Mixed Complementarity Problem Research Articles

Electricity sector impacts of water taxation for natural gas supply under high renewable generation

The future of energy development could be soon restrained by water scarcity. Shale gas, which is highly water-consuming in its extraction, is even more vulnerable. Thus, understanding the power system’s response to water constraints considering uncertainties from variable renewable energy generation becomes critical. This study presents a multi-stage stochastic Mixed Complementarity Problem for the energy sector, considering renewables variability and water scarcity’s impact on natural gas extraction. The power and natural gas sectors were modeled with hourly and daily resolution, respectively. Results demonstrate that stochastic planning enables smoother natural gas production and storage strategies, ensuring consistent operations with a narrower production range. Water taxation affects gas production and the generation mix, leading to electricity price changes. Gas-based electricity generation varies up to 16.2% based on water taxation and wind availability. Furthermore, water taxation of up to 20% can raise gas prices by up to 21.1%. This study underscores the need to integrate water scarcity considerations into energy planning, offering insights into sustainable decision-making, water resource management, reliable electricity generation, and price stability amidst evolving renewable portfolios.

Efficient coordination of top-down and bottom-up models for energy system design: An algorithmic approach

To account for endogeneity effects in, e.g., energy demand, modern bottom-up energy system models are often linked to a top-down model describing the macroeconomic system. Solving such linked models involves iteratively passing solutions from one model to the other and vice versa until convergence is reached, which can be computationally demanding. This paper proposes a coordination algorithm that speeds up convergence for the linkage of the two models in case the bottom-up model is a linear program and the top-down model is a mixed-complementarity problem. The coordination algorithm uses duality theory to select optimal bases from previous iterations to predict the solution of the bottom-up model. If the predicted solution is correct, which is shown to be equivalent to the predicted solution vector being non-negative, the bottom-up problem need not be solved in that iteration, resulting in a time gain. Numerical experiments on an energy system design problem illustrate that our coordination algorithm correctly predicts the bottom-up solution in most iterations, resulting in a significant reduction in overall computation time.

Analyzing the Impact of Black Sea Gas Production on the Turkish Natural Gas Market with a Mixed Complementarity Problem Approach

The discovery of natural gas in Black Sea has been a milestone for the Turkish natural gas market. This study examined the impacts of the Black Sea natural gas production on the Turkish natural gas market using a market equilibrium model, which is simulated with existing data from the Turkish natural gas market and solved using the GAMS software. The findings suggest that if global natural gas and oil prices remain as expected or stronger, Black Sea production will place downward pressure on end-user prices; however, if global market prices are lower than expected, natural gas will not be produced and will not impact the natural gas market under an oligopolistic market structure. Our model adds to the literature by offering an economic analysis of a gas production project through a market equilibrium modelling approach.

Optimal intersection design and signal setting in a transportation network with mixed HVs and CAVs

It is widely recognized that human-driven vehicles (HVs) and connected and autonomous vehicles (CAVs) are expected to coexist and share the urban traffic infrastructure in the transportation network for a long time. To fully utilizes CAVs’ potential to reduce congestion in the transitional period, this study proposes and addresses the intersection design and signal setting problem in the transportation network with mixed HVs and CAVs. Due to the difference in terms of communication technology and autonomous driving technology for HVs and CAVs, three types of intersections have been proposed to amplify the efficiency-improvement benefit from CAVs by separating CAVs from HVs in a temporal or local-spatial dimension: the conventional signalized intersection, the novel signalized intersection with a dedicated CAV phase and dedicated CAV approaches, and the intelligent signal-free intersection. The problem is to determine the spatial layout of different types of intersections in the transportation network, the cycle time, and green time duration for each phase of signalized intersections that minimize the total travel cost, in which the route choice behavior of heterogeneous travelers has been respected based on the user equilibrium principle. A mixed-integer nonlinear programming model is developed to formulate the proposed intersection design and signal setting problem based on the link-node modeling method, in which the path enumeration is avoided. Then, by employing various linearization techniques (e.g., disjunctive constraints, logarithmic transformation, piecewise linearization with logarithmic-sized binary variables and constraints, outer-approximation technique), the proposed model can be further transformed into a relaxed sub-problem in the form of mixed-integer linear programming. A globally optimal solution algorithm embedding with solving a sequence of relaxed sub-problems and nonlinear mixed complementarity problems is proposed to converge to a global optimum. The results of numerical experiments illustrate that the proposed methodology can significantly improve the performance of the whole network. Moreover, it consistently outperforms the optimization model considering only conventional signalized intersections under various CAV market penetration rates.

