Optimal Scheduling of Integrated Demand Response-Enabled Integrated Energy Systems with Uncertain Renewable Generations: A Stackelberg Game Approach

1 Introduction

• The introduction frames integrated demand response as coordinating flexible loads, storage, and energy-conversion equipment through multiple-energy coupling to improve system flexibility and energy-use efficiency.It contrasts chance-constraint programming’s reliability-economy trade-off with the conservatism of robust programming.Source: §1 · p. 2

1.1 Literature review

• Prior studies address renewable consumption, demand response, and game optimization, but the reviewed work does not combine uncertain multiple renewables, adaptable heating comfort and district heating network characteristics, and stakeholder balancing through game theory.The cited game-optimization studies do not consider renewable generations in integrated energy system scheduling.Source: §1.1 · p. 2

1.2 Contributions and paper organization

• The framework coordinates multiple uncertain renewable generations and price-based integrated demand response to balance the interests of integrated energy operators and users.The operator sets energy prices while users adjust consumption plans to minimize their costs.Source: §1.2 · p. 2
• The scheduling model captures district heating network time delay and thermal attenuation, and represents users’ flexible thermal comfort requirements with a predicted mean vote index.Source: §1.2 · p. 2
• Sequence operation theory converts the spinning-reserve chance constraint into a deterministic equivalent, and the paper proves existence and uniqueness of the Stackelberg equilibrium.The authors state that this supports stable and fast solution of the equilibrium.Source: §1.2 · p. 2
• The proposed method is applied to two integrated energy systems, including a real-world system in China, and is reported to promote renewable consumption while keeping consumer thermal comfort within an acceptable range.The paper also reports that it obtains the Stackelberg equilibrium efficiently.Source: §1.2 · p. 2

2.1 Stochastic photovoltaic unit model

• PV output is modeled from solar irradiance, treated as Beta-distributed, together with panel area and conversion efficiency.The model derives PV power using these inputs.Source: §2.1 · p. 3

2.2 Stochastic wind turbine unit model

• Wind speeds follow a Weibull distribution, while turbine output is represented by a piecewise function using rated output and wind-speed limits.The model identifies actual, rated, cut-in, and cut-out wind speeds as relevant quantities.Source: §2.2 · p. 3

2.3 District heating network model

• The district-heating model represents thermal attenuation and flow-related delay, excludes the smaller secondary network, and approximates user heat demand by total heat-exchanger loads.It adopts quality regulation and models transported heat and pipeline heat loss.Source: §2.3 · p. 4

2.4.1 Electric load model; 2.4.2 Heating load model

• Electric demand aggregates fixed and time-shiftable loads, while heating demand follows building thermal dynamics and can be reduced within users’ PMV-based comfort limits.Users are described as more sensitive to heat changes during daytime, so daytime comfort requirements are higher than nighttime requirements.Source: §2.4.1 · p. 4, §2.4.2 · p. 4

• The private IEO maximizes profit by setting prices, while users minimize energy costs by adjusting consumption; IDR prevents monopoly risk despite transactions occurring only between users and the IEO.Source: §3 · p. 5

3.1 Stackelberg game model formulation

• The IEO prioritizes its pricing strategy based on load demands, after which users adjust their demand in response to prices.The transaction process is depicted in Fig. 3.Source: §3.1 · p. 5, Figure 3 · p. 5
• The game comprises the IEO and users, with respective period-t strategy sets {μ_t, γ_t} and {P_SL_t, P_CL_t}.The model denotes the game as a Stackelberg game in equation (14).Source: §3.1 · p. 5
• The game’s profit set is {F1, F2}.These are defined as the set of profits in the game.Source: §3.1 · p. 5

3.2.1 Objective function

• The IEO maximizes profit as total income minus operating costs, including generation, storage, and reserve costs.The objective accounts for costs associated with TP and CHP generation, BESS operation, and reserve capacity.Source: §3.2.1 · pp. 5–6

3.2.2.1 Power balance constraints

• Supply-demand balance must hold over the scheduling horizon, accounting for renewable-energy consumption.Source: §3.2.2.1 · p. 6

3.2.2.2 Ramp rate constraints

• TP and CHP units must satisfy ramp-up and ramp-down limits.Source: §3.2.2.2 · p. 6

