Bi-level Optimization Framework Research Articles

Scenario-based stochastic optimization on the variability of solar and wind for component sizing of integrated energy systems

The inherent intermittency and variability of renewable energy sources present significant challenges to the optimal design and implementation of integrated energy systems (IES). This paper introduces a novel stochastic optimization model that integrates advanced scenario generation and clustering algorithm for renewable energy sources within a multi-objective, bi-level optimization framework. Specifically, the clearness index is employed to represent the stochastic distribution of solar radiation intensity by beta distribution, while wind speed uncertainty is modeled seasonally using the Weibull distribution. Monte Carlo sampling with synchronous back substitution is applied for scenario generation and reduction of solar radiation and wind speed. To address the multi-objective evaluation, the analytic hierarchy process is utilized, and the joint optimization is achieved by combining a region contraction algorithm with stochastic programming. The proposed methodology is validated on an IES featuring various heating devices, incorporating uncertainties in both wind and solar energy. The results indicate that the absorption heat pump-based scheme achieves superior energy-saving performance, achieving an energy rate of 0.4724. Additionally, the compression heat pump-based scheme exhibits excellent economic efficiency and environmental sustainability, with a cost of energy of 0.3639 and a renewable fraction of 0.5536.

Bi-level expansion planning of power system considering wind farms based on economic and technical objectives of transmission system operator

Planning for the expansion of generation and transmission infrastructure, with a focus on integrating wind farms is presented in this study according to the goals of economic efficiency, operational performance, security, and reliability of the transmission system operator. The approach incorporates a bi-level optimization framework, with the upper level outlining the formulation of power system expansion planning to reduce construction and operational costs while adhering to the investment budget limits. In the lower-level model, which economic reliable-secure functioning is formulated for the transmission network. The objective function of the problem aims to minimize operational costs, energy losses, and anticipated energy not supplied, and the voltage security index. The constraints of this problem include AC optimal power flow model, security, and reliability limits. Scheme extracts a convex-linear model for the formulation using the conventional linearization technique. Pareto optimization driven by the aggregation of weighted functions results in a single-objective framework for the lower-level problem, and the Karush-Kuhn-Tucker technique presents a single-level model for the scheme. The approach utilizes stochastic optimization has been adopted to model load, wind farms, and network equipment availability uncertainties. Finally, the study cases yielded numerical findings that showcase the efficacy of the proposed scheme in achieving the desired economic, operational, security, and reliability status within the transmission network. This is particularly true in situations where there is appropriate generation and transmission expansion.

Co-optimization of virtual power plants and distribution grids: Emphasizing flexible resource aggregation and battery capacity degradation

Coordination between virtual power plants and active distribution networks is crucial as these plants increasingly aggregate distributed resources within the power system. This study introduces a bilevel optimization framework to coordinate the scheduling of multiple virtual power plants and an active distribution network using pricing strategies for energy and reserves. The upper-level optimization minimizes total operating costs by incorporating bidding plans of the active distribution network in various markets, its interactions with multiple virtual power plants, and operational costs. The lower-level optimization maximizes revenue for each virtual power plant, considering both battery capacity degradation costs and operational costs of various resources. To facilitate solutions, this research developed a nonlinear transformation method for modeling capacity degradation. Based on the dispatching strategy from the virtual power plant, this study uses the squared difference between energy consumption of equipment for controllable loads and the strategy as the optimization target to derive control strategies for two equipment types. Results show that the framework effectively integrates dispatch and control strategies without oversimplifying the system model, proving its applicability in various scenarios with diverse resource compositions.