Pollution trading as a mathematical game

A game theory approach is used to study a market-based pollution trading strategy among different point sources. Point sources discharge effluents to the same watershed and participate as rational entities with particular objectives, interacting with each other in a hierarchical organization. Each point source must implement technologies to sell credits to other sources or buy credits to compensate for non-compliance with environmental regulations. The objective function of the formulations proposed considers the minimization of total annual costs for each point source. The participants in the pollutant credit trading are related in a Nash Game by a mixed complementarity problem (MCP) and in a Stackelberg Game by a bilevel optimization problem (BLP). Both systems are solved using the disjunctive constraint method. Different scenarios of interaction among the participants are proposed in order to observe the behavior of each decision strategy. These models should generate more reliable results with respect to non-competitive approaches as the interaction of the participants is considered.

A Class of Smoothing Modulus-Based Iterative Methods for Solving the Stochastic Mixed Complementarity Problems

In this paper, we present a smoothing modulus-based iterative method for solving the stochastic mixed complementarity problems (SMCP). The main idea is that we firstly transform the expected value model of SMCP into an equivalent nonsmooth system of equations, then obtain an approximation smooth system of equations by using a smoothing function, and finally solve it by the Newton method. We give the convergence analysis, and the numerical results show the effectiveness of the new method for solving the SMCP with symmetry coefficient matrices.

Stochastic ridesharing equilibrium problem with compensation optimization

In the urban traffic system with ridesharing programs, we develop a generalized stochastic user equilibrium model to formulate travelers’ mode and route choice behavior. To suit more general scenarios, the proposed model takes into consideration travelers’ heterogeneity in terms of car ownership and value of time, and travelers’ limited perceived information based on the stochastic user equilibrium principle instead of Wardrop’s user equilibrium principle. The proposed model is formulated as variational inequalities and an equivalent nonlinear mixed complementarity problem due to the inseparable and asymmetric travel cost functions. Furthermore, we address the decision-making problem of ridesharing compensation from the perspective of traffic managers and policy-makers who want to minimize the total travel cost and vehicular air pollution emissions simultaneously. A bi-objective optimization model and two single-objective optimization models are proposed to formulate this decision-making problem, in which travelers’ mode and route choice behavior has been respected. As a mathematical problem with complementarity constraints, the bi-objective optimization model is solved by an improved Non-Dominated Sorting Genetic Algorithm II to generate a set of Pareto-optimal solutions for policy-makers and allow them to choose desired solutions. Finally, several numerical experiments based on two different scales of networks are conducted to demonstrate the properties of the problem and the performance of the proposed model and algorithm. The results show that rational pricing of ridesharing compensation can indeed mitigate urban traffic congestion and pollution emissions simultaneously. Moreover, by integrating travelers’ choice behavior based on the stochastic user equilibrium principle instead of the user equilibrium principle in the ridesharing compensation optimization model, this study derives a series of more effective decision-making strategies for ridesharing compensation.

The role of demand response in mitigating market power: a quantitative analysis using a stochastic market equilibrium model

Market power is a dominant feature of many modern electricity markets with an oligopolistic structure, resulting in increased consumer cost. This work investigates how consumers, through demand response (DR), can mitigate against market power. Within DR, our analysis particularly focusses on the impacts of load shifting and self-generation. A stochastic mixed complementarity problem is presented to model an electricity market characterised by an oligopoly with a competitive fringe. It incorporates both energy and capacity markets, multiple generating firms and different consumer types. The model is applied to a case study based on data for the Irish power system. The results demonstrate how DR can help consumers mitigate against the negative effects of market power and that load shifting and self-generation are competing technologies, whose effectiveness against market power is similar for most consumers. We also find that DR does not necessarily reduce emissions in the presence of market power.

The interpolating element-free Galerkin method for the p-Laplace double obstacle mixed complementarity problem

The interpolating element-free Galerkin method for the p-Laplace double obstacle mixed complementarity problem

Bayesian Nash Equilibrium in Electricity Spot Markets: An Affine-Plane Approximation Approach

This article proposes a system of stochastic mixed complementarity problems (MCP) for calculating the Bayesian Nash equilibrium (BNE) in an electricity spot market. The generators submit strategic multilevel price-offer functions under incomplete information about their rivals’ marginal cost. This strategic multilevel price offering of each generator is modeled as a standard stochastic bilevel program which is then reformulated as its equivalent MCP. The merger concept is employed and formulated to model multilevel price-offer functions. We then propose the BNE-MCP model which solves the stochastic MCP models of all generators together. A scenario reduction technique is also developed to accurately model the wind and demand uncertainty in the proposed BNE-MCP model. The proposed BNE-MCP model is carefully studied on two illustrative examples. The modified IEEE 14-, 30-, and 57-node systems are also examined in our article.

