Abstract:

Abstract:As massive distributed energy resources (DERs) are integrated into distribution networks (DNs) and the distribution automation facilities are widely deployed, the DNs are evolving to active distribution networks (ADNs). This paper introduces the architecture and main function modules of an integrated distribution management system (IDMS) and its applications in China. This system consists of three subsystems, including the real-time operation and control system (OCS), outage management system (OMS), and operator training simulator (OTS). The OCS has a hierarchical architecture with three levels, including the local controller for DER clusters, the optimization of DNs incorporated with multi-clusters, and the coordination operation of integrated transmission & distribution (T&D) networks. The OMS is developed based on the geographical information system (GIS) and coordinated with OCS. While in the OTS, both the ADN and its host transmission network (TN) are simulated to make the simulation results more credible. The main functions of the three subsystems and their interaction data flows are described and some typical application scenarios are also presented.

Abstract:The distribution control center (DCC) has evolved from a sideshow in the traditional distribution service center to a major centerpiece of the utility moving into the decentralized world. Mostly, this is the place where much of the action is happening due to new forms of energy that are coming into the distribution system. This creates the flexibility of operation and increased complexity due to the need for increased coordination between the transmission control center and DCC. However, the US and European utilities have adapted to this change in very different ways. Firstly, we describe the research works done in a DCC and their evolutions from the perspectives of major US utilities, and those enhanced by the European perspective focusing on the coordination of distribution system operator and transmission system operator (DSO-TSO). We present the insights into the systems used in these control centers and the role of vendors in their evolution. Throughout this paper, we present the perspectives of challenges, operational capabilities, and the involvement of various parties who will be responsible to make the transition successful. Key differences are pointed out on how distribution operations are conducted between the US and Europe.

Abstract:From its commissioning in 2006 to the present, the Spanish Control Centre for Renewable Energies (CECRE) has been a worldwide pioneer and reference centre in the integration of renewable energies in the electrical system. In the last 15 years, CECRE has allowed a high penetration of renewable energies in Spanish electrical system by means of the supervision and control of renewable facilities, while ensuring the power supply security. This paper defines the bases of renewable energy operation in CECRE and introduces forecasting systems for renewable energies. Furthermore, this paper describes the participation of renewable energies in balancing services and congestion management. Finally, new challenges for Spanish transmission system operator (TSO) are presented for the integration of renewable energies along with the energy transition and decarbonization.

Abstract:With the rapid integration of distributed energy resources (DERs), distribution utilities are faced with new and unprecedented issues. New challenges introduced by high penetration of DERs range from poor observability to overload and reverse power flow problems, under-/over-voltages, maloperation of legacy protection systems, and requirements for new planning procedures. Distribution utility personnel are not adequately trained, and legacy control centers are not properly equipped to cope with these issues. Fortunately, distribution energy resource management systems (DERMSs) are emerging software technologies aimed to provide distribution system operators (DSOs) with a specialized set of tools to enable them to overcome the issues caused by DERs and to maximize the benefits of the presence of high penetration of these novel resources. However, as DERMS technology is still emerging, its definition is vague and can refer to very different levels of software hierarchies, spanning from decentralized virtual power plants to DER aggregators and fully centralized enterprise systems (called utility DERMS). Although they are all frequently simply called DERMS, these software technologies have different sets of tools and aim to provide different services to different stakeholders. This paper explores how these different software technologies can complement each other, and how they can provide significant benefits to DSOs in enabling them to successfully manage evolving distribution networks with high penetration of DERs when they are integrated together into the control centers of distribution utilities.

Abstract:Microgrid (MG) is a small-scale, self-sufficient power system that accommodates various distributed energy resources (DERs), controllable loads, and future distribution systems. Networked microgrids (NMGs) are clusters of MGs, which are physically interconnected and functionally coordinated to enhance distribution systems in terms of economics, resilience, and reliability. This paper introduces the architecture and control of NMGs including nanogrid (NG) and MG. To accommodate variable DERs in NMGs, master and distributed control strategies are adopted to manage the high penetration of DERs, where master control focuses on economic operation, while distributed control focuses on reliability and resilience through active power sharing and voltage and frequency regulation. The initial practices of NG, MG, and NMG in the networked Illinois Institute of Technology (IIT) campus microgrid (ICM) and Bronzeville community microgrid (BCM) in the U.S. are presented. The applications of the master and distributed control strategies are illustrated for the networked ICM-BCM to show their benefits to economics, resilience, and reliability.

