Karush-Kuhn-Tucker Research Articles

Distributed event-triggered voltage restoration and optimal power sharing control for an islanded DC microgrid

This paper proposes a distributed voltage restoration and power allocation control scheme for the DC microgrid (MG) system, which only relies on discrete and aperiodic event-triggered communication. Different from most existing control approaches, we attempt to break the hierarchy of secondary control and tertiary optimization, and solve the optimal power regulation and voltage restoration problems simultaneously in the secondary layer. Specifically, a distributed controller with event-triggered communication strategy is developed by not only considering the secondary voltage restoration error but also taking the Karush–Kuhn–Tucker (KKT) condition of tertiary power allocation optimization problem into account. In addition, compared to most distributed secondary control methods with continuous-time communication, our approach can still realize the same objectives only with limited noncontinuous communication, which would benefit us a lot in communication cost saving. An islanded DC MG test system consisting of three DGs is built in both Simulink and the laboratory to validate the proposed control scheme’s effectiveness.

Cold Chain Logistics Network Design for Fresh Agricultural Products with Government Subsidy

This paper investigates the cold chain logistics network design in the first mile for fresh agricultural products with government subsidy, involving the capacity, location of cold storage facilities, and transportation from production areas to cold storage facilities after harvest. A bi-level programming model is formulated considering the quality degradation of fresh agricultural products. Based on the proposition and KKT conditions, a solution method is designed for reformulating the bi-level model into a single-level programming model. Numerical experiments are conducted to verify the proposed model. Experimental results show that the solution method efficiently solves the problem of the cold chain logistics network design for fresh agricultural products with subsidy, and sensitivity analysis provides managerial insights for decision-makers.

Computation bits maximization in data compression-assisted MEC networks

In this article, we study the weighted sum computation bits (WSCB) maximization for a multiuser mobile edge computing (MEC) system while considering data compression at each edge user (EU). Considering the partial offloading scheme and data compression at each EU, we formulate a WSCB maximization problem by jointly optimizing the compression time, the computing and compression frequencies of all the EUs as well as the transmit power and time of EUs when performing task offloading. Since the formulated problem is non-convex and difficult to solve, we first replace the original problem with an equivalent one by using the proof by contradiction. Subsequently we introduce auxiliary variables for transforming the equivalent problem into a convex one that can be optimally solved by using the existing convex tools. Further, we devise the closed-form expressions for the optimal transmit power, compression and computing frequencies at each EU by using Karush-Kuhn–Tucker (KKT) conditions. Simulation results verify that our proposed scheme is better than several baseline schemes from the WSCB perspective.

Optimal bidding strategy of multi-carrier systems in electricity markets using information gap decision theory

This paper proposes a novel optimization framework based on information gap decision theory (IGDT) for strategic participation of multi-carrier systems (MCS) in the electricity market under price uncertainty. The proposed framework helps the operator of MCS to take advantage of flexible generation units such as combined heat and power (CHP) systems, boilers, and battery energy storage systems (BESS) and participate in the electricity market considering the strategies presented by IGDT against the uncertainty in the market clearing price. The proposed optimization framework is developed as a bilevel problem, in which the risk-based bidding strategy of the MCS operator is modeled in the upper-level problem, and the energy market is cleared by the bulk power system operator in the lower-level problem. Karush–Kuhn–Tucker conditions are used to transform the bilevel problem into a single-level optimization problem. Duality theory is later used to approximate the product of power and market clearing price in the MCS operation problem, and the non-linearity enforced by the implementation of IGDT is handled by a linearization method called Triangle method. The studied optimization framework is modeled as a single-level mixed-integer linear programming problem, and CPLEX is used to solve it. The simulations are carried out on the modified IEEE 6-bus transmission network, and the results are analyzed to verify the economic and computation efficiency of the studied model. According to the simulation results, by taking a risk-averse bidding strategy provided by the IGDT, the operation cost of MCS increases 16.6% in order to increase the robustness level of MCS up to 36%, indicating that MCS becomes robust against 36% of the increase in the electricity market price, compared to the case with a risk-neutral bidding strategy being taken.