A decision-making model for joint energy and reserve scheduling of wind power producers with local intraday demand response exchange market

In this paper, a three-stage stochastic bi-level optimization framework is presented for optimal participation of wind power producers (WPPs) in day-ahead (DA), intraday, and balancing markets. In this framework, to leverage demand response (DR) services, a peer-to-peer (P2P) energy trading platform is implemented that allows local load aggregators (LAs) to contribute to the intraday markets improving both LAs’ and WPP’s benefits. Participating in the intraday DR exchange (IDRX) market enables WPP to purchase DR services from LAs, to reduce the penalty cost on the deviation between the day-head bidding and the real-time dispatch. A Stackelberg game for the bi-level decision-making model captures the conflict of interests between the WPP and LAs, in which, the upper level seeks to maximize WPP profit, while the lower level aims to maximize LAs’ economic surplus. The bi-level model is converted into its equivalent single-level mixed-integer quadratic problem (MIQP) employing the Karush-Kuhn-Tucker (KKT) conditions and strong duality theorem. Simulation results show that participation of the WPP in the IDRX market and employing spinning reserve and DR services for compensating the uncertainties are greatly dependent on its risk preferences and increase its expected profit in all conditions, significantly.

Strategic Investments and Portfolio Management in Interdependent Low-carbon Electricity and Natural Gas Markets

Addressing global warming necessitates carbon emissions reduction and renewable energy integration within the energy sector. Gas-Fired Power Plants (GFPP) are appealing to investors due to their low emissions and operational flexibility, which are considered necessary characteristics within low-carbon power systems with increasing renewable energy uncertainty. Investing in GFPPs presents intricate challenges due to the increasingly interdependent electricity and natural gas markets, especially in the light of a low-carbon economy. This work addresses these challenges for a strategic agent by proposing a bi-level optimization framework. The upper-level model derives the optimal electricity portfolio management regarding new investments and strategic biddings, while in the lower-level model, the electricity and gas markets are cleared sequentially under a Carbon Emission Trading Scheme (CETS). Case studies on a Pennsylvania-New Jersey- Maryland (PJM) 5-bus power system and an IEEE 24-bus test system demonstrate the applicability and efficacy of the proposed model in capturing the impact of a transitional integrated market framework on GFPPs investments. Also, introducing stochasticity to the model provides a better insight into the contrasting effects of emission allowance trading and gas prices on investment and bidding strategies.

The Virtual Power Plant Bidding Strategy Model based on Multi-stage Semi-anticipativity Distributionally Robust Optimization

For virtual power plants (VPPs), the strength of their market competitiveness largely depends on their ability to dispatch various distributed resources. An efficient bidding strategy model that can provide superior decision-making solutions is the key to enhancing this competitiveness. This paper focuses on price-making VPP and constructs an efficient three-period VPP bidding strategy model (3P-VPP) with a bi-level optimization framework for the day-ahead energy market by introducing three-period generator output constraints and generator ramping constraints. Furthermore, to address the uncertainty of renewable resources, we introduce the theory of distributionally robust optimization (DRO) and the concept of semi-anticipativity, refining the 3P-VPP model into a multi-stage DRO VPP bidding strategy model based semi-anticipativity (SADRO-VPP). The model possesses dynamic adaptation capabilities, tailoring the optimal decision-making solutions based on the accuracy of the forecast data. When faced with unreliable prediction data, the model will offer a more robust decision-making plan to guard against potential extreme scenarios. Conversely, when the forecast data is credible, the model adeptly utilizes this information to provide more economical decision-making strategies. The experimental results demonstrate that the 3P-VPP model exhibits higher computational efficiency compared to traditional model, while the SADRO-VPP model provides effective scheduling solutions and bidding strategies for VPP.

Multitasking optimization for the imaging problem in electrical capacitance tomography

Electrical capacitance tomography excels in measuring multiphase flows, but it faces challenges in reconstructing high-accuracy images. To tap into the potential of this measurement technique, we model the image reconstruction problem as a bilevel multitasking optimization problem, aiming to enhance imaging quality via effective knowledge transfer and integration across various tasks. This new imaging model consists of a target optimization problem and an auxiliary optimization problem, facilitating knowledge transfer and fusion between tasks. We design a new algorithm to solve the target optimization problem and integrate it with an autoencoder used for the cross-task knowledge transfer, forming a new multitasking optimizer to solve this proposed bilevel multitasking optimization imaging model. The extreme learning machine is developed for the prediction of learnable prior images by introducing a novel bilevel optimization framework that enables concurrent parameter selection and model training. A novel nested algorithm, designed to efficiently address optimization problems structured hierarchically, is developed to solve the training model. Evaluation results show that the new imaging algorithm outperforms popular imaging methods in terms of reconstruction accuracy and robustness, while also demonstrating stable performance. This proposed imaging algorithm achieves cross-task knowledge transfer and fusion, integrates measurement physics and machine learning, mines and integrates effective image priors, adaptively learns model parameters, alleviates the ill-posed property of the image reconstruction problem, increase the automation of the model, and improves imaging quality, robustness and performance stability, which provides a comprehensive solution to mitigate challenges in imaging.