Strategic Investment in Power and Heat Markets: A Nash–Cournot Equilibrium Model

The current efforts to address the expansion planning of power and heat production facilities can lead to suboptimal results from a market perspective because these efforts often implicitly assume that power and heat systems are planned by a single entity. The present study addresses this issue by establishing a Nash–Cournot equilibrium model, where the producers determine the quantity of power and heat that they produce independently and simultaneously, and thereby collectively affect the price. The model considers three types of producers, including power-only, combined power and heat, and heat-only producers, where each producer determines its own expansion investment strategies independently with the aim of maximizing its own profit. An investment equilibrium is obtained by reformulating the equilibrium model as a mixed complementarity problem. Numerical results carried out on two test systems illustrate the impact of transmission congestion and supply-side market power on the investment equilibria.

The access to CETA quotas: Extending CGE models with a market for quota licenses

Abstract We analyze the market dynamics that are caused by tariff-rate quotas, particularly the effects of quota license allocation between heterogeneous commodities at the tariff line level. The allocation is endogenously modeled with a mixed complementarity problem approach for the case of the Comprehensive Economic and Trade Agreement between Canada and the European Union. The model results are compared both with alternative models that resemble pre-existing approaches and with the real trade figures that have been collected since the trade agreement's implementation. Our analysis shows a bias toward more expensive commodities if the shadow value of a quota license manifests in a secondary license market. The same quota can thereby be binding to some commodities but not so for others. This feature of quotas can be crucial for policymakers who are concerned about price effects or who want to understand the effects of lumping together commodities of different quality in one quota.

Strategic investment decisions in an oligopoly with a competitive fringe: An equilibrium problem with equilibrium constraints approach

Modern wholesale electricity markets often have producers who exercise market power. The standard way to model market power in an oligopoly with a competitive fringe is by using Mixed Complementarity Problems (MCPs) and conjectural variations. However, such models can lead to myopic (sub-optimal) behaviour for oligopolists. We first build on existing literature to show that an oligopoly with a competitive fringe where all firms have investment decisions will also lead to myopic behaviour when modelled using MCPs with conjectural variations. To overcome this issue, we develop an Equilibrium Problem with Equilibrium Constraints (EPEC) to model such an electricity market structure. The EPEC models two types of players: price-making firms, who have market power, and price-taking firms, who do not. Our model is the first in the literature to consider an oligopoly with a competitive fringe where all firms have investment decisions. Our results indicate that an EPEC can model investment decisions in an oligopoly with a competitive fringe more credibly and thus overcome the myopic behaviour observed in MCPs. The EPEC found multiple equilibria for investment decisions and firms’ profits. Despite this, market prices and consumer costs were found to remain relatively constant across the equilibria. A further contribution of the modelling approach is that it shows how it may be optimal for price-making firms to accept losses in some time periods in order to disincentivize price-taking firms from investing further into the market. We conclude our paper with a discussion of the computational limitations of our approach.

Improved Gauss Seidel based projection–contraction algorithm for the mixed complementarity problem in contact problem

AbstractThe GSCP (Gauss–Seidel based projection‐contraction) algorithm has been well established in solving elastoplastic mixed complementarity problems (MiCPs) or other problems which have similar mathematical form. However, the stability problems in the GSPC algorithm may cause nonconvergence or program crashes. Thus, the method of constructing and analyzing counterexamples is adopted to study the reasons for the instability of the GSPC algorithm. The R‐GSPC algorithm was developed by modifying the part of the GSPC algorithm that caused the crash to make the algorithm stable. In addition, this paper analyzes the convergence, compatibility, and initial value impact of R‐GSPC, and finds that the R‐GSPC algorithm greatly improves the robustness and stability without losing accuracy. This paper allows the GSPC algorithm to be better extended to more fields and to be suitabled for being introduced into commercial programs.

Location of Tension Cracks at Slope Crests in Stability Analysis of Slopes

Over the conventional limit equilibrium method and limit analysis method, the finite element method is advantageous, especially for slopes involving complex failure mechanisms where the critical slip surfaces cannot be represented by log spirals and other similarities. In the presence of tension cracks at slope crests, however, the finite element method encounters difficulties in convergence while handling Mohr–Coulomb’s yielding surfaces with tensile strength cut-off. Meanwhile, the commonly used load-controlled method for the system of nonlinear equilibrium equations is hard to bring the slope into the limit equilibrium state. The two drawbacks drag down the finite element method in more extensive applications. By reducing the constitutive integration of plasticity with non-smooth yielding surfaces to the mixed complementarity problem, the convergence in numerical constitutive integration is established for arbitrarily large incremental strains. In order to bring the slope to the limit equilibrium state, a new displacement-controlled algorithm is designed for the system of nonlinear equilibrium equations, which is far more efficient than the load-controlled method. A procedure is proposed to locate tension cracks. Corresponding to the Mohr–Coulomb failure criterion with and without tensile strength cut-off, the failure mechanisms differ significantly, while the difference in the factor of safety might be ignorable.