A distributed approach for solving AC–DC multi-objective OPF problem

This paper first proposes an optimization model for Multi-Objective Optimal Power Flow (MO-OPF) problem in AC–DC networks. The considered objective functions are the fuel cost, active power losses, and environmental emission. The AC and DC power flow equations, the converter operating limits, the voltage magnitude limits, and line capacity limits are considered in the proposed AC–DC MO-OPF model. Then an equivalent Gradient Dynamical System (GDS) for the AC–DC MO-OPF model is derived. Each equilibrium point of this GDS is a Karush–Kuhn–Tucker (KKT) point of the original AC–DC MO-OPF model. In Theorem 1, we derive condition under which this KKT point is a saddle point of the original AC–DC MO-OPF problem. The equivalent GDS has the property that it can be solved using a distributed solution algorithm. Every bus in the AC–DC power grid is considered as a local agent in the proposed distributed GDS algorithm, which only obtains information only from its neighbors. In Theorem 2, we prove that proposed distributed approach is locally convergent to the saddle point of the original AC–DC MO-OPF problem. To show the performance of our distributed approach to solve our developed AC–DC MO-OPF model, the simulation results based on several test systems are provided.

Merchant Energy Storage Investment Analysis Considering Multi-Energy Integration

In this paper, a two-stage model of an integrated energy demand response is proposed, and the quantitative relationship between the two main concerns of investors, i.e., investment return and investment cycle and demand response, is verified by the experimental data. Energy storage technology is a key means through which to deal with the instability of modern energy sources. One of the key development paths in the electricity market is the development by energy merchants of energy storage power plants in the distribution network to engage in a grid demand response. This research proposes a two-stage energy storage configuration approach for a cold-heat-power multi-energy complementary multi-microgrid system. Considering the future bulk connections of distributed power generation, the two most critical points of energy storage station construction are the power generation equipment and specific scenarios for serving the community, as well as the purchase and sale price of electricity for serving the community microgrid (which directly affects the investment revenue). Therefore, this paper focuses on analyzing the different impacts caused by these two issues; namely, the two most important concerns for the construction of energy storage configurations. First, the basic model enabling wholesale electricity traders to construct energy storage power plants is presented. Second, for a multi-microgrid system with a complementary cold-heat-power multi-energy scenario, a two-stage optimum allocation model is constructed, whereby the upper model calculates the energy storage allocation problem and the lower model calculates the optimal dispatch problem. The lower model’s dispatch computation validates the upper configuration model’s reasonableness. Finally, the two-layer model is converted to a single-layer model by the KKT condition, and the nonlinear problem is converted to a linear problem with the big-M method. The validity is proved via mathematical examples, and it is demonstrated that the planned energy storage plants by merchants may accomplish resource savings and mutual advantages for both users and wholesale power traders.

Indentation of geometrically exact adhesive beams

We investigate the indentation of thin adhesive beams by a rigid frictionless circular punch in two-dimensions through computations and experiments. A non-linear geometrically exact (GE) beam formulation that allows for arbitrarily large displacements and rotations is employed for the thin beam. The Karush–Kuhn–Tucker condition for unilateral contact is adapted to model adhesive interaction in a nonlinear finite element (FE) framework. In this, contact is enforced through a penalty method and adhesion is incorporated through two different cohesive zone models (CZMs), viz. exponential and bilinear. The indentation of the adhesive GE beam is then studied in terms of the variations of the applied force, the pressure distribution, the cross-sectional rotation and the sizes of the contact area and the adhesive zone for different indentation extents, adhesive strengths, punch radii, beam thicknesses and end supports. We find that the system is largely agnostic to the CZM utilized, as well as the type of end supports unless the contact area spans the beam. At the same time, thinner beams tend to initiate contact with a punch more easily, and are harder to detach. We then demonstrate that thin beams may undergo large rotations even if the indentation is small, which necessitates a nonlinear beam theory. Under such conditions approximating a circular punch by a parabolic one used typically when indentations are small introduces errors. Finally, we conduct experiments on clamped adhesive beams made of two different types of PDMS (polydimethylsiloxane) and various thicknesses, and find a good match with our computations.