Abstract:The smart grid integrates advanced sensors, a two-way communication infrastructure, and high-performance computation-based control. The distribution management systems for smart grid include several functions for manipulating legacy voltage control devices and distributed energy resources through closed-loop volt/var control, leading to wide-area regulation of voltages in the presence of fluctuating power. The other primary distribution network analysis application is concerned with automatic fault location and service restoration following fault events, aiming to provide the grid with autonomous intelligence for self-healing. Communication technologies are vital to enable the computing applications of distribution networks, whether they work in centralized or distributed modes. This paper presents the state of the art in distribution management system architectures and modern workflows showing data exchange, practical parallel implementations designed to handle large amounts of data, in addition to communication standards that serve as interoperability enablers. It demystifies the relationship between different functions developed independently by power system researchers and shows their operation as a complete system, thus placing them in a better context for future research and development.

Abstract:The exchange of information between transmission system operators (TSOs) and distribution system operators (DSOs) is a common practice. However, the evolution of the regulatory frameworks in Europe has increased the need for enhancing TSO-DSO data exchange and interoperability. This paper provides an overview of the TSO-DSO data exchanges and demonstrates the best practices using International Electrotechnical Commission (IEC) common information model (CIM), including the implementation of IEC common grid model exchange standard (CGMES), and discussion of the corresponding advantages, disadvantages, and challenges. Furthermore, this paper evaluates and reports the activities already carried out within European projects, with particular focus on TSO-DSO interoperability. Finally, this paper concludes the need for TSOs and DSOs to rely on standard-based solutions when performing TSO-DSO data exchange, which enables the efficient operation and development of the future power systems.

Abstract:With the development of renewable energy and the changes in the characteristics of power grid, it is becoming increasingly difficult to balance power supply and demand in space and time. In addition, the requirement for improved dispatching capability of power grid is increasing. Therefore, the potential of flexible load dispatching should be realized, which can promote the large-scale consumption of renewable energy and the construction of new power grid. Based on the analysis of existing load dispatching studies and the differences in the characteristics of domestic and foreign load dispatchings, a technical architecture and several key technologies are proposed for load resources to participate in power grid dispatching under the new situation, i.e., the autonomous collaborative control system of load dispatching. This system implements the multi-layer coordinated control of main, distribution and micro grids (load aggregators). Adjustable load resources are aggregated through an aggregator operation platform and connected with a dispatcher load regulator platform to realize real-time data interaction with dispatching agencies as well as the monitoring, control, and marketing of aggregators. It supports the load resources to participate in network-wide dispatching optimization via continuous power adjustment. Several key technologies such as the control mode, load modeling, dispatching strategy, and safety protection are also elaborated. Through the closed-loop control of orderly charging piles and energy storage clusters in the North China Power Grid, the feasibility of the proposed architecture and key technologies is verified. This route has successively supported multiple adjustable load aggregators to participate in the ancillary services market of North China Power Grid for peak-shaving. Finally, the technical challenges of load resources participating in power grid dispatching under the dual carbon goals are discussed and prospected.

Abstract:Today’s power systems are seeing a paradigm shift under the energy transition, sparkled by the electrification of demand, digitalisation of systems, and an increasing share of decarbonated power generation. Most of these changes have a direct impact on their control centers, forcing them to handle weather-based energy resources, new interconnections with neighbouring transmission networks, more markets, active distribution networks, micro-grids, and greater amounts of available data. Unfortunately, these changes have translated during the past decade to small, incremental changes, mostly centered on hardware, software, and human factors. We assert that more transformative changes are needed, especially regarding human-centered design approaches, to enable control room operators to manage the future power system. This paper discusses the evolution of operators towards continuous operation planners, monitoring complex time horizons thanks to adequate real-time automation. Reviewing upcoming challenges as well as emerging technologies for power systems, we present our vision of a new evolutionary architecture for control centers, both at backend and frontend levels. We propose a unified hypervision scheme based on structured decision-making concepts, providing operators with proactive, collaborative, and effective decision support.

