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

Yang Li, Chunling Wang, Guoqing Li, Chen Chen

arXiv:2103.04723v1eess.SPeess.SY

TL;DR

The paper tackles integrated energy system scheduling when renewable generation is uncertain and operator and user interests must be balanced. It combines a Stackelberg game with uncertainty handling, district-heating-network modeling, and flexible thermal comfort, then solves the resulting formulation through deterministic reformulation and optimization. The reported cases achieve equilibrium, promote renewable consumption, reduce consumers’ energy costs, maintain acceptable thermal comfort, and support applicability to a real Chinese system.

  • Problem

    IES scheduling must address uncertain renewable generations and thermal-load models that inadequately represent users’ varying thermal-comfort requirements.

  • Method

    The paper proposes a Stackelberg game framework with price-based IDR, a time-delay and thermal-attenuation DHN model, PMV-based comfort, and SOT-based deterministic reformulation of spinning-reserve chance constraints.

  • Results

    The approach achieves Stackelberg equilibrium, promotes renewable consumption, reduces consumers’ energy costs, and maintains thermal comfort within an acceptable range across two cases.

  • Takeaways & Limitations

    A real integrated energy system in China validates the proposed approach’s applicability to realistic applications.

  • Takeaways & Limitations

    The study ignores the secondary heating network and assumes users’ heat demands are approximately equivalent to total heating loads at heat exchangers.

Abstract

from arXiv · show

In order to balance the interests of integrated energy operator (IEO) and users, a novel Stackelberg game-based optimization framework is proposed for the optimal scheduling of integrated demand response (IDR)-enabled integrated energy systems with uncertain renewable generations, where the IEO acts as the leader who pursues the maximization of his profits by setting energy prices, while the users are the follower who adjusts energy consumption plans to minimize their energy costs. Taking into account the inherent uncertainty of renewable generations, the probabilistic spinning reserve is written in the form of a chance constraint; in addition, a district heating network model is built considering the characteristics of time delay and thermal attenuation by fully exploiting its potential, and the flexible thermal comfort requirements of users in IDR are considered by introducing a predicted mean vote (PMV) index. To solve the raised model, sequence operation theory is introduced to convert the chance constraint into its deterministic equivalent form, and thereby, the leader-follower Stackelberg game is tackled into a mixed-integer quadratic programming formulation through Karush-Kuhn-Tucker optimality conditions and is finally solved by the CPLEX optimizer. The results of two case studies demonstrate that the proposed Stackelberg game-based approach manages to achieve the Stackelberg equilibrium between IEO and users by the coordination of renewable generations and IDR. Furthermore, the study on a real integrated energy system in China verifies the applicability of the proposed approach for real-world applications.

1 Introduction

The paper addresses renewable-generation uncertainty and limited thermal-load flexibility in integrated energy system scheduling through a Stackelberg game framework coordinating the operator and users. It models heating-network dynamics and comfort, transforms the chance constraint into a deterministic form, and evaluates the approach on two systems.

  • Research gaps: Prior IES scheduling studies promoted renewable consumption or demand response but often neglected renewable-generation uncertainty and adaptable thermal-comfort requirements.The review identifies single-type renewable integration and simplified heating-load representations as remaining gaps.
  • Stackelberg framework: The proposed framework coordinates uncertain renewable generations and price-based IDR while balancing the profits of the IEO and energy costs of users.The IEO sets prices as leader, and users adjust consumption plans as followers.
  • Integrated modeling: The scheduling model incorporates district-heating-network time delay and thermal attenuation, together with a PMV-based representation of flexible user thermal comfort.These features extend the treatment of heating loads beyond fixed proportions or fixed temperature ranges.
  • Solution methodology: Sequence operation theory converts the probabilistic spinning-reserve chance constraint into a deterministic equivalent for solving the scheduling model.The resulting formulation supports stable and fast computation of the Stackelberg equilibrium.
  • Evaluation: Case studies show that the approach efficiently obtains Stackelberg equilibrium, promotes renewable consumption, and maintains consumers’ thermal comfort within an acceptable range.The study includes two IES cases, including a real-world integrated energy system in China.

2 Physical modeling of IES

The IES model combines conventional and renewable generation, storage, heat exchange, district heating, and controllable electric and heating loads. Renewable uncertainty, heating-network delay and attenuation, and flexible thermal comfort are represented in the physical models.