Stochastic two-stage coordination of electric vehicles in distribution networks: A multi-follower bi-level approach

As the number of electric vehicles (EVs) continues to rise, it is essential to consider appropriate management strategies for coordinating EVs connected to different buses in the power networks. In light of this, this paper proposes a stochastic two-stage bi-level model for coordinating EVs in a distribution network with charging stations under alternating current optimal power flow (ACOPF) constraints. The scheduling problem is considered to independently minimize the costs of the distribution system operator (DSO) and EVs parked at different charging stations located at various buses of the network. In the proposed model, the DSO as the leader, and all EVs as independent followers, are individual entities who try to follow their priorities and objectives. The amount and price of exchanged power between the DSO and EVs are optimally determined in the proposed model. The proposed bi-level model has been converted to a single-level model using the Karush–Kuhn–Tucker (KKT) conditions. Afterwards, the Big M method is used to convert the non-linear equations that appear due to utilizing the KKT approach. The scenario-based uncertainty modeling is used to model the uncertainty in input data such as day-ahead and real-time market prices, EVs’ initial state of charge (SOC), and arrival/departure time. The centralized unilateral form of the model has also been developed to validate the proposed model. The results indicate that the bi-level model can lead to cost reduction for the EVs.

Universum parametric $$\nu $$-support vector regression for binary classification problems with its applications

AbstractUniversum data sets, a collection of data sets that do not belong to any specific class in a classification problem, give previous information about data in the mathematical problem under consideration to enhance the classifiers’ generalization performance. Recently, some researchers have integrated Universum data into the existing models to propose new models which result in improved classification performance. Inspired by these Universum models, an efficient parametric$$ \nu $$ν-support vector regression with Universum data ($$ {\mathfrak {U}} $$UPar-$$ \nu $$ν-SVR) is proposed in this work. This method, which finds two non-parallel hyperplanes by solving one optimization problem and considers heteroscedastic noise, overcomes some common disadvantages of the previous methods. The$$ {\mathfrak {U}} $$UPar-$$ \nu $$ν-SVR includes unlabeled samples that don’t belong to any class in the training process, which results in a quadratic programming problem. Two approaches are proposed to solve this problem. The first approach derives the dual formulation using the Lagrangian function and KKT conditions. Furthermore, a least squares parametric$$ \nu $$ν-support vector regression with Universum data (named LS-$$ {\mathfrak {U}} $$UPar-$$ \nu $$ν-SVR) is suggested to further increase the generalization performance. The LS-$$ {\mathfrak {U}} $$UPar-$$ \nu $$ν-SVR solves a single system of linear equations, instead of addressing a quadratic programming problem in the dual formulation. Numerical experiments on artificial, UCI, credit card, NDC, and handwritten digit recognition data sets show that the suggested Universum model and its solving methodologies improve prediction accuracy.

Siting and sizing of energy storage for renewable generation utilization with multi-stage dispatch under uncertainty: A tri-level model and decomposition approach

For grids suffering from large-scale renewable generation curtailment, the reasonable allocation of energy storage can smooth renewable generation fluctuation for better utilization. This paper analyzes the optimal non-profit planning of energy storage in a grid rich in renewable generation from the perspective of third-party investors. First of all, to model the prediction errors of wind and solar output, a multi-stage scenario tree of intraday uncertainty is established to describe the stochastic output of wind and photovoltaics. In addition, scenario reduction techniques and nonanticipativity constraints are adopted to realize multi-stage optimization. Second, aimed to improve social welfare with an acceptable cost, a tri-level optimization model is established to describe storage siting and sizing, storage dispatch, and market clearing respectively to reach a high level of renewable generation utilization. Furthermore, Karush–Kuhn–Tucker conditions and linearization techniques are used to reform the tri-level model into a linear bi-level one. Third, a decomposition algorithm and a differential cut are proposed to solve the bi-level problem. The optimum is gradually approached via iterating by adding differential cuts and integer cuts. Finally, the model and method are verified based on a region of the HRP-38 system. The results show that storage investment aimed at social welfare can effectively achieve a much higher goal of renewable generation utilization than individual profit-seeking storage investment.

