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The 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. [ABSTRACT FROM AUTHOR]

Multi-energy systems with carbon capture system (CCS) and power to gas (P2G) can reduce carbon emissions while increasing the consumption of renewable energy. This paper proposed a long-term equilibrium including electricity, gas and carbon markets to study the price effect of the system and its impact on the carbon-gas- electricity sector. The simulation results indicate that the main impact of the system on the electricity sector is to reduce energy curtailment and provide flexible resource for the electric system. P2G used an excess of 22.21 TWh of renewable energy, while CCS used 2.65 TWh of electricity to operate the capture system. The multi-energy system will reduce the market clearing price in the gas market (the annual average gas price decreased from 276.90 to 276.30 yuan/MWh). However, the partial transfer of flexibility to the electricity sector may increase cost in the gas industry, which in turn will be transferred back to the electricity sector again. In addition, through the operation of multi-energy system, carbon elements will circulate in the three sectors. CCS will become a carbon price-maker when electricity scarcity and carbon quota tightening, and increase carbon prices, which will increase the utilization hours of CCS and P2G and their profitability. • A carbon-gas-electricity market coupling model is proposed. • The multi-energy system will reduce the market clearing price in the gas market. • The partial transfer of flexibility may increase cost in the gas industry • CCS will become a carbon price-maker when electricity scarcity. [ABSTRACT FROM AUTHOR]

This paper proposes a logit-based multi-class ridesharing user equilibrium assignment framework that can incorporate different policy measures such as car restrictions, cordon tolling, and subsidization. The framework is formulated as a mixed complementarity problem (MCP). Numerical studies are conducted to illustrate model properties and compare the effects of these measures under different circumstances. The results show that the effectiveness of different policy measures can be greatly influenced by the performance of the transit mode compared with that of the driving mode and the users' preference for traveling by car. The "cordon toll" policy can be better than the "car restriction" policy in terms of the improvement in social surplus when the performance of the competing transit system is poor. Subsidizing ridesharing (transit fares) using the toll income is the most effective when the performance of the competing transit system is poor (good). It is also found that the implementation of the cordon toll policy cannot effectively promote ridesharing when the performance of the transit system is good. [ABSTRACT FROM AUTHOR]

The existence of prosumers who concurrently produce and consume electricity with distributed renewable resources such as photovoltaic generation has been viewed as an effective way of fortifying grid resilience and enhancing sustainability. However, prosumers' self-consumption is impacted by how surplus electricity is priced when feeding it into the grid. From the perspective of fixed cost recovery, this study compares market outcomes when their transactions with the grid are subject to net-metering and net-billing policies considering both the wholesale and retail markets. Under the net-billing policy, prosumers may face a power price lower than the retail rate or the wholesale price when selling their surplus energy into the grid. Compared to the benchmark case, where prosumers are subject to the same price signals as conventional consumers and producers, prosumers would over-consume under the net-billing case. The net-billing policy converges to the net-metering policy when prosumers purchase power from the grid. The analysis also finds that the net-metering policy can result in more wealth transfer from conventional producers and consumers to prosumers when compared to the net-billing policy. • From the perspective of fixed cost recovery, we examine net-metering and net-billing policy. • Compared to the benchmark case, prosumers would self-consume more under the net-billing case. • Net-metering policy results in more wealth transfer among participants compared to the benchmark. [ABSTRACT FROM AUTHOR]

China's natural gas market is focusing on price reform and aims to reconstruct vertically integrated industrial chains in the future. Based on the mixed complementarity problem model of gas markets with nodes in Henan Province, China, as an example, this paper applies numerical modeling to simulate the effects of social welfare and equilibrium prices on nodes in two scenarios: pipeline integration and pipeline separation. The findings reveal the following: (1) Pipeline separation yields greater overall social welfare than pipeline integration, with the welfare shifting from gas producers to consumption markets. (2) Pipeline separation lowers the equilibrium consumption prices by driving competition among gas supply sources. (3) Pipeline separation will increase the contribution of natural gas to primary energy. [ABSTRACT FROM AUTHOR]

