Progressive Hedging Algorithm Research Articles

Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem

Selecting facility locations requires significant investment to anticipate and prepare for disruptive events like earthquakes, floods, or labor strikes. In practice, location choices account for facility capacities, which often cannot change during disruptions. When a facility fails, demand transfers to others only if spare capacity exists. Thus, capacitated reliable facility location problems (CRFLP) under uncertainty are more complex than uncapacitated versions. To manage uncertainty and decide effectively, stochastic programming (SP) methods are often employed. Two commonly used SP methods are approximation methods, i.e., Sample Average Approximation (SAA), and decomposition methods, i.e., Progressive Hedging Algorithm (PHA). SAA needs large sample sizes for performance guarantee and turn into computationally intractable. On the other hand, PHA, as an exact method for convex problems, suffers from the need to iteratively solve numerous sub-problems which are computationally costly. In this paper, we developed two novel algorithms integrating SAA and PHA for solving the CRFLP under uncertainty. The developed methods are innovative in that they blend the complementary aspects of PHA and SAA in terms of exactness and computational efficiency, respectively. Further, the developed methods are practical in that they allow the specialist to adjust the tradeoff between the exactness and speed of attaining a solution. We present the effectiveness of the developed integrated approaches, Sampling Based Progressive Hedging Algorithm (SBPHA) and Discarding SBPHA (d-SBPHA), over the pure strategies (i.e. SAA). The validation of the methods is demonstrated through two-stage stochastic CRFLP. Promising results are attained for CRFLP, and the method has great potential to be generalized for SP problems.

An inexact algorithm for stochastic variational inequalities

We present a new Progressive Hedging Algorithm to solve Stochastic Variational Inequalities in the formulation introduced by Rockafellar and Wets in 2017, allowing the generated subproblems to be approximately solved with an implementable tolerance condition. Our scheme is based on Hybrid Inexact Proximal Point methods and generalizes the exact algorithm developed by Rockafellar and Sun in 2019, providing stronger convergence results. We also show some numerical experiments in two-stage Nash games.

Chance-constrained multi-stage stochastic energy system expansion planning with demand satisfaction flexibility

A classic multi-period stochastic energy system expansion planning (ESEP) model aims to address demand uncertainty by requiring immediate demand satisfaction for all scenarios. However, this approach may result in an expensive system that deviates from the planner’s long-term goals, especially when facing unexpectedly high demand scenarios. To address this issue, we propose a chance-constrained stochastic multi-stage ESEP model that allows for a portion of demand to remain unmet in specific periods while still ensuring complete demand satisfaction during most of the planning horizon, including the final period. This approach provides more time flexibility to build infrastructure and assess needs, ultimately reducing costs and allowing for a broader view of infrastructure planning options. To solve the chance-constrained stochastic model, we introduce a binary-search-based progressive hedging algorithm heuristic, which is particularly useful for large-scale models. We demonstrate the effectiveness and benefits of implementing the chance-constrained model through a case study of Rwanda using real-world data.

A Localized Progressive Hedging Algorithm for Solving Nonmonotone Stochastic Variational Inequalities

The progressive hedging algorithm (PHA) is an effective solution method for solving monotone stochastic variational inequalities (SVIs). However, this validity is based on the assumption of global maximal monotonicity. In this paper, we propose a localized PHA for solving nonmonotone SVIs and show that its validity is based on the weaker assumption of locally elicitable maximal monotonicity. Furthermore, we prove that such assumption holds when the mapping involved in the SVI is locally elicitable monotone or locally monotone. The local convergence of the proposed algorithm is established, and it is shown that the localized PHA has the rate of linear convergence under some mild assumptions. Some numerical experiments, including a two-stage orange market problem and randomly generated two-stage piecewise stochastic linear complementarity problems, indicate that the proposed algorithm is efficient. Funding: This work was supported by the National Natural Science Foundation of China [Grant 12171271]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0017 .

