Stochastic Unit Commitment Problem Research Articles

Design of experiments for the stochastic unit commitment with economic dispatch models

We develop a Design and Analysis of the Computer Experiments (DACE) approach to the stochastic unit commitment problem for power systems with significant renewable integration. For this purpose, we use a two-stage stochastic programming formulation of the stochastic unit commitment-economic dispatch problem. Typically, a sample average approximation of the true problem is solved using a cutting plane method (such as the L-shaped method) or scenario decomposition (such as Progressive Hedging) algorithms. However, when the number of scenarios increases, these solution methods become computationally prohibitive. To address this challenge, we develop a novel DACE approach that exploits the structure of the first-stage unit commitment decision space in a design of experiments, uses features based upon solar generation, and trains a multivariate adaptive regression splines model to approximate the second stage of the stochastic unit commitment-economic dispatch problem. We conduct experiments on two modified IEEE-57 and IEEE-118 test systems and assess the quality of the solutions obtained from both the DACE and the L-shaped methods in a replicated procedure. The results obtained from this approach attest to the significant improvement in the computational performance of the DACE approach over the traditional L-shaped method.

An effective hybrid decomposition approach to solve the network-constrained stochastic unit commitment problem in large-scale power systems

We propose a novel hybrid method to solve the network-constrained stochastic unit commitment problem. We target realistic large-scale instances including hundreds of thermal generation units, thousands of transmission lines and nodes, and a large number of stochastic renewable generation units. This scheduling problem is formulated as a two-stage stochastic programming problem with continuous and binary variables in the first stage and only continuous variables in the second stage. We develop a hybrid solution method that decomposes the original problem into a master problem including unit commitment and dispatch decisions, and decomposed subproblems representing dispatch with transmission constraints per scenario. The proposed decomposition embeds a column-and-constraint generation step within the traditional Benders decomposition framework. The performance of the proposed decomposition technique is contrasted with the solution of the extensive form via branch-and-cut and Benders decomposition available in commercial solvers, and with conventional Benders decomposition variants. Our computational experiments show that the proposed method generates bounds of superior quality and finds solutions for instances where other approaches fail.

Three-Stage Stochastic Unit Commitment for Microgrids Toward Frequency Security via Renewable Energy Deloading

Renewable energy is boosting the deployment of microgrids (MGs) with stochastic and low inertia nature. To improve the operational efficiency of MGs while guaranteeing frequency security, a three-stage stochastic unit commitment problem is proposed, where renewable energy can be deloaded. In the first stage, the diesel generators (DGs) are scheduled, responding to uncertainties of loads and photovoltaic generator output. In the second stage, the outputs of DGs are optimized to reduce the operational cost under uncertain disturbances. In the third stage, a novel PV deloading strategy is proposed to test the frequency security of MGs. The three-stage optimization problem is formulated as a multi-stage stochastic optimization problem with recourse. This problem is then solved using a novel nested Benders decomposition algorithm with both dual cuts and partial primal cuts. Simulations are performed on an AC MG under different PV penetration levels and disturbances. The results verify the effectiveness of the proposed model in balancing operational efficiency, renewable energy utilization, and frequency security.

More Efficient Energy Management for Networked Hybrid AC/DC Microgrids With Multivariable Nonlinear Conversion Losses

Accurate conversion loss models are the keys to guarantee the more efficient operation of networked hybrid ac/dc microgrids (N-ac/dc-MGs). A two-stage stochastic unit commitment (UC) problem is proposed to improve the operational efficiency of N-ac/dc-MGs under uncertain renewable energy generation output and loads. The nonlinear power losses of ac/dc converters and dc/dc converters under different operating modes are formulated as novel multivariate nonlinear functions of both power and voltage. These functions are approximated by linear surrogate models and adopted by the dynamic optimal power flow (OPF), where the voltage and power of the converters can be optimized considering the physical laws among voltages of converters. Embedding the dynamic OPF problem as mixed-integer recourse, a two-stage stochastic programming problem is formulated for the day-ahead operation of N-ac/dc-MGs. To reduce the computational cost, a finite iteration convergent Benders decomposition algorithm is proposed to solve the UC problem. Case studies are performed on hybrid ac/dc MGs and N-ac/dc-MGs. Simulation results reveal that using more accurate conversion loss models, the output commands, especially the dc bus voltage, from the energy management systems will lead to more efficient system operation, and thus, save energy.