This study proposes a modified formulation for the popular fast decoupled load flow method to facilitate automatic incorporation of reactive power flow related constraints, i.e. generator reactive power and in‐phase transformer tap limits. This formulation has been developed by viewing each device constraint as a set of complementary conditions and adopting the complementarity framework to define one equation to represent this set. The original set of load flow equations is augmented with this set of new equations representing the device constraints and solved to get the adjusted solutions directly. This approach is a significant departure from all the earlier approaches. The earlier approaches generally adopt a two‐stage scheme. In the first stage, an approximate unadjusted solution is obtained and in the next stage, adjustments are attempted by ad hoc schemes. The new formulation presented here permits handling both the reactive power related constraints simultaneously. However, it can also be used in situations where only one type of adjustment is needed. Two alternate algorithms in complementarity framework based on mixed complementarity problem and non‐linear complementarity problem formulations are presented here. Simulation results are presented to provide evidence of the impressive performance of the new algorithms as compared with the existing ones. [ABSTRACT FROM AUTHOR]

Abstract A node-to-node (NTN) scheme for modeling three-dimensional contact problems within a scaled boundary finite element method (SBFEM) framework is proposed. Polyhedral elements with an arbitrary number of faces and nodes are constructed using the SBFEM. Only the boundary of the polyhedral element is discretized to accommodate a higher degree of flexibility in mesh transitioning. Nonmatching meshes can be simply converted into matching ones by appropriate node insertions, thereby allowing the use of a favorable NTN contact scheme. The general three-dimensional frictional contact is explicitly formulated as a mixed complementarity problem (MCP). The inherent nonlinearity in the three-dimensional friction condition is treated accurately without requiring piecewise linearization. Contact constraints for non-penetration and stick-slide are enforced directly in a complementarity format. Numerical examples with 1st and 2nd order elements demonstrate the accuracy and robustness of the proposed scheme. Highlights • A node-to-node contact scheme leading to great simplification and robustness in enforcing contact constrains. • Nonmatching meshes are converted to matching meshes using the scaled boundary finite elements. • The surface remeshing is limited to the contact interface and no volume remeshing is required. • The node-to-node contact problems are formulated and solved as mixed complementarity problems. • Robustness and accuracy of the proposed scheme are demonstrated. [ABSTRACT FROM AUTHOR]

Turkey is a key country in order to realize the Southern Gas Corridor (SGC) due to its geographical location. However, as the main transit country within the SGC, Turkey could potentially exert market power with gas transits. Whether Turkey exerts market power or not, is crucial for an economic assessment of the SGC. Hence, the article investigates this issue quantitatively using a global partial equilibrium gas market model. An oligopolistic and a competitive supply structure in the European upstream market in 2030 are considered in the model based on calibrations to historical gas market situations. If the European gas market in 2030 is characterized by an oligopolistic supply, Turkey is able to exert market power resulting in higher prices compared to competitive transits, in particular in South Eastern Europe. In a competitive market structure, however, the importance of the SGC and thus the potential of Turkish transit market power is limited. [ABSTRACT FROM AUTHOR]

Abstract The growing development of gas-fired power plants over the last few decades has enhanced the interaction between power and natural gas sectors. As a result, considering an integrated gas and power network is essential to obtain a more precise and reliable analysis of generation and transmission capacity in the power network. This paper investigates a power generation and transmission expansion planning problem in an integrated and restructured electricity and gas market. To model interactions between the different participants of the markets and to identify the equilibrium point, a mixed complementarity problem approach is applied. This paper exploits Cournot game models to provide a behavioral framework for modeling of imperfect competition among electricity producers. Iran's real natural gas and power systems are used to test the proposed model, and its validation is conducted by comparing the pattern of the model outcomes with Iran's real data. The results indicate that expansion decisions lead to lower electricity price as well as improvement in social indexes such as consumer benefit and social welfare. In addition, it is revealed from the results that higher gas price due to low production capacity can lead to a change in the equilibrium point of the power market. Highlights • Investigating an integrated electricity and gas system by a combined analysis. • Formulating the mixed complementarity problem for the integrated system. • Providing a realistic solution for the expansion planning of Iran's power system. • Power producers' expansion decisions influence other actors' behavior. • Expansion decisions in the power system foster social indexes. [ABSTRACT FROM AUTHOR]