Flexibility Pricing of Grid-Connected Energy Hubs in the Presence of Uncertain Energy Resources

The paper expresses the problem of flexibility pricing in energy hubs (EHs) that are in connection with electricity, heat, and gas networks considering of uncertain energy generation sources. Scheme includes a bilevel formulation. Its upper-level states for modeling of the flexibility services are provided by various resources within the EH. The problem considers maximization of the expected profit of these resources in the flexibility market. The problem constraints include the flexibility model of flexible resources such as storage devices, responsive loads, and controllable distributed generations (DGs). The flexibility model of resources relies on their active and heat power. The lower-level problem calculates energy and flexibility prices and formulates the flexible operation of energy resources considering EHs. Here, constraints include optimal power flow equations in the energy networks; operation model of EHs with power sources, storage devices, and different responsive loads; and flexibility limits of EHs. Also, a linear approximation model is adopted in the suggested design using conventional linearization techniques. Next, the Karush–Kuhn–Tucker (KKT) method is used to derive a single-level model for the problem. The scheme adopts scenario-based stochastic programming (SBSP) so that uncertainties of renewable power, energy price, load, and energy consumption of mobile storage devices are properly modeled. Finally, the results validate the suggested design’s potential in modifying and enhancing the operation, flexibility, and economic situation of energy networks and EHs.

Analysis of strategic bidding of a DER aggregator in energy markets through the Stackelberg game model with the mixed-integer lower-level problem

An open electricity market environment provides sufficient conditions for a distributed energy resource (DER) aggregator to participate in the day-ahead electricity market. Therefore, a one-leader multi-follower Stackelberg game model is developed to analyze the strategic bidding behavior of a DER aggregator and one of the lower-level problems is to check the security of the distribution system (DS), which is elaborated by the mixed-integer second-order cone programming (MISOCP) model considering the active power loss and discrete controls in the DS. Moreover, we first use the Karush–Kuhn–Tucker (KKT)-based reformulation approach to transform the model into a one-leader one-follower model. However, the objective function of the upper-level model cannot be completely linearized by using the strong dual theorem because the active power loss of the DS is considered in the lower-level security check problem. Therefore, we propose a piecewise linear approximation approach based on the data-driven algorithm to linearize the upper-level objective function. Moreover, we apply multivariate linear surrogation to transform this one-leader one-follower model into a single-level MISOCP model. Finally, we verify the correctness of the proposed algorithm by comparing it with the particle swarm optimization (PSO) algorithm in a small integrated transmission and distribution (T&D) system and validate the effectiveness of the proposed algorithm in a practical integrated T&D system.

Ramification of information asymmetry on a green supply chain management with the cap-trade, service, and vendor-managed inventory strategies

The competition in the supply chain business has become considerably challenging. The environment protection is another problem adjacent to the competitive business. The supply chain players follow carbon policies to reduce carbon emissions along with profit maximization. But, asymmetric information sharing among supply chain players affects supply chain profit. This information asymmetry may be profitable for a few supply chain players but not beneficial for an entire supply chain and it may cause a non-coordination within them. A green supply chain management is developed with carbon tax & cap and trade regulations, where information asymmetry is reduced by the technology support through a radio frequency identification. This study finds the best service and distance between radio frequency identification readers to reduce asymmetry, and the optimum order quantity to obtain the maximum total profit for coordination or non-coordination policy. The objective function becomes a non-linear function with inequality constraint and the optimum solutions are derived using Kuhn–Tucker conditions. The optimum values of the objective function are found by using two subproblems because of the alternative scenarios of the cap and trade policy. Results indicate that a non-coordination policy is a better strategy than a coordination policy when information and product security play a major role in a green supply chain management. The effects of changes in cost parameters are identified via sensitivity analysis. The profit of the non-coordination policy is 6.07% more than the coordination profit.

Backscatter-enabled CR-NOMA based cooperative V2X communication with imperfect CSI

Non orthogonal multiple access (NOMA) and backscatter communications are considered to be promising technology due to their applications in large scale communications and spectral efficiency. Cognitive radio (CR) also has distinct advantage in terms of allowing secondary users to share spectrum with the primary users. Thus CR and NOMA provides a new way to enable vehicle to everything (V2X) communication through cooperation, in terms of the achievable rate (throughput), which is further enhanced with use of backscatter devices. Moreover, in V2X it is critical to inspect imperfect channel state information (CSI) because, mobility of vehicles leads to more channel estimation uncertainties. Therefore, CR based approach which is supported by NOMA with imperfect CSI, called as B-CR-NOMA is proposed in which several vehicles connect with the base stations (BSs) through different road side units (RSUs) and backscatter devices. To optimize the performance of the proposed approach, a problem of optimal (maximization) of throughput has been formulated. Moreover, the non-convexity associated with the signal to interference plus noise ratio (SINR) is removed by the use of Lagrangian dual transformation by introducing auxiliary variables, which is further solved under Karush–Kuhn–Tucker (KKT) conditions along with using sub-gradient techniques. The exceptional attainment of the throughput for the proposed approach in comparison to the contemporary algorithms in terms of: power as well as number of RSU, number of vehicle users and channel estimation error, accentuate the capability of the proposed approach. The fortitude of B-CR-NOMA is further substantiated in terms of fairness (fairness index), which is measured and compared based on the channel CSI: perfect and imperfect. Also, the effect of the auxiliary variables and iterations on the throughput for the proposed approach is also analyzed. Along with this, two key metrics: average outage probability and average fairness index, which measures the cooperative V2X communication, are also considered for in depth analysis of the proposed approach.