Abstract:Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data in these tasks are typically represented in Euclidean domains. Nevertheless, there is an increasing number of applications in power systems, where data are collected from non-Euclidean domains and represented as graph-structured data with high-dimensional features and interdependency among nodes. The complexity of graph-structured data has brought significant challenges to the existing deep neural networks defined in Euclidean domains. Recently, many publications generalizing deep neural networks for graph-structured data in power systems have emerged. In this paper, a comprehensive overview of graph neural networks (GNNs) in power systems is proposed. Specifically, several classical paradigms of GNN structures, e.g., graph convolutional networks, are summarized. Key applications in power systems such as fault scenario application, time-series prediction, power flow calculation, and data generation are reviewed in detail. Furthermore, main issues and some research trends about the applications of GNNs in power systems are discussed.

Abstract:A nearest-neighbor-based detector against load redistribution attacks is presented. The detector is designed to scale from small-scale to very large-scale systems while guaranteeing consistent detection performance. Extensive testing is performed on a realistic large-scale system to evaluate the performance of the proposed detector against a wide range of attacks, from simple random noise attacks to sophisticated load redistribution attacks. The detection capability is analyzed against different attack parameters to evaluate its sensitivity. A statistical test that leverages the proposed detector is introduced to identify which loads are likely to have been maliciously modified, thus, localizing the attack subgraph. This test is based on ascribing to each load a risk measure (probability of being attacked) and then computing the best posterior likelihood that minimizes log-loss.

Abstract:Heat exchanger systems (HXSs) or heat recovery steam generators (HRSGs) are commonly used in 100 kW to 50 MW combined cooling, heating, and power (CCHP) systems. Power flow coupling (PFC) is found in HXSs and is complex for researchers to quantify. This could possibly mislead the dispatch schedule and result in the inaccurate dispatch. PFC is caused by the inlet and outlet temperatures of each component, gas flow pressure variation, conductive medium flow rate, and atmosphere condition variation. In this paper, the expression of PFC is built by using quadratic functions to fit the nonlinearity of thermal dynamics. While fitting the model, the environmental condition needs prediction, which is calculated using phase space reconstruction (PSR) Kalman filter. In order to solve the complex quadratic dispatch model, a hybrid following electricity load (FEL) and following thermal load (FTL) mode for reducing the dimension of dispatch model, and a feasible zone analysis (FZA) method are proposed. As a result, the PFC problem of CCHP system is solved, and the dispatch cost, investment cost, and the maximum power requirements are optimized. In this paper, a case in Jinan, China is studied. The PFC model is proven to be more precise and accurate compared with traditional models.

Abstract:Accurate short-term prediction of overhead line (OHL) transmission ampacity can directly affect the efficiency of power system operation and planning. Any overestimation of the dynamic thermal line rating (DTLR) can lead to the lifetime degradation and failure of OHLs, safety hazards, etc. This paper presents a secure yet sharp probabilistic model for the hour-ahead prediction of the DTLR. The security of the proposed DTLR limits the frequency of DTLR prediction exceeding the actual DTLR. The model is based on an augmented deep learning architecture that makes use of a wide range of predictors, including historical climatology data and latent variables obtained during DTLR calculation. Furthermore, by introducing a customized cost function, the deep neural network is trained to consider the DTLR security based on the required probability of exceedance while minimizing the deviations of the predicted DTLRs from the actual values. The proposed probabilistic DTLR is developed and verified using recorded experimental data. The simulation results validate the superiority of the proposed DTLR compared with the state-of-the-art prediction models using well-known evaluation metrics.

Abstract:The rapidly increasing wind power penetration presents new challenges to the operation of power systems. Improving the accuracy of wind power forecasting is a possible solution under this circumstance. In the power forecasting of multiple wind farms, determining the spatio-temporal correlation of multiple wind farms is critical for improving the forecasting accuracy. This paper proposes a spatio-temporal convolutional network (STCN) that utilizes a directed graph convolutional structure. A temporal convolutional network is also adopted to characterize the temporal features of wind power. Historical data from 15 wind farms in Australia are used in the case study. The forecasting results show that the proposed model has higher accuracy than the existing methods. Based on the structure of the STCN, asymmetric spatial correlation at different temporal scales can be observed, which shows the effectiveness of the proposed model.