In this paper, three novel approaches are presented for the clearing of the European day-ahead electricity market, incorporating the majority of the currently tradable products, namely the simple hourly orders, the block orders, the complex orders, and the “Prezzo Unico Nazionale” (PUN) orders. The first approach employs a Master Problem and a Sub-Problem sequentially solved within an iterative process, whereas two single-model approaches are also formulated for the efficient handling of the market orders’ clearing conditions and the handling of non-intuitive bilateral exchanges. In the single-model approaches, the clearing conditions of the block, complex, and PUN orders are explicitly incorporated in the Sub-Problem, by utilizing a Mixed Complementarity Problem formulation. The three approaches are compared in terms of solution efficiency (social welfare) and computational requirements. A sensitivity analysis is performed for all three approaches, in order to evaluate the efficiency of the attained solution with respect to the method used for the designation and subsequent elimination of all attained Paradoxically Accepted Orders. [ABSTRACT FROM AUTHOR]

This paper proposes a new node-to-node (NTN) scheme for modeling two-dimensional contact problems within a scaled boundary finite element method (SBFEM) framework. The analysis is formulated and solved directly as a mathematical programming problem known as a mixed complementarity problem (MCP). Polygon elements with arbitrary number of edges and nodes are constructed in the SBFEM modeling. Only the edges of these polygon elements are discretized thereby accommodating highly non-uniform discretization. Non-matching meshes are easily converted into matching meshes by appropriate node insertions, thereby allowing the use of the advantageous node-to-node contact scheme. Contact constraints of non-penetration and friction are described directly in a complementarity format that enables, for instance, the satisfaction of non-penetration without having to resort to user-specified control parameters. Numerical examples using 1st and 2nd order scaled boundary elements are presented to highlight the quality of the elements used through satisfaction of various patch tests, and the accuracy and robustness of the overall approach. [ABSTRACT FROM AUTHOR]

The ever-increasing demand for power leads to increased loading on the power system. It is necessary to determine the maximum loadability of power systems so that a sufficient margin can be maintained. Maximum loadability can be determined by solving the power flow equations successively while increasing the loading gradually; conventionally, the unadjusted power flow problem is solved. This also helps in determining the voltage profile with respect to loading. However, determination of loadability margin by the solution of adjusted power flow problem is challenging in the presence of certain limits such as the limits of tap-changing and phase-shifting transformers, and the operating limits of Flexible AC Transmission Systems (FACTS) controllers. In the course of determination of maximum loading, it is likely that these limits are encountered. A novel method for systematic handling of these limits is presented in this paper. A Mixed Complementarity Problem formulation is proposed for inclusion of the limits of tap-changing and phase-shifting transformers, and FACTS controllers. The proposed method is tested on the IEEE 300 bus system and the WSCC 9 bus system in the presence of FACTS controllers. [ABSTRACT FROM AUTHOR]

This paper presents the hard-linking between a bottom-up model of energy sectors and a top-down Computable General Equilibrium (CGE) model, for an oil-exporting country - Saudi Arabia - where domestic energy prices are set by the government. The integrated model is solved with a Jacobi algorithm where the bottom-up and the top-down model are formulated separately and solved iteratively until convergence. The model is used to assess the sectoral and economy-wide consequences of domestic energy price reforms. Removing energy price controls triggers fuel switching and energy efficiency gains, increases crude oil exports and leads to a massive deployment of solar photovoltaic. As a result of price reforms, at the 2030 horizon, Saudi Arabia's Gross Domestic Product (GDP) increases by 3 to 6% and energy-related CO 2 emissions are reduced by one third. • Energy use and exports play a key role in most of oil-exporting economies. • Hybrid models are needed to study energy policy shifts in oil-exporting countries. • The paper designs the first hard-linked top-down bottom-up model with price controls. • A Jacobi algorithm finds the equilibrium of the integrated economic model. • The model fully captures the economic effects of energy price reforms in Saudi Arabia. [ABSTRACT FROM AUTHOR]

Capacity remuneration mechanisms exist in many electricity markets. Capacity mechanism designs do not explicitly consider the effects of refurbishment of existing generation units in order to increase their reliability. This paper presents a stochastic mixed complementarity problem to examine the impact of refurbishment on electricity prices and generation investment. Capacity payments are found to increase reliability when refurbishment is not possible, while capacity payments and reliability options yield similar results when refurbishment is possible. Final costs to consumers are similar under the two mechanisms with the exception of the initial case of overcapacity. [ABSTRACT FROM AUTHOR]