An analysis on Turkey’s natural gas hub development through a game-theoretic model application

ABSTRACT Turkey’s natural gas hub development has been under the spotlight recently as its long-term gas supply contracts expire between 2021 and 2026. This study is an effort to investigate end-user prices and shippers’ behaviors in the natural gas hub development process through four noncooperative game models representing different maturity levels of the Turkish gas market. These models, comprising mixed complementarity problems, are solved by GAMS software using the Turkish gas market’s data. Our findings suggest that although increasing hub maturity yields decreasing end-user prices owing to the availability of more supply options and enhanced competition, a shift from long-term contracts to spot trading may result in an increase in the end-user prices due to strategic withholding by importers. Our models are pioneering as they are the first to depict the Turkish gas market with regard to shippers’ behaviors.

A compact MLCP-based projection recurrent neural network model to solve shortest path problem

ABSTRACT We develop a projection recurrent neural network (RNN) to obtain the solution of the shortest path problem (SPP). Our focus on the paper is to give a compact single-layer structure RNN model to solve the SPP. To present the RNN model, we utilise a mixed linear complementarity problem (MLCP). Moreover, the developed RNN is proved to be globally stable. Finally, some numerical simulations are stated to show the performance of the presented approach. We compare the results with some other methods.

Data envelopment analysis and stochastic equilibrium analysis for market power investigation in a bi-level market

This study investigates how the exercise of market power affects the economic efficiency and revenue sharing in a bi-level market, which integrates upstream contract market and downstream wholesale market. We consider price volatility and identify Nash equilibrium restricted by the data envelopment analysis (DEA)-estimated capacity constraint. This study develops stochastic mixed complementarity problems (MiCP) models embedded with stochastic programming (SP) and robust optimization (RO) to address price uncertainty in a bi-level market. We investigate the exercise of market power: (1) wholesalers as leaders or (2) bilateral market power between manufacturers and wholesalers. A numerical study of the power generation market is conducted to validate the proposed stochastic MiCP models. The results show that price uncertainties in the upstream contract market have a significant effect in a bi-level market, and thus controlling price volatility in the upstream contract market will improve productivity and support policies. In addition, we suggest the wholesalers should be market leaders and the manufacturers should negotiate profit-sharing contracts with the wholesalers in a deregulated system.

Dimension extending technique for constitutive integration of plasticity with hardening–softening behaviors

Existence and uniqueness of solution of constitutive integration of non-associative plasticity has been an open issue until it was partly solved in Zheng et al. (2020), where the constitutive integration of elastic-perfect plasticity is reduced to a Mixed Complementarity Problem (MiCP), a special case of finite-dimensional variational inequalities, and the qualitative properties of the MiCP have been well established even for non-smooth yield surfaces and non-associated flow rules. The algorithm for the MiCP, called GSPC, has been proved globally convergent. In order to handle plasticity with hardening and softening behaviors, hardening functions are deemed the same position as stress components. In this way, hardening/softening plasticity reduces to elastic-perfect plasticity, and the GSPC is ideally suited to hardening/softening plasticity with no need to revise. The application of the proposed procedure to the Modified Cam-Clay plasticity is demonstrated. Comparisons are made with the return mapping (R-M) algorithm, indicating that those examples causing R-M to fail to converge can be easily solved using GSPC.

Has the inter-regional transmission expansion promoted the low-carbon transition of China’s power sector?

China is facing unprecedented pressure to reduce carbon emissions than ever before. Significant differences do exist among regions in resource endowment, emission potential and environmental capacity. The introduction of inter-regional transmission lines is of great significance in alleviating environmental pressure and regulating the power mix. However, it is uncertain whether and to what extent the electricity sector has been locked into fossil fuel, as well as the regional differences in low-carbon transition. This study investigates whether the transmission grid expansion represented by more ultra-high voltage (UHV) lines construction will mitigate carbon emissions through a mixed complementarity problems equilibrium analysis. Through simulating the influence of various carbon emission caps on inter-regional transmission lines, it explores the regional differences in low-carbon transition in the power sector, and investigates the regions’ reliance on fossil-fuel from the perspective of carbon lock-in. We propose the following suggestions: (1) Mitigating the reliance on fossil fuel-fired power may relieve the low-carbon transition pressures of the central, eastern, and southern regions; (2) The inter-regional transmission lines help to regulate the distribution of carbon emissions by adjusting emission cost; and (3) The effect of the inter-regional transmission lines on mitigating the carbon lock-in of regional power sector will be weakened when facing a tight carbon emission cap.