Abstract:Formulating accurate dynamic load models is critical for power system analysis, control, and planning. In this paper, a generic dynamic load model is proposed. The dynamic power response of the load is directly approximated as the superposition of various mathematical functions that produce a dynamic response. Basic physical principles of the dynamic process are reflected in the mathematical functions utilized in the proposed model. First, different stages of the dynamic process are detected based on the continuity of derivatives of the measurement. Second, a complete set of mathematical functions that produce the dynamic responses in electric devices are formulated. Third, a parsimonious set of mathematical functions is selected at each stage by performing feature selection using nonlinear principal component analysis. The proposed model is further formulated based on the parsimonious set of mathematical functions at each stage. Finally, the parameters of the proposed model corresponding to different system events are solved. Based on the formulated model, its possible application in event detection is further analyzed. The proposed model is easy to implement given limited data measurement. Various tests on different system event data are performed to validate the effectiveness of the proposed model. The results show that the proposed model has excellent accuracy and robustness for different system events.

Abstract:We present the ferroresonance overvoltage mitigation concerning the power systems of the grid-connected wind energy conversion systems (WECSs). WECS is considered based on a doubly-fed induction generator (DFIG). Ferroresonance overvoltage associated with a single-pole outage of the line breaker is mitigated by fast regulating the reactive power using the static compensator (STATCOM). STATCOM controller is introduced, in which two incorporated proportional-integral (PI) controllers are optimally tuned using a modified flower pollination algorithm (MFPA) as an optimization technique. To show the capability of the proposed STATCOM controller in mitigating the ferroresonance overvoltage, two test cases are introduced, which are based on the interconnection status of the power transformer used with the grid-connected DFIGs. The results show that the ferroresonance disturbance can occur for the power transformers installed in the wind farms although the transformer terminals are interconnected, and neither side of the transformer is isolated. Furthermore, as a mitigation method of ferroresonance overvoltage, the proposed STATCOM controller succeeds in improving the system voltage profile and speed profile of the wind turbine as well as protecting the system components against the ferroresonance overvoltage.

Abstract:Sufficient fault ride-through (FRT) of large wind power plants (WPPs) is essential for the operation security of transmission system. The majority of studies on FRT do not include all disturbances originating in the transmission system or the disturbances irrelevant to the operation security. Based on the knowledge of power quality, this paper provides a guide to stakeholders in different aspects of FRT for wind turbines (WTs) and WPPs. This paper details the characteristics of the most common disturbances originated in the transmission system, how they propagate to the WT terminals, and how they impact the dynamic behavior of a large WPP. This paper shows that the details of the voltage disturbances, not only in the transmission system, but also at the WT terminals, should be taken into consideration. Moreover, a detailed representation or characterization of voltage dips is important for FRT studies, despite that the simplified models used in the literature are insufficient. This paper strongly recommends that distinct events and additional characteristics such as the phase-angle jump and oscillations in the transition segments should be considered in FRT analysis.

Abstract:Aggregate thermostatically controlled loads (ATCLs) are a suitable candidate for power imbalance on demand side to smooth the power fluctuation of renewable energy. A new control scheme based on an improved bilinear aggregate model of ATCLs is investigated to suppress power imbalance. Firstly, the original bilinear aggregate model of ATCLs is extended by the second-order equivalent thermal parameter model to optimize accumulative error over a long time scale. Then, to ensure the control performance of tracking error, an improved model predictive control algorithm is proposed by integrating the Lyapunov function with the error transformation, and theoretical stability of the proposed control algorithm is proven. Finally, the simulation results demonstrate that the accuracy of the improved bilinear aggregate model is enhanced; the proposed control algorithm has faster convergence speed and better tracking accuracy in contrast with the Lyapunov function-based model predictive control without the prescribed performance.

Abstract:Identification and classification of DC faults are considered as fundamentals of DC grid protection. A sudden rise of DC fault current must be identified and classified to immediately operate the corresponding interrupting mechanism. In this paper, the Boltzmann machine learning (BML) approach is proposed for identification and classification of DC faults using travelling waves generated at fault point in voltage source converter based high-voltage direct current (VSC-HVDC) transmission system. An unsupervised way of feature extraction is performed on the frequency spectrum of the travelling waves. Binomial class logistic regression (BCLR) classifies the HVDC transmission system into faulty and healthy states. The proposed technique reduces the time for fault identification and classification because of reduced tagged data with few characteristics. Therefore, the faults near or at converter stations are readily identified and classified. The performance of the proposed technique is assessed via simulations developed in MATLAB/Simulink and tested for pre-fault and post-fault data both at VSC1 and VSC2, respectively. Moreover, the proposed technique is supported by analyzing the root mean square error to show practicality and realization with reduced computations.