A novel model for the clearing of European day-ahead electricity markets, with all associated types of orders, namely simple hourly orders, block orders, complex orders and PUN (“Prezzo Unico Nazionale”) orders is presented in this paper. An iterative algorithm is employed for the market clearing, bearing an inner iteration process for the handling of Paradoxically Accepted block and MIC orders, and an outer iteration process for the handling of PUN orders. During the iterative process a Master Problem and a PUN subproblem are sequentially solved and coordinated, for the efficient handling of all order clearing conditions and market and/or system constraints. The clearing conditions of each order type are explicitly incorporated in the PUN subproblem, formulating a Mixed Complementarity Problem bearing both primal and dual decision variables. The PUN subproblem concerns the sequential clearing of 24 hourly subproblems, employing a special technique for handling the possible inefficiencies due to the myopic nature of the hourly subproblem solutions, leading to enhanced behavior in terms of social surplus with respect to the state-of-the-art day-ahead market clearing solver. The model and the algorithmic process are evaluated in a west-European test case, demonstrating minimal computational requirements and proving its excellent escalation capabilities. [ABSTRACT FROM AUTHOR]

In this paper two mathematical programming models are presented and compared for the solution of the European electricity day-ahead market with adjustable block orders. The concept of “adjustable” products is initially introduced, as products that can be partially cleared; thus, they do not bear the inherent indivisibilities of the classical block orders that are tradable in the European markets. Also, the clearing conditions of these adjustable products are analytically described. Due to their flexibility, the market clearing can adjust their clearing status accordingly, in order to match supply offers with demand bids avoiding non-convexities in the problem solution surface. This leads to the elimination of paradoxically accepted and rejected block orders. These adjustable products are modeled within the context of the European electricity day-ahead market problem, which is formulated as two different mathematical programming models: a Mixed Complementarity Problem (MCP) and a Linear Programming model (LP). The two models are evaluated in terms of computational efficiency using the pan-European zonal power system, by considering an increasing number of adjustable products per product type. The MCP model can formulate more adjustable product types, by efficiently formulating respective non-linear mixed pricing rules, but it is computationally demanding; on the other hand, the LP model is computational efficient, but it is not flexible enough to handle an extended variety of adjustable supply/demand block orders. [ABSTRACT FROM AUTHOR]

Natural gas gained significant attention due to its low carbon emissions and competitive prices in North America relative to other energy sources. The Annual Energy Outlook 2015 projects the U.S. as a net exporter by 2017. Recently, Mexico launched its energy reform, aiming to expand domestic production by opening the market to private investors. The success or failure of these policy changes will impact the development of the natural gas market in North America. To analyze possible pathways of the Mexican energy reform, we develop the North American Natural Gas Model (NANGAM). NANGAM is a long-term partial equilibrium model that allows for endogenous infrastructure expansion and non-linear cost functions. NANGAM is calibrated using the most recent data available from U.S., Canadian, and Mexican sources. We find that, in order to reduce pipeline imports, Mexico depends on economic incentives that lower barriers to infrastructure investment and keep production costs at competitive levels. If reforms to guarantee these incentives are not successfully implemented, growing gas demand in Mexico will be satisfied by further supply from Texas and neighboring states. This will cause a ripple-effect of increasing production in other regions in the U.S. and a shift in trade flows across the continent. [ABSTRACT FROM AUTHOR]

This paper presents a bilevel model for describing the interdependence of decisions between a retailer/aggregator and the distributed resources owners that the former serves and represents. Decision making is performed in a hierarchical fashion. The retailer performs the energy management of various resources indirectly; price signals are used as incentives for inducing the entities owning the distributed resources in a behavior that optimizes the operation and economic profit of the microgrid that they compose. The two-level interaction is formulated mathematically as a mathematical programming problem with complementarity constraints, while nonlinearities are replaced by equivalent linear expressions. The results of a characteristic case study indicate the efficiency and scalability of the proposed model. [ABSTRACT FROM PUBLISHER]

It is estimated that 76% of commuters are driving to work alone while each of them experiences a 38-h delay annually due to traffic congestion. Ridesharing is an efficient way to utilize the unused capacity and help with congestion reduction, and it has recently become more and more popular due to new communication technologies. Understanding the complex relations between ridesharing and traffic congestion is a critical step in the evaluation of a ridesharing enterprise or of the effectiveness of regulatory policies or incentives to promote ridesharing. The objective of this paper is to introduce a mathematical framework for the study of the ridesharing impacts on traffic congestion and to pave the way for the analysis of how people can be motivated to participate in ridesharing, and conversely, how congestion influences ridesharing activities. We accomplish this objective by developing a new traffic equilibrium model with ridesharing, and formulating the model as a mixed complementarity problem (MiCP). We provide conditions on the model parameters under which there exists one and only one solution to this model. The computational results show that when the congestion cost decreases or the ridesharing inconvenience cost increases, more travelers would become solo drivers and thus less people would participate in ridesharing. On the other hand, when the ridesharing price increases, more travelers would become ridesharing drivers. [ABSTRACT FROM AUTHOR]