Incorporating catastrophe insurance in power distribution systems investment and planning for resilience enhancement

Severe damage and high economic losses caused by frequent extreme disasters are becoming unbearable for distribution system operators (DSOs) under climate change. In order to cope with extreme disasters, this paper proposes an investment portfolio and planning framework of multi-resource for resilience enhancement. In the investment portfolio and planning framework, insurance is incorporated to mitigate high economic losses as a risk sharing and resilience enhancement supplementary strategy, and co-optimized with security investment from the perspective of DSOs. Firstly, the insurance investment is modelled based on Gordon-Loeb model according to insurance industry principles and formulated as constraints by piecewise linearization techniques. Secondly, the investment portfolio and planning framework proposed in this paper is modeled as a scenario-based two-stage stochastic mixed integer linear optimization program (MILP). In the first stage, DSO makes investment and planning decisions including energy storage (ESS) allocation, line hardening strategy and insurance investment. Both operation strategies of normal operation scenarios and extreme disaster scenarios under climate change are optimized in the second stage. Finally, progressive hedging (PH) algorithm is applied to solve the two-stage stochastic MILP model. The proposed model and algorithm are tested on the modified IEEE 33-bus test system. The results verify the validity of investment portfolio and planning framework, and the results show that portfolio investment considering insurance can realize resilience enhancement in both load shedding and economic losses.

A Risk-Averse Adaptive Stochastic Optimization Method for Transactive Energy Management of a Multi-Energy Microgrid

This paper proposes a new energy management method for a multi-energy microgrid (MEMG) which supplies both electrical and thermal energies. Based on the transactive energy (TE) concept, the problem is formulated as a Stackelberg game-theoretic bi-level optimization model. The MEMG operator optimizes the energy scheduling and pricing strategies at the upper level, and the industrial, commercial and residential agents optimize their energy trading strategies at the lower level. The equivalent single-level mixed-integer linear program (MILP) reformulation is then derived for computational tractability. To coordinate the strategies made in the day-ahead and intra-day energy markets, an adaptive stochastic optimization (SO) approach is adopted, by which a day-ahead stochastic MILP and an intra-day deterministic model are formed. Furthermore, the conditional value-at-risk (CVaR) measure is incorporated in the day-ahead stage for risk aversion of the MEMG operator towards uncertainties. To solve the models, an adaptive Progressive Hedging (PH) algorithm is developed to decompose the day-ahead stochastic MILP into multiple scenario-based subproblems which can be solved in parallel, and an outer approximation (OA) algorithm is adopted in the intra-day stage to linearize the bilinear objective function. Finally, the simulation results verify the effectiveness of the proposed energy management method and the efficiency of the solution algorithms.

Optimal allocation of distributed renewable generations in low-carbon distribution system considering impact of natural disasters

With global climate change, increasingly frequent natural disasters have brought great challenges to the safe and reliable power supply and low-carbon transition of power distribution systems. Most of the existing researches on the distribution system under the impact of natural disasters only focus on the improvement of power supply reliability, but have not consider the impact of disaster severity and disaster response measures on carbon emissions. In order to juggle the load restoration and carbon emission mitigation of distribution system under natural disasters, this paper proposes an optimal allocation method of distributed renewable generations (DRGs) considering carbon emission for multi-scenario natural disasters based on the framework of cyber-physical-social system in energy (CPSSE), and establishes a three-stage optimization model of pre-disaster prevention-disaster attack-post-disaster restoration. For the purpose of ensuring the practicability and robustness of the allocation results, the disaster scenario is modeled and the selection method of the worst fault scenario under disaster is proposed. The progressive hedging algorithm (PHA) is adopted to solve the proposed multi-scenario optimization problem. Finally, the simulation results indicate that the proposed method can restore more lost load at a lower cost of carbon emissions.

A Resilience-Oriented Multi-Stage Adaptive Distribution System Planning Considering Multiple Extreme Weather Events

Climate change may increase the risk of an area being hit by multiple extreme weather events, which brings significant challenges for distribution system planners in an increasing renewable penetration era. There is an urgent need for planning approaches to be more flexible and allow for adaptive adjustments in the future to hedge against high uncertainties in extreme weather event scenarios. In this work, we propose a resilience-oriented distribution system planning approach that considers multiple extreme weather events. A multi-stage hybrid-stochastic-and-robust formulation is developed to model decisions not only for initial investments, but also for adaptive investments and emergent operations in response to particular extreme events, meanwhile considering both long-term and short-term uncertainties. Our model is solved by a novel progressive hedging algorithm that is embedded with a nested column-and-constraint generation method. Case studies demonstrate the benefits of the proposed approach in making flexible and affordable planning decisions to protect distribution systems against multiple extreme weather events.

