and threfore there is no start-up cost for the CHP unit [25] where: which is subject to the following constraints: ,, ,1 min 0 min , , HST, HST, HST, HST, where (2) and (3) represent the operational costs of conventional thermal and CHP units. Constraints (4) and (5) link the start-up u gt and shut-down v gt binary variables to the binary commitment variables x gt . Constraint (6) enforces the minimum and maximum generation capacity limits. Constraints (7) and (8) enforce the minimum up-and down-time requirements. Constraint (9) enforces ramp-up and -down limits. Constraints (12)- (15) are, respectively, the electric output constraint, the ramp rate constraint, and the spinning reserve constraint for CHP units. Constraint (16) enforces the output limits of the renewable power plant. Equation (17) (25) and (26) are the constraints of power and heat balance at the system level. Constraint (27) enforces the limits of the transmission line.

A. Uncertainty Modelling of Renewable Energy
In this section, an ambiguity set characterising renewable power based on the Wasserstein measure is presented, where the empirical distribution of renewable output is used as a reference to estimate its actual probability density function. Given the historical sampled dataset 12 , , ..., ,   the empirical distribution N is a stepwise function, which can be expressed as follows According to the law of large numbers, it is proven that the reference distribution will finally converge to the real distribution with increasing input data. The set of ambiguities for the forecast renewable power can be defined as Combined with the above formulas, the comprehensive set of ambiguity of the contingency probability can be further expressed as , , where C t  and T t  are introduced to control the size of the ambiguity set, which represents the conservativeness of the system. Considering the probability of contingency, the unit and system constraints (6), (12), (25)

. Optimal Dispatch Model Incorporating the DNE Limit
When incorporating the DNE limit into the aforementioned optimal dispatch model, the goal of the power system is to adjust the power for rescheduling so that the accommodated power of renewable energy can meet the DNE limit. Assume that the system uncertainty at time t is , t  which satisfies ,.

Formulation of a Two-Stage Distributionally Robust Optimal Dispatch Model
In general, this paper presents a two-stage distributionally robust optimal dispatch model of spinning reserves and DNE limits in IES considering the uncertainty of renewable energy and system contingencies. The two-stage decision-making structure can be rewritten in the following min-max-min form

s.t. (4)-(27), (31)-(46) and (48)-(51).
Due to the quadratic form of the operation cost in the objective function, we could alternatively use the piecewise linear expression to approximate the quadratic function for computational tractability.
The objective function of the second stage is to minimise the regulation cost of the unit: The maximisation problem of formula (53) is transformed into the equivalent minimisation problem (55)  After solving (57), the upper bound of the worst-case scenario and the objective function in the set of ambiguities can be found. This problem can be solved by using Karush-Kuhn-Tucker (KKT) conditions as follows: where k  and j  are the Lagrange multipliers. Once the KKT conditions are applied, the entire optimisation problem is transformed into a single-level mixed integer linear programming (MILP) model, which can be directly solved by commercial solvers such as CPLEX and Gurobi.

A. Case Study Setup
The integrated test energy system is illustrated in Fig. 1, which consists of an IEEE 33-bus power distribution network and a modified 33-bus heating network.
It generally contains 3 CHP units, 2 energy storage systems (ESS), 2 electricity boilers (EB) and 2 heat storage tanks (HST), 1 wind farm and 1 solar photovoltaic (PV) plant. Detailed information on the topological structure and load data can be found in [26]. The CHP unit (CHP 1) with flexible and adjustable energy conversion ratio is considered a slack bus. Various cost coefficients and energy prices can be derived from [27]. The confidence interval of the Wasserstein ball is set to 95 % and the control parameters are set as C 10, To investigate the influence of renewable and load uncertainty on the dispatch results, we assume that the fluctuation range covers 20 % of the forecasted value, which is shown in Fig. 2. The proposed MILP model is solved by GAMS/CPLEX on a desktop computer with a quadcore 3.6 GHz processor and 12 GB memory.

43
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 29, NO. 3,2023 For comparison, in this paper, three scenarios are set for analysis to verify the effectiveness and reliability of the proposed model:  Case 1: The distributionally robust optimal dispatch model of IES without considering the uncertainty of the probability of the contingency and the limit of the DNE, which is named "DROD";  Case 2: The distributionally robust optimal dispatch model of IES considering the uncertainty of the probability of contingency but without accounting for the limit of the DNE, which is named "DROD-CP";  Case 3: The distributionally robust optimal dispatch model of IES collaboratively considering both the uncertainty of the probability of contingency and the DNE limit, which is named "DROD-CP-DNE".