Multi-objective two-stage stochastic unit commitment model for wind-integrated power systems: A compromise programming approach

In this paper, we introduce a compromise programming (CP) framework for solving a multi-objective two-stage stochastic unit commitment problem characterized by high penetration of wind power. The proposed framework aims at finding best-compromise Pareto efficient on/off schedules, accounting for wind and power demand uncertainties: such solutions must trade off the three objectives of operating cost, CO2 emissions, and wind power curtailment in accordance to the decision maker preferences. To achieve this, we introduce a practical procedure to compute the ideal and Nadir points associated to the multi-objective two-stage stochastic unit commitment problem and propose a linearized ℓ1 norm-based compromise program to design best-compromise on/off schedules that correspondingly minimize and maximize their weighted distances to the ideal point and to the Nadir point, considering the preference weights assigned by the decision maker to each of the three objective functions. The proposed CP framework is applied to a case study related to the New England IEEE-39 bus test system. The results show that, compared to the schedule obtained through the traditional minimization of operating cost, the designed best-compromised schedules considerably improve CO2 emissions reduction and wind power curtailment performance by conservatively sacrificing operating cost performance.

Stochastic scheduling of generating units with weekly energy storage: A hybrid decomposition approach

We propose a solution method for the large-scale stochastic unit commitment (SUC) problem with weekly-dispatched energy storage and significant weather-dependent stochastic generating capacity. Weekly storage facilities that mostly charge during weekends and discharge during weekdays require a weekly scheduling of generating units, which result in a large-scale optimization problem. This SUC problem is formulated as a two-stage stochastic model and we use the conditional value-at-risk as a risk measure. Using a Benders framework, the proposed solution method decomposes the problem into a mixed-integer linear master problem and linear and continuous subproblems. The master problem corresponds to the first-stage decisions throughout the week and includes all the commitment (binary) variables and their corresponding constraints. The subproblems correspond to the actual dispatch of the generating units on a weekly basis. Based on the success of column-and-constraint generation algorithms to solve robust optimization problems, we improve the low communication between the master problem and the subproblems in the standard Benders decomposition by adding primal variables and constraints from the subproblems to the master problem, which provides a better approximation of the recourse function. Our computational experiments demonstrate the effectiveness of the proposed decomposition method using an instance of the South Carolina synthetic system with 90 generating units under 40 scenarios.

On the Structure of Decision Diagram–Representable Mixed-Integer Programs with Application to Unit Commitment

Despite the successful applications of decision diagrams (DDs) to solve various classes of integer programs in the literature, the question of which mixed-integer structures admit a DD representation remains open. The present work addresses this question by developing both necessary and sufficient conditions for a mixed-integer program to be DD-representable through identification of certain rectangular formations in the underlying sets. This so-called rectangularization framework is applicable to all bounded mixed-integer linear programs, providing a notable extension of the DD domain to continuous problems. As an application, the paper uses the developed methods to solve stochastic unit commitment problems in energy systems. Computational experiments conducted on benchmark instances show that the DD approach uniformly and significantly outperforms the existing solution methods and modern solvers. The proposed methodology opens new pathways to solving challenging mixed-integer programs in energy systems more efficiently.

Economic Value of Energy Storages in Unit Commitment With Renewables and Its Implication on Storage Sizing

Energy storage unit (ESU) is playing an increasingly important role in load shifting and uncertainty mitigation. This paper aims to quantify the value of ESU in the unit commitment (UC) with renewable generation. By treating the power and energy capacities of ESU as continuous parameters, the stochastic UC problem is cast as a multi-parametric mixed-integer linear program (mp-MILP), whose optimal value function (OVF) gives the relation between storage capacity and the daily UC cost in an analytical manner. It encompasses abundant sensitivity information, and its surface can be easily visualized. The reduced cost compared with the benchmark case without storage can be regarded as the value of ESU. As a potential application, the OVF is used to formulated an optimal storage sizing problem that maximizes the ratio between the reduced operation cost and the investment cost, ensuring the minimum time of cost recovery. The solution consists of two steps: the first step constructs the OVF parameterized in storage capacity; the second step reformulates the fractional storage sizing program into an MILP, leveraging the expression of the OVF and variable transformations. Case studies conducted on the IEEE 9-bus system and IEEE 118-bus system illustrate how the OVF offers useful reference for power system operation and planning. The proposed optimal sizing model shows advantages in efficient utilization of investment in terms of payback time and rate of return.