RETRACTED: Robust bidding strategy of interconnected multi-carrier systems in the electricity markets under the uncertainty in electricity load

This paper presents a robust bi-level model for optimal bidding of interconnected multi-carrier systems (MCSs) in the wholesale electricity markets under the uncertainty in electricity load. The proposed model implements a robust optimization method (ROM) to model the risk-based bidding of interconnected MCSs under the uncertainty in load in the upper-level problem while including the energy and reserve markets in the lower-level problem. The studied MSCs include combined heat and power (CHP) units, boilers, central air conditioning (CAC) systems, and battery storage units to provide local energy and make the MCSs’ operator capable of participating in the reserve market to gain profit. ROM models the electricity demand and provides different bidding strategies (i.e., risk-seeking, risk-neutral, and risk-averse) for the participation of MCSs’ operator in the energy and reserve markets. The studied bi-level model is converted to a single-level mixed-integer linear programming problem using the Karush–Kuhn–Tucker conditions and strong duality and then solved using the CPLEX solver of the General Algebraic Modeling System. Numerical experiments reveal that the interconnection of MSCs, along with the integration of storage units, enables the MSCs’ operator to control the uncertainty in the electricity load and increase the energy efficiency in order to optimally bid in the day-ahead energy and reserve markets. Results show that by enabling the MCSs to locally trade energy, the total operation cost of MCSs in the risk-seeking, risk-neutral, and risk-averse modes decreases by 5.65%, 4.28%, and 3.49%. Moreover, with the integration of battery storage systems into the interconnected operation of MCSs, the total operation cost of systems in the risk-seeking, risk-neutral, and risk-averse modes decreases by 7.65%, 6.99%, and 6.40%, respectively.

A bi-level scheduling strategy for integrated energy systems considering integrated demand response and energy storage co-optimization

In the context of China's “double carbon” commitment to the world, the introduction of integrated demand response mechanism and compressed air energy storage system into the traditional energy system is important to improve its structure, promote the interaction of multiple heterogeneous energy sources, and improve energy conversion efficiency. As such, in this study, we constructed an upper-layer capacity configuration model aiming to optimize the operating costs of compressed air energy storage systems during the planning period and subsequently integrated a heterogeneous energy network operation and China's certified emission reduction constraints under various operating scenarios. We established a lower-level optimal scheduling model with the optimization objective of minimizing the energy purchase, operation, maintenance, integrated demand response subsidy, and carbon trading costs. Finally, the KKT condition and the Big M method were used to solve this two-tier optimization problem. The comparative analysis of the three schemes showed that, under the consideration of the integrated demand response mechanism and the equipped compressed air energy storage system, the total operating cost of the system was reduced by an average of 2.28 %, the carbon emission was reduced by an average of 7.17 %, the profit from the sale of surplus carbon emission credits increased by an average of 16.7 %, and the energy storage capacity was reduced by approximately 39 %. Thus, this method effectively reduced the capacity of the energy storage system and significantly improved its reliable, economic, and low-carbon operation.