3.2.2.3 Pipeline temperature constraints

• Supply- and return-water temperatures must remain within pipeline allowable limits.Source: §3.2.2.3 · p. 6

3.2.2.4 Real-time price (RTP) constraints

• Real-time electricity and thermal prices are constrained according to energy-policy and fairness considerations.The constraints specify minimum, maximum, and average values for both price types.Source: §3.2.2.4 · p. 6

3.2.2.5 Spinning reserve constraints

• Spinning reserve is imposed as a chance constraint at confidence level ϑ, using expected renewable generation; other unit and CHP coupling constraints are referenced and.The passage defines the probability of an event and expected renewable generation in the reserve formulation.Source: §3.2.2.5 · p. 6

3.3.1 Objective function

• Users minimize energy cost with a thermal-comfort penalty for heating-load cuts; the study uses a unified penalty coefficient across users [27].The study also analyzes how different penalty coefficients affect cuttable heating loads.Source: §3.3.1 · p. 7

3.3.2.1 Time-shiftable electric load constraints

• Time-shiftable electric load is constrained by its share of total electric load and by minimum and maximum load limits.The formulation also includes an equality constraint and lower- and upper-bound multipliers.Source: §3.3.2.1 · p. 7

3.3.2.2 Cuttable heating load constraints

• Cuttable heating load is capped to preserve basic heating needs, subject to a minimum heating-load value.The constraints include lower and upper bounds on cuttable heating load.Source: §3.3.2.2 · p. 7

3.3.2.3 PMV index constraints

• PMV index constraints quantify the minimum heating load, with the supplied constraint text specifying 0.5 for 7:00–20:00.Source: §3.3.2.3 · p. 7

4.1 Deterministic transformation of chance constraint

• SOT represents renewable outputs with probabilistic sequences and converts the spinning-reserve chance constraint into a deterministic equivalent.WT and PV sequences are combined by addition-type convolution to obtain the joint renewable-output sequence.Source: §4.1 · pp. 7–8

4.2 Conversion of KKT optimality condition

• The lower-level problem is transformed into KKT optimality conditions, and the resulting nonlinear complementary constraints are linearized using the Big-M method.The KKT conditions are incorporated into the upper level to convert the bi-level formulation into a mixed-integer quadratic programming form.Source: §4 · p. 7, §4.2 · pp. 8–9

4.3.1 Proof of existence

• The Stackelberg game has an equilibrium when the IEO profit and user cost are nonempty and continuous in both strategy sets and user cost is concave in users’ strategy.The proof concludes that user cost is concave in the users’ load decisions.Source: §4.3.1 · p. 9

4.3.2 Proof of uniqueness

• A unique equilibrium exists under the stated conditions when users have a unique cost minimum for a given IEO strategy and the IEO has a unique profit maximum for given user strategies.The proof establishes a unique user optimum and a unique IEO profit maximum over the defined domain.Source: §4.3.2 · p. 9

4.4 Solution process

• The solution process builds the game, generates renewable-output sequences with SOT, transforms the chance constraint, and applies KKT and Big-M reformulations before solving with CPLEX.The process then outputs the optimal integrated-energy-system scheduling solution.Source: §4.4 · pp. 9–10

Case study

• The evaluation uses two cases: a modified IEEE 30-bus system with two 6-bus district heating systems and a real integrated energy system in China.The real-system case is intended to verify applicability to real-world applications.Source: §5 · p. 10
• Four modes are compared, and mode 3—joint optimization of the operator and users with district heating network characteristics and integrated demand response—is adopted.Modes 1, 2, and 4 independently optimize the operator or users; modes 1 and 2 set heating loads at users’ optimal thermal-comfort state (PMV=0).Source: §5 · p. 10
• In modes 1 and 2, electricity and thermal prices are constant and proportional to electric and heating loads, respectively.Mode 1 excludes user costs and district heating network characteristics; mode 2 includes district heating network characteristics.Source: §5 · p. 10
• Mode 4 independently optimizes users while considering district heating network characteristics.Source: §5 · p. 10

5.1 Case 1: Integrated IEEE 30-bus system and two 6-bus DHSs

• The case couples a modified IEEE 30-bus system with two 6-bus DHSs, each HE serving 10 buildings, over a 24-hour horizon with 1-hour periods.A typical winter day is used for subsequent analysis.Source: §5.1 · pp. 10–11