In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Since the 90s various policies have been applied for supporting the development of Renewable Energy Sources (RES), including quota or amount-based systems and price-based systems (feed-in tariffs or "FiT"). In both cases, there is a political stress when there is a need to increase the renewable uplift charge rates (out-of-market mechanism), in order to finance the RES projects. This issue is resolved by adopting a novel market framework, in which the demand entities' clearing price entails the whole cost they are willing to pay for their participation in the energy market, including energy prices, reserve prices and the RES uplift price. A Mixed Complementarity Problem is utilized for clearing the market, in which the demand clearing prices are implicitly defined by mixing the explicit prices for energy, reserves and the RES uplift. The model retains the consistency of the supply (energy and reserves) and demand cleared quantities with the respective bids and the clearing prices, and attains a significant decrease of the payments through the relevant uplift accounts. The efficiency of the proposed model is demonstrated on a 24-h day-ahead market simulation using the IEEE RTS-96, defining endogenously the RES uplift under a system-wide FiT and a Green Certificate mechanism. [ABSTRACT FROM AUTHOR]

Highlights: [•] A modified NR load flow which automatically includes generator Q limits is proposed. [•] Modelling Q–V relation at a PV bus using Mixed Complementarity Problem framework. [•] Simultaneously solving conventional load flow equations along with the merit function. [•] Extensive testing and comparison of the proposed method with conventional methods. [ABSTRACT FROM AUTHOR]

The concept of adjustable combinatorial products within the scope of the day-ahead electricity market is introduced in this paper. These adjustable products comprise simple and linked profile block offers/bids, with a variable energy quantity profile during the block periods. In the proposed formulation, these blocks can be partially cleared, namely, the indivisibility of the classical profile blocks that are present in European markets is relaxed. This flexibility allows the elimination of paradoxically accepted and rejected blocks and, thus, the elimination of iterative heuristic procedures for their handling. Such modeling is not possible as a mixed-integer linear programming problem, due to the inherent indivisibility denoted by the use of binary variables. Additionally, advanced profile bids are formulated for demand entities, called “joint demand profile block bids” in this paper, which involve not only inter-temporal relations, but also spatial relations with each other. This two-dimensional relation between demand profile blocks is novel and is deemed feasible with the model presented here. These advanced profile orders are incorporated in a day-ahead market problem, formulating a mixed complementarity problem model, which is solved using commercial solvers. The model is validated using the IEEE RTS-96 system; nevertheless, its scalability to large real-world power systems is straightforward. [ABSTRACT FROM PUBLISHER]

Due to the unbundling of power generation and transmission in the restructured power system, the coordination between generation expansion planning and transmission expansion planning has become a new challenge for both generation companies and transmission companies. In this article, a single-stage deterministic model is proposed to study the strategic interaction between generation and transmission enterprises. The Cournot model is used to simulate the expansion behaviours for generation and transmission enterprises. Equilibrium is obtained using the approach for the mixed complementarity problem, and the proposed model is applied to a three-bus system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bimatrix game using the robust optimization approach, in which each player may neither exactly estimate his opponent's strategies nor evaluate his own cost matrix accurately while he may estimate a bounded uncertain set. We obtain computationally tractable robust formulations which turn to be linear programming problems and then solving a robust optimization equilibrium can be converted to solving a mixed complementarity problem under the l ∩ l-norm. Some numerical results are presented to illustrate the behavior of the robust optimization equilibrium. [ABSTRACT FROM AUTHOR]