A novel hybrid grey wolf optimization algorithm and artificial neural network technique for stochastic unit commitment problem with uncertainty of wind power

The unit commitment (UC) is highly complex to solve the increasing integrations of wind farm due to intermittent wind power fluctuation in nature. This paper presents a hybrid methodology to solve the stochastic unit commitment (SUC) problem depending on binary mixed integer generator combination with renewable energy sources (RESs). In this combination, ON/OFF tasks of the generators are likewise included to satisfy the load requirement as for the system constraints. The proposed hybrid methodology is the consolidation of grey wolf optimization algorithm (GWOA) and artificial neural network (ANN), hence it is called the hybrid GWOANN (HGWOANN) technique. Here, the GWOA algorithm is used to optimizing the best combination of thermal generators depending on uncertain wind power, minimum operating cost and system constraints – that is, thermal generators limits, start-up cost, ramp-up time, ramp-down time, etc. ANN is utilized to capture the uncertain wind power events, therefore the system ensures maximal application of wind power. The combination of HGWOANN technique guarantees the prominent use of sustainable power sources to diminish the thermal generators unit operating cost. The proposed technique is implemented in MATLAB/Simulink site and the efficiency is assessed with different existing methods. The comparative analysis demonstrates that the proposed HGWOANN approach is proficient to solve unit commitment problems and wind integration. Here, the HGWOANN method is compared with existing techniques such as PSO, BPSO, IGSA to assess the overall performance using various metrics viz. RMSE, MAPE, MBE under 50 and 100 count of trials. In the proposed approach, the range of RMSE achieves 9.26%, MAPE achieves 0.95%, MBE achieves 1% in 50 count of trials. Moreover, in 100 count of trials, the range of RMSE achieves 7.38%, MAPE achieves 1.91%, MBE achieves 2.87%.

Optimal Operation Scheduling Considering Cycle Aging of Battery Energy Storage Systems on Stochastic Unit Commitments in Microgrids

As renewable penetration increases in microgrids (MGs), the use of battery energy storage systems (BESSs) has become indispensable for optimal MG operation. Although BESSs are advantageous for economic and stable MG operation, their life degradation should be considered for maximizing cost savings. This paper proposes an optimal BESS scheduling for MGs to solve the stochastic unit commitment problem, considering the uncertainties in renewables and load. Through the proposed BESS scheduling, the life degradation of BESSs is minimized, and MG operation becomes economically feasible. To address the aforementioned uncertainties, a scenario-based method was applied using Monte Carlo simulation and the K-means clustering algorithm for scenario generation and reduction, respectively. By implementing the rainflow-counting algorithm, the BESS charge/discharge state profile was obtained. To formulate the cycle aging stress function and examine the life cycle cost (LCC) of a BESS more realistically, the nonlinear cycle aging stress function was partially linearized. Benders decomposition was adopted for minimizing the BESS cycle aging, total operating cost, and LCC. To this end, the general problem was divided into a master problem and subproblems to consider uncertainties and optimize the BESS charging/discharging scheduling problem via parallel processing. To demonstrate the effectiveness and benefits of the proposed BESS optimal scheduling in MG operation, different case studies were analyzed. The simulation results confirmed the superiority and improved performance of the proposed scheduling.

Stochastic Unit Commitment Problem, Incorporating Wind Power and an Energy Storage System

This paper presents a modified formulation for the wind-battery-thermal unit commitment problem that combines battery energy storage systems with thermal units to compensate for the power dispatch gap caused by the intermittency of wind power generation. The uncertainty of wind power is described by a chance constraint to escape the probabilistic infeasibility generated by classical approximations of wind power. Furthermore, a mixed-integer linear programming algorithm was applied to solve the unit commitment problem. The uncertainty of wind power was classified as a sub-problem and separately computed from the master problem of the mixed-integer linear programming. The master problem tracked and minimized the overall operation cost of the entire model. To ensure a feasible and efficient solution, the formulation of the wind-battery-thermal unit commitment problem was designed to gather all system operating constraints. The solution to the optimization problem was procured on a personal computer using a general algebraic modeling system. To assess the performance of the proposed model, a simulation study based on the ten-unit power system test was applied. The effects of battery energy storage and wind power were deeply explored and investigated throughout various case studies.

A Mixed-Integer and Asynchronous Level Decomposition with Application to the Stochastic Hydrothermal Unit-Commitment Problem

Independent System Operators (ISOs) worldwide face the ever-increasing challenge of coping with uncertainties, which requires sophisticated algorithms for solving unit-commitment (UC) problems of increasing complexity in less-and-less time. Hence, decomposition methods are appealing options to produce easier-to-handle problems that can hopefully return good solutions at reasonable times. When applied to two-stage stochastic models, decomposition often yields subproblems that are embarrassingly parallel. Synchronous parallel-computing techniques are applied to the decomposable subproblem and frequently result in considerable time savings. However, due to the inherent run-time differences amongst the subproblem’s optimization models, unequal equipment, and communication overheads, synchronous approaches may underuse the computing resources. Consequently, asynchronous computing constitutes a natural enhancement to existing methods. In this work, we propose a novel extension of the asynchronous level decomposition to solve stochastic hydrothermal UC problems with mixed-integer variables in the first stage. In addition, we combine this novel method with an efficient task allocation to yield an innovative algorithm that far outperforms the current state-of-the-art. We provide convergence analysis of our proposal and assess its computational performance on a testbed consisting of 54 problems from a 46-bus system. Results show that our asynchronous algorithm outperforms its synchronous counterpart in terms of wall-clock computing time in 40% of the problems, providing time savings averaging about 45%, while also reducing the standard deviation of running times over the testbed in the order of 25%.