Abstract:The integration of renewable distributed generation (RDG) into distribution networks is promising and increasing nowadays. However, high penetration levels of distributed generation (DG) are often limited as they may have an adverse effect on the operation of distribution networks. One of the operation challenges is the interaction between DG and voltage-control equipment, e.g., an under-load tap changer (ULTC), which is basically designed to compensate for voltage changes caused by slow load variations. The integration of variable DGs leads to rapid voltage fluctuations, which can negatively affect the tap operation of ULTC. This paper investigates the impact of high penetration levels of RDG on the tap operation of ULTC in distribution networks through simulations. Various mitigation techniques that can alleviate this impact are also examined. Among these techniques, constant power-factor mode is regarded as the best trade-off between the simplicity and effectiveness of minimizing the number of tap operations. Simulations are performed on a Canadian benchmark rural distribution feeder using OpenDSS software.

Abstract:Modern fault-resilient microgrids (MGs) require the operation of healthy phases during unbalanced short-circuits to improve the system reliability. This study proposes a differential power based selective phase tripping scheme for MGs consisting of synchronous and inverter-interfaced distributed generators (DGs). First, the differential power is computed using the line-end superimposed voltage and current signals. Subsequently, to make the scheme threshold-free, a power coefficient index is derived and used for identifying faulted phases in an MG. The protection scheme is tested on a standard MG operating in either grid-connected or islanding mode, which is simulated using PSCAD/EMTDC. The efficacy of the scheme is also assessed on the OPAL-RT manufactured real-time digital simulation (RTDS) platform. Further, the performance of the proposed protection scheme is compared with a few existing methods. The results show that the selective tripping of faulted phases in MGs can be achieved quickly and securely using the proposed scheme.

Abstract:Controlled islanding plays an essential role in preventing the blackout of power systems. Although there are several studies on this topic in the past, no enough attention is paid to the uncertainty brought by renewable energy sources (RESs) that may cause unpredictable unbalanced power and the observability of power systems after islanding that is essential for back-up black-start measures. Therefore, a novel controlled islanding model based on mixed-integer second-order cone and chance-constrained programming (MISOCCP) is proposed to address these issues. First, the uncertainty of RESs is characterized by their possibility distribution models with chance constraints, and the requirements, e.g., system observability, for rapid back-up black-start measures are also considered. Then, a law of large numbers (LLN) based method is employed for converting the chance constraints into deterministic ones and reformulating the non-convex model into convex one. Finally, case studies on the revised IEEE 39-bus and 118-bus power systems as well as the comparisons among different models are given to demonstrate the effectiveness of the proposed model. The results show that the proposed model can result in less unbalanced power and better observability after islanding compared with other models.

Abstract:The energy consumption of buildings accounts for approximately 40% of total energy consumption. An accurate energy consumption analysis of buildings can not only promise significant energy savings but also help estimate the demand response potential more accurately, and consequently brings benefits to the upstream power grid. This paper proposes a novel physical-data fusion modeling (PFM) method for modeling smart buildings that can accurately assess energy consumption. First, a thermal process model of buildings and an electrical load model that focus on building heating, ventilation, and air conditioning (HVAC) systems are presented to analyze the thermal-electrical conversion process of energy consumption of buildings. Second, the PFM method is used to improve the accuracy of the energy consumption analysis model for buildings by modifying the parameters that are difficult to measure in the physical model (i.e., it effectively modifies the electrical load model based on the proposed PFM method). Finally, case studies involving a real-world dataset recorded in a high-tech park in Changzhou, China, demonstrate that the proposed method exhibits superior performance with respect to the traditional physical modeling (TPM) method and data-driven modeling (DDM) method in terms of the achieved accuracy.

Abstract:Residential flexible resource is attracting much attention in demand response (DR) for peak load shifting. This paper proposes a scenario for multi-stage DR project to schedule energy consumption of residential communities considering the incomplete information. Communities in the scenario can decide whether to participate in DR in each stage, but the decision is the private information that is unknown to other communities. To optimize the energy consumption, a Bayesian game approach is formulated, in which the probability characteristic of the decision-making of residential communities is described with Markov chain considering human behavior of bounded rationality. Simulation results show that the proposed approach can benefit all residential communities and power grid, but the optimization effect is slightly inferior to that in complete information game approach.