The integration of the spot electricity markets in Europe shall lead to multi-area power exchanges that will substitute the local markets. In such scheme, market prices are affected by physical (e.g., network) constraints, yet they should sometimes follow regulatory policy rules, which do not necessarily reflect or depend on physical characteristics. In some cases, complex pricing rules should be implemented, which impose price discrimination for supply and demand entities within the same area. The methodology presented in this paper enables the balancing of supply and demand in a multi-area market considering energy and reserve bids, under complex pricing rules, which mix energy and reserve prices. A demand bid corresponds to the whole cost a demand entity is willing to pay for its participation in the energy market, including the cost for the procurement of the necessary reserves. The approach attains price integration of energy and reserves markets, simultaneous settlement of energy and reserves, and significant decrease of the payments through the uplift accounts. The main principle is the formulation of a mixed complementarity problem for the system equilibrium conditions, in which supply and demand are associated to explicitly or implicitly defined prices, which may be different even in the same zone. [ABSTRACT FROM AUTHOR]

Market clearing has always been an issue of great interest and research as liberalized electricity markets evolved over time in many countries. As trading in electricity evolves rapidly, multi-area power exchanges appear to substitute the local markets. The tie-lines constitute a significant parameter in multi-area power exchanges, since congestion leads to price differentiation. Prices are affected by physical (e.g., network) constraints, yet they should sometimes follow regulatory policy rules, which do not necessarily reflect or depend on physical characteristics. Until now, all approaches in clearing a multi-area power dispatch (or a multi-area market) are based on a zonal or nodal pricing model, which is applied uniformly to both production and demand within the same zone (or at each node). These approaches are not able to deal with complex pricing rules, which impose price discrimination for supply or demand entities within the same area. This paper presents a mathematical approach for the solution of a multi-area dispatch, in which production and demand of the same area may be cleared in different prices. The main principle is the formulation of a mixed complementarity problem for the system equilibrium conditions, in which supply and demand are associated to explicitly or implicitly defined prices. Illustrative implementations and test results for a simple five-zone system and the 73-bus IEEE RTS-96 are presented. [ABSTRACT FROM AUTHOR]

The mixed complementarity problem (denoted by MCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In the paper, based on a perturbed mid function, we contract a new smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P0 function are discussed. Then we presented a predictor-corrector smoothing Newton algorithm to solve the MCP with a P0-function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the local superlinear convergence of the method is proved under some suitable assumptions. [ABSTRACT FROM AUTHOR]

Abstract: A new smoothing Broyden-like algorithm for solving the mixed complementarity problems (denoted by MCP) is proposed in this paper. The algorithm considered here first uses a floating genetic algorithm to quickly calculate superior results, which are close to the precise solution of the equivalent optimization problems for the mixed complementarity problems, and then by taking the results as the initial values in the following smoothing Broyden-like algorithm, which is based on a perturbed mid function and makes use of the line search rule of D.-H. Li (2000) , we obtain a satisfactory approximate solution. In addition, the existence and continuity of a smooth path for solving the mixed complementarity problem with a -function are also discussed. The new smoothing Broyden-like algorithm combines the advantages of quasi-Newton method of local superlinear convergence and a floating genetic algorithm in group search, and global convergence. Numerical results show that the new method is feasible and effective. [Copyright &y& Elsevier]

Abstract: We present a smooth approximation for the generalized Fischer–Burmeister function where the 2-norm in the FB function is relaxed to a general -norm (), and establish some favorable properties for it — for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000)  to the solution set. In particular, we investigate the influence of on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with has better numerical performance than the one with . [Copyright &y& Elsevier]

Abstract: In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily ) mixed complementarity problems based on well-known Chen–Harker–Kanzow–Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established. [Copyright &y& Elsevier]

Abstract: The mixed complementarity problem (denote by MCP(F)) can be reformulated as the solution of a smooth system of equations. In the paper, based on a perturbed mid function, we propose a new smoothing function, which has an important property, not satisfied by many other smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the rate of convergence of the method is proved under some suitable assumptions. [Copyright &y& Elsevier]

In this article, we investigate robust optimization equilibria with two players, in which each player can neither evaluate his opponent's strategy nor his own cost matrix accurately while may estimate a bounded set of the strategy or cost matrix. We obtain a result that solving this equilibria can be formulated as solving a second-order cone complementarity problem under an ellipsoid uncertainty set or a mixed complementarity problem under a box uncertainty set. We present some numerical results to illustrate the behaviour of robust optimization equilibria. [ABSTRACT FROM AUTHOR]