Reserve Model of Energy Storage in Day-Ahead Joint Energy and Reserve Markets: A Stochastic UC Solution

With many favorable advantages including fast response ability in particular, utility-level energy storage systems (ESS) are being integrated into energy and reserve markets to help mitigate uncertain renewable resources and fluctuant demands. This paper discusses a stochastic unit commitment (UC) model to explore capabilities of ESSs in providing valuable grid services by simultaneously joining energy and reserve markets. The proposed reserve model of ESSs presents the following features: (i) two constraints are proposed to formulate ESS's reserve provision ability in each hour via six operation modes, namely increasing/reducing the level of charge, switching to discharge, increasing/reducing the level of discharge, switching to charge; (ii) as an energy-limited asset, constraints on ESS's reserve deliverability across multiple hours is introduced to hold enough state of charge (SOC) headroom and floor room, guaranteeing that reserves cleared in individual hours are continuously deliverable over multiple successive hours without violating operation limits. Moreover, a scenario-based UC model, which addresses uncertainties of renewables and demands, is adopted to evaluate reserves provided by ESSs and generators. To effectively solve the stochastic UC problem, the progressive hedging algorithm with heuristic approaches is discussed. A 6-bus system and a modified IEEE 118-bus system are used to illustrate effectiveness of the proposed approaches.

Impact of off-nominal frequency values on the generation scheduling of small-size power systems

This paper studies the impact of off-nominal steady-state frequency values on the generation scheduling of small-size power systems. For that, a security-constrained stochastic unit commitment problem that explicitly considers frequency variations and primary frequency control is proposed. The frequency of the system under normal operating conditions is characterized by a probability distribution that can be estimated from real-world measurements. The reserve scheduling is considered to face uncertain variations of renewable power and demand. A set of contingencies modeling generating unit failures is considered to ensure a N-1 security criterion. Numerical results obtained from a realistic case study show that modeling off-nominal steady-state frequency values in the scheduling model reduces the probability of experiencing high frequency deviations and unserved demand after contingency.

Sensitivity Analysis of the Impact of the Sub- Hourly Stochastic Unit Commitment on Power System Dynamics

Subhourly modeling of power systems and the use of the stochastic optimization are two relevant solutions proposed in the literature to address the integration of stochastic renewable energy sources. With this aim, this paper deals with the effect of different formulations of the subhourly stochastic unit commitment (SUC) problem on power system dynamics. Different SUC models are presented and embedded into time domain simulations (TDS) through a cosimulation platform. The objective of the paper is to study the combined impact of different frequency control/machine parameters and different SUC formulations on the long-term dynamic behaviour of power systems. The analysis is based on extensive Monte Carlo TDS (MC-TDS) and a variety of scenarios based on the New England 39-bus system.

Asynchronous Lagrangian scenario decomposition

We present a distributed asynchronous algorithm for solving two-stage stochastic mixed-integer programs (SMIP) using scenario decomposition, aimed at industrial-scale instances of the stochastic unit commitment (SUC) problem. The algorithm is motivated by large differences in run times observed among scenario subproblems of SUC instances, which can result in inefficient use of distributed computing resources by synchronous parallel algorithms. Our algorithm performs dual iterations asynchronously using a block-coordinate subgradient descent method which allows performing block-coordinate updates using delayed information, while candidate primal solutions are recovered from the solutions of scenario subproblems using heuristics. We present a high performance computing implementation of the asynchronous algorithm, detailing the operations performed by each parallel process and the communication mechanisms among them. We conduct numerical experiments using SUC instances of the Western Electricity Coordinating Council system with up to 1000 scenarios and of the Central Western European system with up to 120 scenarios. We also conduct numerical experiments on generic SMIP instances from the SIPLIB library (DCAP and SSLP). The results demonstrate the general applicability of the proposed algorithm and its ability to solve industrial-scale SUC instances within operationally acceptable time frames. Moreover, we find that an equivalent synchronous parallel algorithm would leave cores idle up to 80.4% of the time on our realistic test instances, an observation which underscores the need for designing asynchronous optimization schemes in order to fully exploit distributed computing on real world applications.