B. Analysis of Numerical Results
The expected system spinning reserves (up and down reserves), renewable energy utilisation ratio (wind farm and PV plant), and load-serving ratio under the three cases are compared and analysed. The numerical results under the three scenarios are given in Table I.  It can be seen that, when considering the uncertainty of system faults, the overall spinning reserve capacity of the DROD-CP and DROD-CP-DNE model is higher than that of the DROD model. This is because when there is an unexpected equipment failure in the system, the DROD-CP and DROD-CP-DNE models will increase the demand for spinning reserve capacity and improve the utilisation ratio of renewable energy, while ensuring that the demand of the system stays as stable as possible. If equipment failure is neglected as in the DROD model, when the system encounters an emergency shutdown, it will result in load shedding or renewable curtailment, and thus the DROD model has the lowest load-serving ratio. For the DROD-CP-DNE model, the DNE limit is further incorporated into the DROD-CP model, which further enhances the utilisation ratio of renewable power in the system. Figures 3 and 4 outline the detailed results of the energy dispatch of Case 3 in the power distribution network. The centred optimal dispatch values of versatile resources are given in Fig. 3, while the maximum power variation around the centred values is shown in Fig. 4 in the presence of uncertainties. It can be seen from the figure that there is no amount of wind and PV power curtailment in the proposed dispatch scheme, which shows that the DROD-CP-DNE model can ensure the effective absorption of renewable energy and provide reliable energy supply for users under the influence of the uncertainty of renewable generation and contingencies.  6 illustrate the centred dispatch results as well as its maximum amount of variation for the heat network. Since CHP 2 is more economical than CHP 1, and CHP 3, it takes up the majority part to supply the heat load, while CHP 1 only maintains the minimum thermal output. Moreover, since the power load is larger than the heat load for most of the time, the CHP unit with flexible and adjustable energy conversion ratio (CHP 1) should be responsible for consuming the extra electric load.
Furthermore, the power capacity of the maximum allowable wind power access for the system is analysed under the three scenarios, and the relevant simulation results are statistically presented in Fig. 7. Figure 7 presents the allowable range of access to wind power for a wind farm. As depicted, the DROD-CP-DNE model allows for a wider range of wind power access compared to the DROD-CP and DROD models. This is because the DROD-CP model takes into account the randomness of equipment failure, resulting in an increased demand for renewable energy and indirectly allowing for a greater amount of wind power to be connected to the system compared to the DROD model. Furthermore, the DROD-CP-DNE model considers the wind power absorption rate based on the DROD-CP model, thus improving the system's ability to cope with uncertainty of wind power and further expanding the allowable range of access to wind power.  To study the impact on system operation under different control parameters T , t  we mainly focus on Case 3, the DROD-CP-DNE model, and relevant experimental results are shown in Table II   the conservatism of the proposed model is enhanced, leading to a consistent increase of the operating cost of the system, and the renewable consumption power is also increased. Therefore, the reliability and economics of the model can be controlled by adjusting the parameters. V. CONCLUSIONS In this paper, a distributionally robust cooperative optimal dispatch model for spinning reserve and DNE limits of IES is presented considering the uncertainty of renewable energy and system contingencies. Using Wasserstein measure theory, an adjustable ambiguity set of both renewable power and system/equipment faults is developed. Furthermore, given the day-ahead generation cost and real-time regulation cost, a two-stage distributionally robust optimal dispatch model is formulated and transformed into a single-level mixed integer linear problem (MILP) using duality theory and KKT conditions. The simulation results of the modified IEEE-33 test system are verified and the concluding remarks are summarised as follows: 1. By accounting for the system contingency uncertainty, the distributionally robust cooperative model can simultaneously optimise the spinning reserves, improve the utilisation rate of renewable energy, and enhance the robustness of the system; 2. The proposed collaborative optimisation model can greatly broaden the feasible range of renewable energy integration into the system by co-optimising the do-not-exceed limits, thereby achieving a more favourable trade-off between the overall operation cost and the utilisation of renewable power; 3. The reliability and economy of the system can be adapted by tuning the adjustable parameters of the ambiguity set of contingency probability; It should be noted that the computational complexity of the proposed distributionally robust optimal model is not investigated in this paper. Future work will focus on the solution efficiency of the proposed model for large-scale integrated energy systems.