Strategic operation of electric vehicle in residential microgrid with vehicle-to-home features

The urgent need for innovative energy solutions to facilitate the transition to renewable energy and electric vehicles (EVs) is particularly critical in developing countries, where high emissions and grid reliability challenges persist. This study investigates an off-grid residential renewable energy system with stationary storage like battery and hydrogen, alongside an EV as mobile storage. A bi-level optimization framework is established considering stationary and mobile energy storage options and a strategy is proposed to implement Vehicle-to-Home (V2H) in off-gird settings. Three cases are studied, with optimal component sizing determined using an optimization tool integrated with MATLAB. Firstly, to determine the most cost-effective stationary storage alternative, Case 1 uses hydrogen and Case 2 employs a battery as storage. Then Case 3 integrates the optimal stationary storage with mobile storage using V2H technology enabling the EV to support household energy needs during peak times. Results highlight that combining stationary and mobile storage (Case 3) offers a 15.92% reduction in Total Net Present Cost (TNPC) and a 13.9% reduction in Levelized Cost of Energy (LCOE) compared with stationary storage alone. This off-grid system (Case 3) also outperforms the conventional grid economically and environmentally, offering 2.13× TNPC reduction and 7,813 kg/yr carbon emissions mitigation.

A new bi-level model for the false data injection attack on real-time electricity market considering uncertainties

State Estimation (SE) plays a critical role in various applications of power systems including the electricity market. False Data Injection Attacks (FDIAs) are capable of manipulating the power system SE process aiming to gain financial profits by the players. In this paper, the potential economic influences of FDIA on SE and consequently the Real-Time (RT) electricity market are examined. Therefore, a new bi-level optimization framework is proposed to realistically model the FDIAs from the attacker perspective. At the upper-level, the attacker intends to intelligently choose the most critical congested transmission lines and manipulate their associated meter readings to maximize the expected financial profitability. At the lower-level, the RT market-clearing model is formulated under the influence of FDIA. Considering the incomplete attacker's prior knowledge of the network topology i.e., connection/disconnection of transmission lines, the angles of the phase-shifting transformers, and the real-time load uncertainty, the proposed formulation is extended to a new bi-level scenario-driven stochastic model. The developed attack model, which must remain undetectable by the system operator in all scenarios, is solved using the Karush–Kuhn–Tucker (KKT) approach. The correctness and effectiveness of the proposed optimization scheme have been validated in GAMS software using CONOPT and OQNLP solvers based on the IEEE 14-bus test system and also comparative results are presented to demonstrate the merits of the proposed formulation.

Aggregator-supported strategy for electric bus fleet charging: A hierarchical optimisation approach

The increasing adoption of electric buses in cities presents an opportunity for these vehicles to serve as mobile energy storage systems, providing valuable grid services through their substantial battery capacity. However, the complex interplay of fleet management and energy market participation requires that public transportation operators (PTOs) use adequate optimisation models and algorithms to make the best operational decisions. To address these issues, our research proposes an aggregator-supported charging strategy using a bi-level optimisation framework for electric bus fleets. The main goal is to achieve a balance between reducing PTO operational costs and maximising aggregator revenues, while ensuring reliable bus system operation. The results indicate that a coordinated charging strategy, including energy arbitrage and reserve provision, can significantly reduce operational costs, although potential limitations arise from associated battery degradation costs. Furthermore, managing fleets for reserve services appears to be more lucrative than engaging in energy arbitrage schemes. In this context, this study emphasizes the role of aggregator entities in managing the energy flexibility of electric bus fleets and enhancing the sustainability and cost-effectiveness of urban public transportation.