Collaborative stochastic expansion planning of cyber‐physical system considering extreme scenarios

AbstractIn this paper, a collaborative stochastic expansion planning model of cyber‐physical system (CPS) with resilience constraints is proposed. The model can reduce the coupling risk and enhance the resilience under extreme scenarios, from the perspective of the structural/functional coupling between the physical and cyber systems. The model is to collaboratively optimize expansion transmission line siting, expansion communication fibre siting, and service routing distribution to minimize total investment cost. Constraints are divided into conventional constraints and resilience constraints. In the resilience constraints, the impact of cyber failures on the dispatch strategy is analyzed, and the load shedding in stochastic scenarios is limited to guarantee the resilience of the planning scheme. Particularly, to reach a stable solution for the stochastic planning, a mixed scenario set is proposed based on the probabilistic typhoon model and the complex network theory. Finally, the model is solved by the progressive hedging (PH) algorithm. The case studies of the IEEE RTS‐79 test system demonstrate that, compared with the traditional independent planning model, the model proposed in this paper performs better in limiting load loss and enhancing resilience under extreme scenarios.

Portfolio optimization of generic energy storage-based virtual power plant under decision-dependent uncertainties

Distributed energy resources (DERs) can be integrated into a smart and aggregated entity, namely a virtual power plant (VPP). This integration is beneficial to facilitate their flexibility utilization and optimally allocate their capacity to participate in competitive electricity markets. While various types of uncertainties exist in the market operation of VPP, such as the stochastic power of renewable energy resources (RES), etc., the literature generally overlooked the decision-dependent uncertainties (DDUs) with the operation of generic energy storage (GES). To be risk-hedging, this manuscript proposes a novel approach for risk-averse portfolio optimization of VPP with special attention to DDUs. To achieve tractable optimization, we propose a two-stage chance-constrained stochastic optimization model and three reformulations of different types of uncertainties. Furthermore, a modified and iterative progressive hedging algorithm is further proposed to handle the conditional value at risk (CVaR) and DDU items and efficiently solve the problem with large-scale GES portfolios. Exhaustive case studies based on real-world data demonstrate that VPP has to face the consequence of a significant real-time penalty (6.36 % loss in average profit) and losses in GES power (about 6.5 MW) when overlooking the DDUs' consequences. Moreover, it is more beneficial for greedy decision-makers to incorporate the DDUs' impact into their less risk-aversion strategy. Additionally, comparative results prove the validity of the proposed algorithm and its superiority in solving problems with large-scale GES portfolios and high-performance computing systems (over 30 % CPU time saved).

A Two-stage Stochastic Mixed-integer Programming Model for Resilience Enhancement of Active Distribution Networks

Most existing distribution networks are difficult to withstand the impact of meteorological disasters. With the development of active distribution networks (ADNs), more and more upgrading and updating resources are applied to enhance the resilience of ADNs. A two-stage stochastic mixed-integer programming (SMIP) model is proposed in this paper to minimize the upgrading and operation cost of ADNs by considering random scenarios referring to different operation scenarios of ADNs caused by disastrous weather events. In the first stage, the planning decision is formulated according to the measures of hardening existing distribution lines, upgrading automatic switches, and deploying energy storage resources. The second stage is to evaluate the operation cost of ADNs by considering the cost of load shedding due to disastrous weather and optimal deployment of energy storage systems (ESSs) under normal weather condition. A novel modeling method is proposed to address the uncertainty of the operation state of distribution lines according to the canonical representation of logical constraints. The progressive hedging algorithm (PHA) is adopted to solve the SMIP model. The IEEE 33-node test system is employed to verify the feasibility and effectiveness of the proposed method. The results show that the proposed model can enhance the resilience of the ADN while ensuring economy.