Abstract: Various models of traffic equilibrium problems (TEPs) with nonadditive route costs have been proposed in the last decade. However, equilibria of those models are not easy to obtain because the variational inequality problems (VIPs) derived from those models are not monotone in general. In this paper, we consider a TEP whose route cost functions are nonadditive disutility functions of time (with money converted to time). We show that the TEP with the disutility functions can be reformulated as a monotone mixed complementarity problem (MCP) under appropriate conditions. We then establish the existence and uniqueness results for an equilibrium of the TEP. Numerical experiments are carried out using various sample networks with different disutility functions for both the single-mode case and the case of two different transportation modes in the network. [Copyright &y& Elsevier]

We consider a generalized mixed complementarity problem (MCP) with box constraints and multi-valued cost mapping. We introduce a concept of an upper Z-mapping, which generalizes the well-known concept of the single-valued Z-mapping and involves the diagonal multi-valued mappings, and suggest an extension of the Jacobi algorithm for the above problem containing a composition of such mappings. Being based on its convergence theorem, we establish several existence and uniqueness results. Some examples of the applications are also given. [ABSTRACT FROM AUTHOR]

Suppliers in competitive electricity markets regularly respond to prices that change hour by hour or even more frequently, but most consumers respond to price changes on a very different time scale, i.e., they observe and respond to changes in price as reflected on their monthly bills. In this paper, we examine mixed complementarity programming models of equilibrium that can bridge the speed of response gap between suppliers and consumers yet adhere to the principle of marginal cost pricing of electricity. We develop a computable equilibrium model to estimate ex ante time-of-use (TOU) prices for a retail electricity market. It is intended that the proposed models would be useful 1) for jurisdictions (e.g., Ontario) where consumers' prices are regulated, but suppliers offer into a competitive market, 2) for forecasting forward prices in unregulated markets, and 3) in evaluation and welfare analysis of the policies regarding regulated TOU pricing compared to regulated single pricing. [ABSTRACT FROM AUTHOR]

Abstract: The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we propose a new smoothing Broyden-like method for the solution of the mixed complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. [Copyright &y& Elsevier]

Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the local superlinear convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations. Moreover, they are also weaker than convergence conditions of the linearization (Josephy-Newton) method. For the special case of optimality systems with primal-dual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions. [ABSTRACT FROM AUTHOR]

This paper shows how bottom-up activity analyses within a dynamic computable general equilibrium framework can be undertaken for the longer-term analysis of energy and climate policies using the model SCREEN [25]. In particular we demonstrate for the case of Switzerland how the impact of policy measures to reduce the carbon intensity of the energy sector can be assessed with such a model for various socio-economic and environmental dimensions (e.g., C02 emissions, GDP, employment, foreign exchange rate). The results can provide valuable insights for the appropriate design of energy or climate policies that allow for the targeted fostering of a more sustainable energy development. [ABSTRACT FROM AUTHOR]

In this work, we study an alternative approach for capping and pricing carbon emissions in electric markets: the cap-and-trade paradigm with re-trade of allowances. We model the electric market (generators and allowances' auctioneer) as a two stage stochastic capacity expansion equilibrium problem, where we allow future investment and re-trading of emission permits among generators. The model studies generation and future investments in the electric sector in two regimes of demand: deterministic and stochastic. The configuration enforces the reduction of carbon emissions by setting a carbon budget, which allows to assess the impact of green policies and pledges concerning an electric system. We use the proposed model to analyze the Chilean electric sector under a cap-and-trade paradigm as an alternative to the existing carbon tax. We show that the Chilean pledge regarding emissions reductions does not encourage a shift to greener technologies. Moreover, we characterize two strategies to comply with the renewable targets by mid-century. On the one hand, a stringent carbon budget that induces high price of carbon permits and phases out coal-based generators. On the other hand, a less stringent target which significantly encourages investment in renewable technologies, but with low remaining shares of coal-based electric generation towards 2050. • A two-stage stochastic capacity investment equilibrium model with CO2 markets. • CO2 markets for allowance allocation and re-trade with endogenous prices. • Current Chilean GHG intensity pledge does not drive a shift in the generation mix. • E_ective carbon prices should be at least ten times larger than the Chilean tax. • Strategies to comply with the renewable energy targets by mid-century. [ABSTRACT FROM AUTHOR]