Optimizing local electricity markets: A bi-level primal-dual approach for integrating price-based demand response and distribution locational marginal pricing

This paper addresses the conundrum of optimizing electricity consumption patterns in response to fluctuations in demand and price, a task managed by both load aggregators and the distribution system operator (DSO). The conventional approaches in the literature to integrate demand response (DR) into optimal power flow (OPF) problems typically overlook the price responsiveness of consumers or simplify power flow equations to account for price-elastic demand.In this paper, we strive to close this gap by introducing a bi-level primal-dual optimization framework that incorporates aggregators' objectives into the market-clearing process while preserving the precision of the OPF equations. At the upper level, the model seeks to minimize both the total payment and peak load. The lower level, structured as a second-order conic (SOC) problem, aims to reduce overall generation costs subject to constraints of the second-order conic (SOC) branch flow model (BFM). The two levels interact through the optimal demand response and distribution locational marginal prices (DLMPs). The principles of convex duality principles together with the strong duality constraint are leveraged to transform the market-clearing problem into a single primal-dual problem. We also circumvent the non-linearity that stems from the DR payment term by incorporating discretizing loads and the big-M method, thus converting the problem into a mixed-integer SOC (MI-SOC) formulation. The merits of the proposed MI-SOC framework are validated through case studies conducted on the IEEE 33-bus test system, showcasing its potential to enforce price-elasticity demand response in distribution system's electricity markets.

A bi-level optimization framework for the power-side virtual power plant participating in day-ahead wholesale market as a price-maker considering uncertainty

This paper develops a novel bilevel decision-making framework for a power-side virtual power plant (VPP) comprising refined power to gas (P2G), wherein the upper-level is the cost minimization objective of VPP, while the lower-level is social cost minimization objective of wholesale electricity market. Meanwhile, wind power uncertainty is handled by distributionally robust optimization method. By utilizing Karush-Kuhn Tucker optimality condition and strong duality theory, the bilevel model is interpreted as a tractable single-level mixed integer programming model. Simulation results show that: relative to the models with removing uncertainty and refined P2G device, 1) the proposed approach realizes outstanding economic benefit by obtaining an increase with a rate by 7.47 % and 105.09 % in total actual profit, respectively. Regarding environmental benefit, CO2 emission volume experiences a 0.89 % decline and 0.90 % increase, and the CO2 intensity of the proposed strategy is the lowest with the value of 193. 25kg/MWh. 2) The proposed model can achieve the economic and low-carbon dispatching of the VPP. Wind power curtailment in the proposed model witnesses a trivial increase and reduced by 53.19 %, respectively. 3) The model exhibits merits due to its trade-off between robustness and computational complexity.

A data-driven approach to urban charging facility expansion based on bi-level optimization: A case study in a Chinese city

This paper addresses the optimization of urban charging facilities expansion in response to the growing charging demand and proposes a data-driven bi-level optimization framework. The proposed framework combines a mixed-integer programming model for siting and capacity planning, an improved sparrow search algorithm (ISSA) for hyperparameter optimization of the CNN-BiLSTM model, and the application of Benders decomposition with Lagrangian relaxation using Gurobi for solving the model. The proposed approach is empirically evaluated through a case study conducted in an administrative area of a Chinese city, using comprehensive charging records and other relevant information. The results demonstrate that the ISSA consistently outperforms other algorithms, showing an average improvement of 23.97 % in root mean square error, 19.25 % in average absolute percentage error, and 1.50 % in the determination coefficient. The modified Diebold-Mariano test confirms the significant advantage of ISSA. Based on the model analysis, the study recommends constructing 20 new charging stations in the case city, with more than half equipped with fast chargers. This strategy ensures that 95 % of the area can find a charging station within 5 min while also ensuring the operator's profitability. The research findings provide effective support for investment decisions related to charging facilities.