A three-stage resilient dispatch of mobile emergency generators in a distribution system against hurricanes

Mobile emergency generators (MEGs) have been widely used to enhance the resilience of distribution systems. However, the traditional two-stage MEG dispatch strategies focus on only the prepositioning and reallocation before and after hurricanes, neglecting the resilient operation during hurricanes. Therefore, this paper proposes a novel three-stage MEG dispatch model to enhance the resilience of distribution systems. To improve the survivability of distribution systems during hurricanes, the fault isolation operations by remote-controlled switches (RCSs) and forced cut-off of wind power are coordinated with MEG dispatch, which is prepositioned as a preparedness measure before hurricanes. Based on the realization of fault scenarios during hurricanes, MEG reallocation and fault isolation by manual switches (MSs) are conducted to boost the restoration of the distribution system after hurricanes. Considering the uncertainties of wind power cut-off and line faults, the proposed method is incorporated into a stochastic dispatch model, which takes the conditional value-at-risk (CVaR) as an objective function to improve robustness in extreme cases. In addition, the computational burden with multiple scenarios is reduced by the modified progressive hedging algorithm (PHA). Finally, the proposed method is validated on a modified IEEE 33-bus distribution system. Simulation results show that the survivability and restoration of the distribution system can be enhanced by the proposed approach.

A stochastic programming approach to surgery scheduling under parallel processing principle

Parallel processing is a principle which enables simultaneous implementation of anesthesia induction and operating room (OR) turnover with the aim of improving OR utilization. In this article, we study the problem of scheduling surgeries for multiple ORs and induction rooms (IR) that function based on the parallel processing principle under uncertainty. We propose a two-stage stochastic mixed-integer programming model considering the uncertainty in induction, surgery and turnover durations. We sequence patients and set appointment times for surgeries in the first stage and assign patients to IRs at the second stage of the model. We show that an optimal myopic policy can be used for IR assignment decisions due to the special structure of the model. We minimize the expected total cost of patient waiting time, OR idle time and IR idle time in the objective function. We enhance the model formulation using bounds on variables and symmetry-breaking constraints. We implement a novel progressive hedging algorithm by proposing a penalty update method and a variable fixing mechanism. Based on real data of a large academic hospital, we compare our solution approach with several scheduling heuristics from the literature. We assess the additional benefits and costs associated with the implementation of parallel processing using near-optimal schedules. We examine how the benefits are inflated by increasing the number of IRs. Finally, we estimate the value of stochastic solution to underline the importance of considering uncertainty in durations.

Hydrogen-Based Networked Microgrids Planning Through Two-Stage Stochastic Programming With Mixed-Integer Conic Recourse

Networked microgrids that integrate the hydrogen fueling stations (HFSs) with the on-site renewable energy sources (RES), power-to-hydrogen (P2H) facilities, and hydrogen storage could help decarbonize the energy and transportation sectors. In this paper, to support the hydrogen-based networked microgrids planning subject to multiple uncertainties (e.g., RES generation, electric loads, and the refueling demands of hydrogen vehicles), we propose a two-stage stochastic formulation with mixed integer conic program (MICP) recourse decisions. Our formulation involves the holistic investment and operation modeling to optimally site and configure the microgrids with HFSs. The MICP problems appearing in the second-stage capture the nonlinear power flow of networked microgrids system with binary decisions on storage charging/discharging status and energy transactions (including the trading of electricity, hydrogen, and carbon credits to recover the capital expenditures). To handle the computational challenges associated with the stochastic program with MICP recourse, an augmented Benders decomposition algorithm (ABD) is developed. Numerical studies on 33- and 47-bus exemplary networks demonstrate the economics viability of electricity-hydrogen coordination on microgrids level, as well as the benefits of stochastic modeling. Also, our augmented algorithm significantly outperforms existing methods, e.g., the progressive hedging algorithm (PHA) and the direct use of a professional MIP solver, which has largely improved the solution quality and reduced the computation time by orders of magnitude. Note to Practitioners—This paper proposes an optimal planning model for electricity-hydrogen microgrids with the renewable hydrogen production, storage, and refueling infrastructures. Our planning model is extended under a two-stage stochastic framework to address the multi-energy-sector uncertainties, e.g., RES generation, electric loads, and the refueling demands of hydrogen vehicles. The first-stage problem is to optimize the siting and sizing plan of microgrids. Then, in the second-stage problem, the coordinated scheduling of electricity and hydrogen supply systems is modeled as second-order conic programs (SOCPs) to accurately capture the power flow representation under stochastic scenarios. Also, the logical constraints with binary variables are introduced to describe the energy transactions and storage operations, which results in an MICP recourse structure. Note that the stochastic MICP formulation could be very challenging to compute even with a moderate number of scenarios. One challenge certainly comes from integer variables that cause the problem nonconvex. Another challenge follows from the fact that the strong duality of SOCPs might not hold in general. To mitigate those two challenges, we prove that the continuous relaxation of our recourse problem has strong duality, and make use of that continuous relaxation and other enhancements to design an augmented decomposition algorithm. As revealed by our numerical tests, the proposed decomposition method outperforms PHA in both the solution quality and computational efficiency. Comparing to the PHA, our ABD method often achieves tighter bounds with trivial optimality gaps. Also, it could reduce the computation time by orders of magnitude. With the help of advanced analytical tool, the proposed planning framework can be readily implemented in real-world applications.