Abstract:The doubly-fed induction generator (DFIG) is considered to provide a low-reactance path in the negative-sequence system and naturally comply with requirements on the negative-sequence reactive current in emerging grid codes. This paper shows otherwise and how the control strategy of converters plays a key role in the formation of the active and reactive current components. After investigating the existing control strategies from the perspective of grid code compliance and showing how they fail in addressing emerging requirements on the negative-sequence reactive current, we propose a new coordinated control strategy that complies with reactive current requirements in grid codes in the positive- and negative-sequence systems. The proposed method fully takes advantage of the current and voltage capacities of both the rotor-side converter (RSC) and grid-side converter (GSC), which enables the grid code compliance of the DFIG under unbalanced three-phase voltages due to asymmetrical faults. The mathematical investigations and proposed strategy are validated with detailed simulation models using the Electric Power Research Institute (EPRI) benchmark system. The derived mathematical expressions provide analytical clarifications on the response of the DFIG in the negative-sequence system from the grid perspective.

Abstract:This paper presents a novel inter-cluster direct current (DC) capacitor voltage balancing control scheme for the single-star configured modular multilevel cascaded converter (MMCC)-based static synchronous compensator (STATCOM) under unbalanced grid voltage. The negative-sequence component of grid voltage at the point of common connection (PCC) causes unbalanced active power flow in the phase limbs of converter. This leads to the imbalance of DC voltages of the sub-module capacitors across the MMCC phases, and consequently, the malfunction of converter. The proposed solution is to inject both negative-sequence current (NSC) and zero-sequence voltage (ZSV) into the phase limbs of MMCC. A quantification factor Q_F is used to achieve the sharing of inter-cluster active power between the NSC and ZSV injection methods. Accurate determination of the quantification factor has been presented. In addition to maintaining the DC voltages of sub-module capacitor across the MMCC phases balanced, it also prevents the overcurrent and overvoltage of converter by injecting NSC and ZSV with the right proportion. The control scheme is validated on a 3.54 kV 1.2 MVA power system using MMCC-based STATCOM with 3-level bridge cells as sub-modules. The results show that the proposed scheme provides superior effectiveness in eliminating the voltage imbalance of DC capacitor in the phase limb while maintaining low voltage and current ratings.

Abstract:In this paper, an adaptive interval type-2 fuzzy controller is proposed for variable-speed and variable-pitch wind turbines. Because of attractive features of the well-known wind turbine baseline controller, the proposed controller acts as an augmented controller and works in parallel to the baseline controller. As typical variable-speed wind turbines have different controllers for different operation regions, for each operation region, a dedicated interval type-2 fuzzy controller is designed. Because of the uncertainty in wind speed measurement, modern control techniques try to estimate this value. However, in contrast to these modern control techniques, the proposed controller is independent of the wind speed estimation. Thus, there is a better saving in cost and computational burden. To evaluate the effectiveness of the proposed controller, simulations are conducted with wind profiles which span all operation regions. Results show that, compared with the baseline controller, the proposed controller enhances power generations and reduces mechanical loads concurrently.

Abstract:The paper analyses the coordinated hydro-wind power generation considering joint bidding in the electricity market. The impact of mutual bidding strategies on market prices, traded volumes, and revenues has been quantified. The coordination assumes that hydro power generation is scheduled mainly to compensate the differences between actual and planned wind power outputs. The potential of this coordination in achieving and utilizing of market power is explored. The market equilibrium of asymmetric generation companies is analyzed using a game theory approach. The assumed market situation is imperfect competition and non-cooperative game. A numerical approximation of the asymmetric supply function equilibrium is used to model this game. An introduced novelty is the application of an asymmetric supply function equilibrium approximation for coordinated hydro-wind power generation. The model is tested using real input data from the Croatian power system.

Abstract:An analytic method is proposed to compute the price-reserve offer curve at the consumer level in hierarchical direct load control. The convexification of the consumer reserve provision is examined, and the analytic expression of the optimal solution within each critical region is derived. Then, based on multi-parametric programming, a combinatorial enumeration method in conjunction with efficient reduction and pruning strategy is proposed to compute the optimal response of consumers in the whole price space. Numerical tests along with an application example in the bi-level aggregator pricing problem demonstrate the merit of this method.