5.1.1 Analysis of renewable energy consumption

• Renewable consumption rises from modes 1 to 3, and mode 3 completely absorbs renewable power output in this case.The study attributes the increase to considering district-heating-network characteristics and integrated demand response.Source: §5.1.1 · p. 11, Figure 6 · p. 11, Table 4 · p. 11

5.1.2 Analysis of the characteristics of district heating network

• District-heating-network characteristics shift equivalent heating load over time, reducing overlap between heavy heating demand and high wind output; pipeline thermal attenuation has no significant scheduling impact.The study attributes the limited attenuation impact to the system’s small scale.Source: Figure 7 · p. 11, §5.1.2 · p. 12, Table 5 · p. 12

5.1.3 Analysis of integrated demand response

• IDR shifts electricity toward off-peak periods and reduces heating without crossing its lower limit, trading some acceptable thermal comfort for lower costs and greater renewable consumption.Reduced heating demand also lowers electric output from heat-set CHP units, promoting renewable consumption.Source: Figure 10 · p. 12, Figure 8 · p. 12, §5.1.3 · p. 13

5.1.4 Impact of penalty factors on cuttable heating load

• Cuttable heating load reaches its maximum at penalty factor 0.6 and minimum at 1; lower penalties especially increase nighttime curtailment, highlighting the role of comfort preferences.Nighttime thermal sensation is described as less sensitive, and the penalty factor represents users’ demand for thermal comfort.Source: Figure 12 · p. 13

5.1.5 Analysis of confidence levels

• Required spinning reserve increases with confidence level, improving reliability to some extent while increasing system operating cost.The study identifies confidence-level selection as a trade-off between reliability and economy.Source: Figure 13 · p. 13, §5.1.5 · p. 13

5.1.6 Economy analysis

• Mode 3 places both IEO profit and users’ energy cost between modes 2 and 4, demonstrating a Stackelberg equilibrium of stakeholder benefits.The comparison attributes the balance to simulation of the stakeholders’ interactive decisions as they pursue their respective objectives.Source: §5.1.6 · p. 14, Table 6 · p. 14

5.2 Case 2: a real integrated energy system in China

• The Jilin Province system comprises 10 HEs serving 2,000 households, with 100 m2 average heating area per household and 5.17 MW total TP and CHP capacity.Power and heating load data and real-time price parameters are drawn from a real Jilin grid; PV and WT outputs are each rated at 300 kW, and BESS capacity ranges from 200 to 900 kW·h.Source: §5.2 · p. 14

5.2.1 Analysis of renewable energy consumption

• Renewable energy consumption is lowest under mode 1, highest under mode 3, and intermediate under mode 2; mode 3 consumes all renewable energy.The authors attribute the increased consumption to the district heating network characteristics and IDR, consistent with Case 1.Source: Figure 14 · p. 14, §5.2.1 · pp. 14–15

5.2.2 Economy analysis

• The approach achieves a Stackelberg equilibrium between the IEO’s profits and users’ energy costs in the real system, indicating financial balance among stakeholders.The authors present this as evidence of applicability to real-world systems.Source: Table 9 · p. 15, §5.2.2 · p. 15

5.2.3 Analysis of calculation efficiency

• Table 10 presents calculation times under different operating modes.Source: Table 10 · p. 15

6 Conclusion

• The leader–follower Stackelberg game achieves equilibrium between the benefits of the IEO and users.The framework schedules integrated energy systems by coordinating renewable generation and integrated demand response.Source: §6 · p. 15
• The approach promotes renewable consumption and reduces consumers’ energy costs while keeping thermal comfort within an acceptable range.These outcomes are attributed to modeling district heating network characteristics and users’ flexible thermal comfort requirements in IDR.Source: §6 · p. 15
• The solution approach stably and quickly obtains the optimal solution under Stackelberg equilibrium, and two cases verify the theoretical models and solution methodology.Source: §6 · p. 15
• A real integrated energy system study in China supports the approach’s applicability to realistic applications.Future work will extend the approach to integrated electrical–heat–gas energy management and consider the secondary heating network, which was ignored in this paper.Source: §6 · p. 15