Stochastic Approximation Methods for the Two-Stage Stochastic Linear Complementarity Problem

The two-stage stochastic linear complementarity problem (TSLCP), which can be regarded as a special and important reformulation of two-stage stochastic linear programming, has arisen in various fields, such as stochastic programming, game theory, traffic equilibrium, and theoretical economics. Considerable effort has been devoted to designing numerical methods for solving TSLCPs. A popular approach is to integrate the progressive hedging algorithm (PHA) as a sub-algorithm into a discretization framework. In this paper, aiming to solve large-scale TSLCPs, we propose two kinds of stochastic methods: the stochastic approximation method based on projection (SAP) and the dynamic sampling SAP (DS-SAP), both of which offering more direct and improved control of the computational costs of the involved subproblems, especially compared with the PHA. In particular, the linear complementarity subproblems are solved inexactly during each iteration, and the convergence analysis of both SAP and DS-SAP with an inexactness criterion is presented. Moreover, numerical implementations and practical applications demonstrate the efficiency of our proposed methods.

A Modified Progressive Hedging Approach for Multistage Intraday Trade of EV aggregators

The growing prominence of electric vehicle (EV) aggregators in the modern power system is drawing more attention towards modeling their behavior in the short-term electricity markets. The demand-side flexibility offered by the EVs can be leveraged to reduce their charging costs. In this paper, the participation of an EV aggregator in the intraday and balancing market is modeled as a multistage stochastic programming problem. The computational complexity introduced by the peculiarities of the intraday market is solved by a progressive hedging algorithm (PHA), a scenario-based decomposition technique. A randomized scenario sampling approach is implemented to accelerate the PHA which is further improved with a parallel randomized PHA. Finally, an asynchronous version of the parallel randomized PHA is leveraged to speed up the multistage model of EV aggregator trading. We compare the computation time of the modified versions of the PHA algorithm with the conventional PHA for the proposed EV aggregator model. Furthermore, we also show the value of EV aggregator trading in the intraday and balancing markets by comparing its cost to baseline models.

Restoration Strategy for Active Distribution Systems Considering Endogenous Uncertainty in Cold Load Pickup

Cold load pickup (CLPU) phenomenon is identified as the persistent power inrush upon a sudden load pickup after an outage. Under the active distribution system (ADS) paradigm, where distributed energy resources (DERs) are extensively installed, the decreased outage duration can induce a strong interdependence between CLPU pattern and load pickup decisions. In this paper, we propose a novel modelling technique to tractably capture the decision-dependent uncertainty (DDU) inherent in the CLPU process. Subsequently, a two-stage stochastic decision-dependent service restoration (SDDSR) model is constructed, where first stage searches for the optimal switching sequences to decide step-wise network topology, and the second stage optimizes the detailed generation schedule of DERs as well as the energization of switchable loads. Moreover, to tackle the computational burdens introduced by mixed-integer recourse, the progressive hedging algorithm (PHA) is utilized to decompose the original model into scenario-wise subproblems that can be solved in parallel. The numerical test on modified IEEE 123-node test feeders has verified the efficiency of our proposed SDDSR model and provided fresh insights into the monetary and secure values of DDU quantification.