PFR ancillary service in low‐inertia power system

A high level of renewable resource penetration could lead to displacement of conventional synchronous generators from dispatch, and consequently, reduce the system inertia. The decline in the system inertia will increase the primary frequency response (PFR) need requiring to maintain grid frequency stability. This study proposes a new market design to co-optimise energy, inertia and PFR while considering uncertainties in renewable energy productions. A full dynamic model of Electric Reliability Council of Texas (ERCOT) interconnection is used to quantify the PFR requirement for the system inertia. The proposed method explicitly incorporates frequency dynamics and uncertainties in energy production from renewable resources in the scheduling process, which is formulated as a stochastic unit commitment problem. A case study of ERCOT grid demonstrated that the proposed stochastic scheduling of energy, inertia and PFR could yield a more cost-effective solution than the traditional deterministic formulation for the grids with a high penetration of renewable resources.

Tractable Stochastic Unit Commitment for Large Systems During Predictable Hazards

Many foreseeable natural hazards, including extreme weather events, lead to an outage of multiple transmission lines. Although such outages can be predicted in advance, there is a great deal of uncertainty in these predictions. To appropriately use the failure estimations in power system scheduling, this paper formulates a stochastic unit commitment (SUC) problem with explicit modeling of the predicted outages. The formulated problem, however, is extremely computationally-demanding, as the uncertainty is placed on the binary status of transmission lines. This paper, then, develops a computationally efficient algorithm to solve the formulated SUC for large-scale systems. The algorithm employs generation shift factors to enable rapid calculation of power flows. Additionally, flow canceling transactions are used to model multiple line outages without having to recalculate shift factors. Finally, critical constraints are iteratively detected and added to the problem. This approach substantially reduces the size of the problem, which helps computational tractability. The effectiveness of the developed algorithm is demonstrated through simulation studies on the Texas 2000-bus test system. The algorithm is used to minimize the lost load during a hypothetical hurricane. The results show that the algorithm is computationally tractable and can effectively identify a preventive dispatch, leading to a substantial reduction in power outages.

Reliability constraint stochastic UC by considering the correlation of random variables with Copula theory

This essay performs a reliability constraint stochastic model for unit commitment problem by considering generation and transmission constraints with high wind penetration and volatility of load demands. This query is expressed as a MILP that is based on the linear direct current model. The proposed approach models uncertainty of wind generators output power, load demand fluctuations and stochastic elements outage of the system like generators and transmission lines. In this paper, stochastic interdependence between random variables like wind speed and load demand is recognized. To establish the probability distribution of these correlated random variables, Copula theory is applied. Correlation structure between wind speed of different locations and a group of loads existing in the same area is investigated and studied based on historical data. For representing these uncertainties in the stochastic unit commitment problem, possible scenarios are generated with Monte Carlo simulation method. The reliability constraints are utilized in each scenario to evaluate the feasibility of solutions from a reliability point. The introduced stochastic UC is executed on the RTS 96-bus test system. Numerical results demonstrate the advantages of implementing stochastic programming on the UC problem by taking into account the intermittent behavior of wind energy and load inconstancy.

Stochastic hydro-thermal unit commitment via multi-level scenario trees and bundle regularization

For an electric power mix subject to uncertainty, the stochastic unit-commitment problem finds short-term optimal generation schedules that satisfy several system-wide constraints. In regulated electricity markets, this very practical and important problem is used by the system operator to decide when each unit is to be started or stopped, and to define how to generate enough energy to meet the load. For hydro-dominated systems, an accurate description of the hydro-production function involves non-convex relations. This feature, combined with the fine time discretization needed to represent uncertainty of renewable generation, yields a large-scale mathematical optimization model that is nonlinear and has mixed-integer variables. To make the problem tractable, a novel solution strategy, based on multi-horizon scenario trees, is proposed. The approach deals in a first level with the integer decision variables representing whether units are on or off. Once units are committed, the expected operational cost is minimized by solving a continuous second-level problem, which is separable by scenarios. The coordination between the two decision levels is done by means of a bundle-like variant of Benders decomposition that proves very efficient for the considered setting. To assess the quality of the optimal commitment on out-of-sample scenarios, a new simulation technique, based on certain sustainable pseudo-distance is proposed. For the numerical experiments, a mix of hydro, thermal, and wind power plants extracted from the Brazilian power system is considered. The results confirm the interest of the approach, particularly regarding a more efficient management of hydro-plants, because non-convex operational regions are considered by the model.