  • The studied IES includes CHP and thermal power units, photovoltaic panels, wind turbines, battery storage, heat exchangers, and controllable loads.
  • 2.1 Stochastic photovoltaic unit model: Photovoltaic generation is modeled from solar irradiance using a Beta distribution and a conversion relationship involving PV radiation area and efficiency.
  • 2.2 Stochastic wind turbine unit model: Wind generation is modeled using a Weibull wind-speed distribution and a piecewise power curve defined by rated, cut-in, and cut-out wind speeds.
  • 2.3 District heating network model: The district heating network model represents transported heat, pipeline losses, and water-flow delay while adopting quality regulation and omitting the smaller secondary network.The transported heat depends on water flow and supply-return temperature difference, while heat loss depends on pipeline length and thermal resistance.
  • 2.4 User load models: Electric demand aggregates fixed and time-shiftable loads, while heating demand can be reduced within users’ acceptable thermal-comfort range using PMV-based constraints.Actual heating load is defined as the initial load minus the cuttable heating load.

3 Proposed Stackelberg game model for IES scheduling

The Stackelberg model assigns the IEO as leader and users as followers to coordinate pricing and energy consumption. The IEO maximizes profit, while users minimize energy costs in response to prices.

  • The IEO sets energy prices to maximize profit, and users adjust their energy-consumption plans to minimize energy costs.
  • IDR prevents monopoly risk in the private IEO setting by providing an important means to prevent the IEO from suppressing users.

3.1 Stackelberg game model formulation

The proposed Stackelberg game models the IEO as a profit-maximizing leader that sets energy prices and users as cost-minimizing followers that adjust consumption. The formulation includes operational, reserve, heating-network, and real-time pricing constraints.

  • The IEO sets energy prices to maximize profit, while users adjust energy consumption plans to minimize energy costs.
  • The transaction process prioritizes pricing according to load demands, followed by users’ demand responses to price information.
  • The game is represented by participant sets, strategy sets for the IEO and users, and their respective profit functions.
  • The leader’s constraints cover TP, CHP, and BESS power outputs, probabilistic spinning reserve, DHN pipeline temperatures, and real-time pricing.

3.2 Stackelberg game model of IEO

The IEO maximizes profit from energy transactions while accounting for generation, reserve, balance, ramping, temperature, pricing, and renewable-uncertainty constraints. Spinning reserve is modeled as a chance constraint based on renewable-generation probabilities.

  • 3.2.1 Objective function: The IEO objective maximizes profit, defined as total income minus generation, storage, and reserve operating costs.
  • 3.2.1 Objective function: Income depends on real-time electricity and thermal prices, while costs include TP and CHP generation, BESS operation, and reserve capacities.
  • 3.2.2 Constraints: Supply-demand balance accounts for TP and CHP output, renewable consumption, BESS charging or discharging, and load.
  • 3.2.2 Constraints: TP and CHP units must satisfy ramp-down and ramp-up rate limits during scheduling.
  • 3.2.2 Constraints: Pipeline supply and return temperatures and real-time electricity and thermal prices are restricted to allowable and policy-based ranges.
  • 3.2.2 Constraints: The spinning reserve requirement is expressed as a chance constraint with confidence level  using the expected renewable generation and reserve capacities.

3.3 Stackelberg game model of users

The user-level model minimizes energy costs while incorporating thermal-comfort penalties and flexible electric and heating demand. Time-shiftable and cuttable heating loads are bounded by operational limits and PMV-based comfort requirements.

  • 3.3.1 Objective function: Users minimize energy cost while accounting for electricity, thermal energy, and penalties associated with reduced heating comfort.
  • 3.3.1 Objective function: The thermal-comfort penalty factor represents users’ requirements, and its penalty cost increases when heating load reduction lowers comfort.
  • 3.3.2 Constraints: Time-shiftable load is constrained by its proportion of total electric load, minimum and maximum values, and associated multipliers.
  • 3.3.2 Constraints: Cuttable heating load cannot exceed its maximum allowable reduction while preserving the minimum heating-load requirement.

4 Model solving

The solution converts renewable-generation uncertainty and the leader-follower game into a tractable optimization model. Sequence operation theory handles the chance constraint, while KKT and Big-M transformations yield a mixed-integer quadratic program solved by CPLEX.