The second-order cone mixed complementarity problems (SOCMCPs) can directly present the KKT conditions of the second-order cone programming, and have broad range of applications. We establish the generalized lower-order penalty algorithm in this article for solving the SOCMCPs. By using the proposed algorithm, the SOCMCP is converted to asymptotic lower-order penalty equations (LOPEs). Under the assumption that the involved function possesses the ξ -monotone, we show that when the penalty parameter tends to positive infinity, the solution sequence of the asymptotic LOPEs converges to the solution of the SOCMCP at an exponential rate. Numerical results are reported to examine the efficiency of the proposed algorithm. • Based on asymptotic lower-order penalty equations, the generalized lower-order penalty algorithm is proposed to solve the SOCMCPs. • Under the assumption of ξ -monotone for the considered function, the unique solvability of the SOCMCP and asymptotic generalized lower-order penalty equations are obtained, and the solution sequence of asymptotic generalized lower-order penalty equations converges to the optimal solution of the SOCMCP at an exponential rate when the penalty parameter tends to positive infinity. • Numerical examples indicate the efficiency of the proposed algorithm based on lower-order penalty equations, and it has better performance than smoothing F-B algorithm in some cases. [ABSTRACT FROM AUTHOR]

Whether shipping market affects international iron ore trade arouses a lot of discussions in recent years. To evaluate the impact of shipping, we build a mixed complementarity-based equilibrium model to understand the strategic behavior and interaction among major players, including importers, exporters, and carriers. Different from models found in other resource trades, this one considers the heterogeneity of iron ores and the production capacity allocation of exporters. In this three-party equilibrium model, importers and exporters compete in a Cournot fashion, incorporating the endogenous freight rate derived from a carrier module. This model is an instance of Discretely Constrained Mixed Complementarity Problem (DC-MCP), for which we provide a solution procedure based on Mixed Integer Nonlinear Program (MINLP) reformulation and convexification techniques. Our model performs better compared with those with given shipping cost and production capacity constraint. In addition, we apply the model to analyze the impact of shipping on the iron ore international trade through a numerical study. The results show that shipping can slightly dampen the negative impact on iron ore trade volume from an unexpected reduction in importer budget. For different exporters, this dampening effect varies by freight rate. [ABSTRACT FROM AUTHOR]

An elastic domain with multiple yield surfaces brings about huge troubles to numerical computations due to singularities, while non-associative plasticity leads further to the open issue of whether constitutive integration is well posed. This study reduces the constitutive integration of non-associative plasticity with multiple yield surfaces to a mixed complementarity problem, represented by MiCP, a special case of finite-dimensional variational inequalities. By means of the projection–contraction algorithm for finite-dimensional variational inequalities and the idea of the Gauss–Seidel method, a new projection–contraction algorithm for MiCP is designed and denoted by GSPC. Applying the monotonicity of the mapping of the MiCP, GSPC is proved convergent theoretically for associative plasticity. For non-associative plasticity, the sufficient condition for GSPC to be convergent is also established if the tension part of the Mohr–Coulomb elastic domain is cut off. Typical examples are designed to illustrate GSPC is highly efficient, accurate and stable, some of which cannot be solved by the conventional return-mapping method. In all the cases, the computational efficiency of GSPC is apparently above the mapping return method. [ABSTRACT FROM AUTHOR]

Policymakers in the USA have provided various mechanisms to grow the domestic biofuel industry. One of the most significant policies in the USA is the volume mandate specified within the Renewable Fuel Standard ( RFS). There are a number of other overlapping factors that impact the use of biofuels, namely the so-called blend wall, or the 10% blend limit of ethanol in gasoline, along with a complex system of tax credits. All of these policies directly affect the value of a Renewable Identification Number ( RIN), the tradable compliance certificate created as part of the RFS. Regulators track RIN prices carefully because they are a measure of the cost of compliance. In this work a mixed complementarity problem ( MCP) is presented to combine these market dynamics into one model. This tool was specifically designed for policymakers to compare scenarios and study the effects on key market variables including RIN, gasoline, and diesel prices, along with production quantities for a number of different finished blended fuels. Our results suggest that RIN prices will increase with an increase in the volume of biofuel mandated by the Environmental Protection Agency ( EPA); however the behavior of the different RIN prices depends on how the biofuel volumes are assigned among all the subcategories. Under scenarios investigated in this study, it is likely that a primary compliance strategy is to blend more biodiesel into diesel fuel. This behavior would increase the price of the D4 RIN but the premium could be mitigated by reinstating the biodiesel production tax credit. © 2015 Society of Chemical Industry and John Wiley & Sons, Ltd [ABSTRACT FROM AUTHOR]