Real-time energy management strategy for flexible traction power supply system

Energy management strategies (EMS) for the flexible traction power supply system (FTPSS) typically depend on the accurate predictive model or probabilistic estimation of traction load and renewable energy generation. However, it can be challenging due to the ultra-short-term dramatic fluctuation in traction load profile. To address the issue, a model-free EMS is proposed to achieve the power dispatch in real time based on the current state. Firstly, a sequential stochastic optimization model is built to optimize the long-term operating cost and load fluctuation. Through the relaxation of time-coupling constraints and incorporation of the Lyapunov optimization technique, the problem is decomposed into individual optimization problems at each time slot. Considering the complementary characteristic of hybrid energy storage system, a bi-level optimization framework is constructed to separately control powers of the supercapacitor and battery with the hierarchical sequence method. More importantly, a fully informative control strategy with the known future load profile and renewable energy generation is adopted as the evaluation benchmark. The gap between the proposed method and the benchmark can be demonstrated to fall within a certain range of below 10% by mathematical derivation and extensive cases.

Modeling the power trading strategies of a retailer with multi-microgrids: a robust bi-level optimization approach

ABSTRACT The primary contribution of this paper lies in the development of a bi-level optimization framework designed to model a retailer’s power trading strategies within the context of multi-microgrids. Another noteworthy aspect of this research is managing the risk associated with the retailer’s decision-making process, achieved by addressing the unpredictability of market prices through a robust bi-level structure. In this model, the upper-level problem focuses on the retailer’s decision-making, while the lower-level problem addresses microgrid considerations. Furthermore, a load-shifting strategy is integrated into the microgrid problem to enhance the profitability-to-cost ratio for both the retailer and the microgrids. The results demonstrate a 3% increase in the retailer’s profits and a significant 30% reduction in the microgrids’ costs using the load-shifting strategy. In the most unfavorable scenario, the retailer experiences a profit reduction of 21.67%, while the maximum increase in the microgrids’ costs reaches 238.3%.

Charging autonomous electric vehicle fleet for mobility-on-demand services: Plug in or swap out?

This paper compares two prevalent charging strategies for electric vehicles, plug-in charging and battery swapping, to investigate which charging strategy is superior for electric autonomous mobility-on-demand (AMoD) systems. To this end, we use a queueing-theoretic model to characterize the vehicle waiting time at charging stations and battery swapping stations, respectively. The model is integrated into an economic analysis of the electric AMoD system operated by a transportation network company (TNC), where the incentives of passengers, the charging/operating shift of TNC vehicles, the operational decisions of the platform, and the planning decisions of the government are captured. Overall, a bi-level optimization framework is proposed for charging infrastructure planning of the electric AMoD system. Based on the proposed framework, we compare the socio-economic performance of plug-in charging and battery swapping, and investigate how this comparison depends on the evolving charging technologies (such as charging speed, battery capacity, and infrastructure cost). At the planning level, we find that when choosing plug-in charging, increased charging speed leads to a transformation of infrastructure from sparsely distributed large stations to densely distributed small stations, while enlarged battery capacity transforms the infrastructure from densely distributed small stations to sparsely distributed large stations. On the other hand, when choosing battery swapping, both increased charging speed and enlarged battery capacity will lead to a smaller number of battery swapping stations. At the operational level, we find that improved charging speed leads to increased TNC profit when choosing plug-in charging, whereas improved charging speed may lead to smaller TNC profit under battery swapping. The above insights are validated through realistic numerical studies.

Identification of optimal locations of adaptive traffic signal control using heuristic methods

Adaptive Traffic Signal Control (ATSC) adjusts signal timings to real-time traffic measurements, increasing operational efficiency within a network. However, ATSC is both expensive to install and operate making it infeasible to deploy at all signalized intersections within a network. This study presents a bi-level optimization framework that applies heuristic methods to identify a limited set of locations for ATSC deployment within an urban network. At the upper-level, the Population Based Incremental Learning (PBIL) algorithm is employed to generate, evaluate, learn, and update different ATSC configurations. The lower-level uses the delay-based Max-Pressure algorithm to simulate the ATSC configuration within a microsimulation platform. The study proposes improvements to the PBIL algorithm by considering constraints on the maximum number of intersections for ATSC deployment and incorporates prior information about the intersection performance (i.e., informed search). Simulation results on the traffic network of State College, PA reveal that the proposed PBIL algorithm consistently outperforms baseline methods that select locations only based on queue-lengths or delays in terms of reducing overall network travel times. The study also reveals that intersections experiencing the highest delays or longest queues are not always the best candidates for ATSC. Moreover, applying ATSC at all intersections does not always provide the best performance; in fact, ATSC applied to some locations could increase travel times by contributing additional congestion downstream. Additionally, the modified PBIL algorithm with the informed search strategy is more efficient at identifying promising solutions suggesting it can be readily applied to more generalized optimization problems.