  • 4 Model solving: Sequence operation theory converts the probabilistic spinning-reserve chance constraint into a deterministic equivalent form.
  • 4.2 Transformation of bilevel programming: The transformed model represents the Stackelberg game as bilevel programming with separate upper- and lower-level decision makers.
  • 4.1 Deterministic transformation of chance constraint: PV and WT outputs are represented by probabilistic sequences, and their joint renewable output is obtained through addition-type convolution.
  • 4.1 Deterministic transformation of chance constraint: The joint renewable probabilistic sequence is organized by discretization step size and sequence length in Table 2.
  • 4.2 Transformation of bilevel programming: KKT optimality conditions replace the lower-level problem, and the Big-M method linearizes the resulting complementary constraints.
  • 4.3 Existence and uniqueness of equilibrium: The approach establishes equilibrium under continuity and concavity conditions, with uniqueness requiring unique user and IEO optima.
  • 4.4 Solution process: The workflow builds the game, generates renewable sequences, transforms the chance constraint, applies KKT and Big-M reformulations, and solves the final model with CPLEX.

5 Case study

The case studies evaluate the proposed approach under four operating modes, comparing independent and joint optimization with differing consideration of users, district-heating-network characteristics, and integrated demand response.

  • Case-study design: Two cases test the approach on a modified IEEE 30-bus system with two 6-bus district heating systems and on a real integrated energy system in China.The second case examines applicability to real-world applications.
  • Compared operating modes: Four modes compare independent optimization without users’ costs or DHN characteristics, independent optimization with DHN characteristics, joint optimization with DHN characteristics and IDR, and independent user optimization.Mode 3 is the study’s adopted joint-optimization mode.
  • Compared operating modes: Mode 3 jointly optimizes IEO and users while incorporating DHN characteristics and IDR.

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

The first case studies a modified IEEE 30-bus system coupled with two 6-bus district heating systems. Results show that DHN characteristics and IDR increase renewable consumption while balancing IEO profit and users’ energy costs.

  • System and inputs: The test system combines a modified IEEE 30-bus system with two 6-bus district heating systems, each heat exchanger supplying 10 buildings.
  • System and inputs: The 24-hour scheduling cycle is divided into 1-hour periods, with supply-pipeline parameters specified for the district heating networks.
  • Renewable-energy consumption: At the 95% confidence level, renewable consumption increases progressively from modes 1 to 3, with mode 3 completely absorbing renewable power outputs in this case.
  • District-heating effects: DHN characteristics create a time-shifting effect in equivalent heating loads, reducing overlap between heavy heating-load periods and high wind-generation periods.
  • Pricing and reserves: Real-time pricing shifts electric demand from peak to off-peak periods, while higher confidence levels continuously increase required spinning reserve and operating cost.
  • Integrated demand response: Mode 3 reduces heating loads in different periods without violating the lower limit, trading some thermal comfort for lower energy costs within an acceptable range.
  • Economic analysis: Mode 3 places both IEO profit and users’ energy cost between the corresponding outcomes of modes 2 and 4, indicating a Stackelberg equilibrium between their benefits.

5.2 Case 2: a real integrated energy system in China

The second case applies the scheduling approach to a real integrated energy system in Jilin Province, China. The results report complete renewable-energy consumption under mode 3 and a financial balance between IEO and users.

  • Real-system setting: The real system contains 10 heat exchangers, each serving 200 households with an average heating area of 100 m2 per household.
  • Real-system setting: The real system uses power and heating-load data from a Jilin Province grid and real-time electricity and thermal price parameters based on real prices.
  • Renewable-energy consumption: Renewable energies are completely consumed under mode 3, while mode 1 absorbs the least and mode 2 lies between modes 1 and 3.
  • Economic applicability: The proposed approach achieves a Stackelberg equilibrium between IEO profits and users’ energy-consumption costs in the real system.
  • Economic applicability: The real-system results demonstrate applicability of the presented method to realistic applications.

6 Conclusion

The paper concludes that its Stackelberg-game scheduling framework balances IEO and user benefits by coordinating renewable generation and IDR. Across two cases, it promotes renewable consumption, reduces users’ energy costs, preserves acceptable thermal comfort, and applies to a real Chinese system.

  • Main conclusions: The leader-follower Stackelberg game achieves an equilibrium between the benefits of the IEO and users.
  • Main conclusions: Considering DHN characteristics and flexible thermal comfort in IDR promotes renewable consumption and reduces consumers’ energy costs while keeping thermal comfort within an acceptable range.
  • Solution performance: The developed solution approach stably and quickly obtains the optimal IEO and user solution under Stackelberg equilibrium.
  • Validation: Tests on two cases verify the performance of the adopted theoretical models and solution methodology.
  • Scope and future work: A real integrated energy system in China validates the approach’s applicability to realistic applications, while future work will extend it to integrated electrical–heat–gas systems.
  • Scope and future work: The secondary heating network is ignored in this paper, and a more realistic scenario is identified for future consideration.
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