Stochastic medical tourism problem with variable residence time considering gravity function

Medical tourism is a recent term in healthcare logistics referring to travel of patients to receive health services and spending leisure time in a destination country. This transferring of patients leads to access high-quality health services which are cheaper than the original country of patients. During this travel, passengers who are the patients from another country, have this opportunity for complimentary entertainment packages (e.g., pleasure tours) in the aftercare period. As far as we know, the term of medical tourism is rarely studied in healthcare logistics and such services are highly important for developing countries. Such facts motivate us to develop a practical optimization model for the Medical Tour Centers (MTCs) for allocation of patients to hospitals in proper time and creation of memorable aftercare time for them. In this regard, the main aim of the proposed model is to maximize the total profit of MTCs through optimal allocation of patients to hospitals while considering an aftercare tour for the passengers. To make the proposed model more realistic, the optimal residence time in attractive places is simulated by a time-dependent gravity function. To address the uncertainty of medical tourism problem, a scenario-based two-stage stochastic optimization approach is extended to encounter different sources of uncertainty existing in surgical success, medical time, restoration restrictions, and the attraction of tourist places. Another novelty of this work is to propose an innovative hybrid meta-heuristic for large-scale instances, which is a combination of Progressive Hedging Algorithm (PHA) and Genetic Algorithm (GA). The model is analyzed by different test problems for small, medium, and large-scale instances where the hybrid meta-heuristic algorithm could solve them with an average gap of 3.4% in comparison with the commercial solver. The results revealed the importance of tourist opinion and public preferences in medical and pleasure tours, respectively, to improve the economic growth in this sector in developing countries.

Maximizing the utilization of existing grids for renewable energy integration

This paper presents a new model to maximize the utilization of existing transmission system infrastructure by optimally sizing and siting the future developments of variable renewable energy sources (VRES). The model tries to maximize the integration of VRES in power systems with minimum expected energy curtailment without relying on new investments in the transmission systems. The proposed model is formulated as a linear stochastic programming optimization problem where VRES output scenarios are generated such that their spatio-temporal correlations are maintained. The Progressive Hedging Algorithm (PHA) with bundled scenarios is utilized to solve the proposed model for large-scale cases. The proposed model is tested on the modified Garver 6-bus and IEEE 118-bus test systems, and its results are compared with the results of the conventional VRES integration model. These results and comparisons illustrate the effectiveness of the proposed approach in terms of maximizing VRES integration and enhancing computational performance.

Resilience enhancement strategy for multi-energy systems considering multi-stage recovery process and multi-energy coordination

With the increasing frequency of extreme weather events (EWEs) in recent years, the resilience of multi-energy systems (MESs) has received considerable attention. To improve the ability of MESs to withstand and recover rapidly from an EWE, this paper proposes a comprehensive resilience enhancement operation strategy for MESs considering the multi-stage recovery process and multi-energy coordination. Wherein, multi-stage recovery process throughout the pre-event stage, system disturbance stage, fast fault isolation stage, and remote-controlled service restoration stage are comprehensively considered and modeled in detail. Multiple resilience enhancement measures are adopted, and their coupling relationships at different stages are taken into account. In addition, an integrated energy flow model is developed to minimize the demand curtailment through coordination among different energy subsystems. The overall problem is established as a stochastic mixed-integer linear programming (MILP) model considering uncertainties of fault components. A customized progressive hedging algorithm (PHA) is further developed to reduce the computational burden caused by multiple scenarios. Case studies verify the effectiveness of the proposed model and solution technique. Simulation results reveal that the combination of multiple restoration measures is essential to reduce the impacts of EWEs, and the coordination among different energy carriers can effectively improve the resilience of the overall MESs.