A Bi-level model for district-fairness participatory budgeting: Decomposition methods and application

Participatory budgeting is one of the most well-known and widespread participatory programs implemented in many municipalities worldwide. Targeting poorer regions to receive a greater per capita amount of spending than wealthier regions is the most important transformational aspect of participatory budgeting. However, current approaches do not provide a precise method for achieving social justice through participatory budgeting. This paper proposes a bi-level mixed-integer non-linear optimization framework under the partial cooperation assumption to promote social justice in participatory budgeting programs. In addition, single-level reformulation and linearization techniques are presented, along with valid inequalities that speed up their resolution procedure. The single-level linear problem is solved using the Benders decomposition algorithm to find global optimality. To improve the computational performance of the proposed model on large-scale instances, a hierarchical iterative, evolutionary algorithm is proposed based on the hybrid binary particle swarm optimization and gravitational search algorithm. To illustrate the capability of the proposed model, computational experiments were conducted on both adapted examples from the literature and real-world, large-scale cases implemented in recent years in Warsaw, the capital of Poland. The results show that the proposed model is significantly more efficient, affordable, and faster than other methods presented in the literature, such as the Greedy rule and ε-district-fair lottery. In addition, the proposed model is fully operational for real-world societal problems.

Metalearning Generalizable Dynamics from Trajectories.

We present the interpretable meta neural ordinary differential equation (iMODE) method to rapidly learn generalizable (i.e., not parameter-specific) dynamics from trajectories of multiple dynamical systems that vary in their physical parameters. The iMODE method learns metaknowledge, the functional variations of the force field of dynamical system instances without knowing the physical parameters, by adopting a bilevel optimization framework: an outer level capturing the common force field form among studied dynamical system instances and an inner level adapting to individual system instances. Apriori physical knowledge can be conveniently embedded in the neural network architecture as inductive bias, such as conservative force field and Euclidean symmetry. With the learned metaknowledge, iMODE can model an unseen system within seconds, and inversely reveal knowledge on the physical parameters of a system, or as a neural gauge to "measure" the physical parameters of an unseen system with observed trajectories. iMODE can be generally applied to a dynamical system of an arbitrary type or number of physical parameters and is validated on bistable, double pendulum, Van der Pol, Slinky, and reaction-diffusion systems.

Tactical price-setter scheduling of a network-constrained integrated energy system under high penetration of solar distributed generations

The growing penetration of solar distributed generations (SDG) in integrated natural gas, electricity and heating systems will be challenging as these networks are intricately entangled. This study investigates the tactical price-setter behavior of an embedded network-constrained integrated energy system (NIES) in the wholesale electricity market (WEM) via a bi-level optimization framework under high penetration of solar distributed generations (SDG). The proposed NIES consists of an 8-node district heating system (DHS), a 20-node natural gas network (NGN) and IEEE 33-bus active distributions system. The first optimization level models the NIES operator that deploys thermal and electrical flexibilities as well as line-pack capability of the NGN pipelines to minimize the energy provision cost. Furthermore, it also strategically submits offers/bids in WEM as a price-setter. At this stage, the second level optimization is the WEM operator that receives offers/bids and clears the WEM (modeled by IEEE 6-node transmission network), aiming to reach maximum communal satisfaction. The intermittent SDG production is dealt with via the robust optimization (RO) approach. The results emphasize that the flexible energy technologies (line-pack, electrical and thermal flexibilities) can enhance the capabilities of NIES in influencing market price, and that leads to an 8.77% lower overall operational costs.