A KKT Conditions Based Transceiver Optimization Framework for RIS-Aided Multiuser MIMO Networks

In many core problems of signal processing and wireless communications, Karush-Kuhn-Tucker (KKT) conditions based optimization plays a fundamental role. Hence we investigate the KKT conditions in the context of optimizing positive semidefinite matrix variables under nonconvex rank constraints. More explicitly, based on the properties of KKT conditions, we optimize a reconfigurable intelligent surface (RIS) aided multi-user multi-input multi-output (MU-MIMO) network. Specifically, we consider the capacity maximization and sum mean square error (MSE) minimization problems of both the RIS-aided MU-MIMO uplink (UL) and downlink (DL) under multiple weighted power constraints and rank constraints. As for the RIS-aided MU-MIMO UL, the optimal structures of the signal covariance matrices are derived based on the KKT conditions. Furthermore, an efficient procedure is designed for solving the capacity maximization and sum mean square error (MSE) minimization problems. Then the UL-DL dualities are exploited for solving the capacity maximization and MSE minimization problems of the RIS-aided MU-MIMO DL based on the results of the UL optimization. Hence in the proposed framework, the phase shifting matrix of the RIS is jointly optimized with the signal covariance matrices for both the UL and DL. Our simulation results demonstrate the performance advantages of the proposed framework.

Strategic Pricing of Electric Vehicle Charging Service Providers in Coupled Power-Transportation Networks

Electric vehicles (EVs) are developing at a rapid pace. The EV charging station (EVCS) plays an indispensable role in supplying EV charging demand and promoting the transportation electrification. This work proposes a pricing method to maximize the profit of EVCS owners, charging service providers. First, the tri-level pricing framework is established where EVCSs set charging prices at the upper level, EVs make route and charging options at the middle level, and the electricity price is cleared in power distribution network at the lower level. Second, the singlelevel optimal pricing model is established with traffic flow assigned, power generation scheduled and electricity price explicitly contained via KKT conditions. The proposed model considers the impact of EVCS configuration on traffic assignment and pricing strategy. The operation risk brought by uncertainties is managed via Conditional value at risk. Third, a decomposition solution algorithm is proposed, which tackles the computational burdens and overcomes the infeasibility of existing algorithms. Finally, the real-world urban network-based case studies verify the effectiveness and merits of the proposed method.

Low-carbon planning for park-level integrated energy system considering optimal construction time sequence and hydrogen energy facility

With the increasing concern about global energy crisis and environmental pollution, the integrated renewable energy system has gradually become one of the most important ways to achieve energy transition. In the context of the rapid development of hydrogen energy industry, the proportion of hydrogen energy in the energy system has gradually increased. The conversion between various energy sources has also become more complicated, which poses challenges to the planning and construction of park-level integrated energy systems (PIES). To solve this problem, we propose a bi-level planning model for an integrated energy system with hydrogen energy, considering multi-stage investment and carbon trading mechanism. First, the mathematical models of each energy source and energy storage in the park are established respectively, and the independent operation of the equipment is analyzed. Second, considering the operation state of multi-energy coordination, a bi-level planning optimization model is established. The upper level is the capacity configuration model considering the variable installation time of energy facilities, while the lower level is the operation optimization model considering several typical daily operations. Third, considering the coupling relationship between upper and lower models, the bi-level model is transformed into a solvable single-level mixed integer linear programming (MILP) model by using Karush–Kuhn–Tucker (KKT) condition and big-M method. Finally, the proposed model and solution methods are verified by comprehensive case studies. Simulation results show that the proposed model can reduce the operational cost and carbon emission of PIES in the planning horizon, and provide insights for the multi-stage investment of PIES.

Bi-level transmission expansion planning considering prohibited operating zones and multi-fuel units

Transmission expansion planning (TEP) provides a solution to economically and securely expand the transmission network. In this context, the bi-level frameworks are receiving increased attention considering a DC or (linear/convex) AC economic dispatch at the lower level. However, the prohibited operating zones (POZ) and multi-fuel units (MFU) are ignored to be modeled at the lower level, while it plays a vital role in the output results. To this end, this article presents a bi-level TEP model that considers POZ and MFU. Incorporating these two issues into the lower level’s economic dispatch problem introduces more complexity to the optimization problem but provides more realistic optimal plans. Unlike the typical approaches to solving bi-level programming (BP), the formulated problem cannot be solved by Karush–Kuhn–Tucker (KKT) conditions or strong duality theory (SDT) due to binary variables at the lower level. To this end, an iterative-based reformulation-and-decomposition (R&D) technique is utilized to find the optimal solutions to the bi-level mixed-integer programming (BMIP) problem. The presented methodology is applied to the IEEE 118-bus system. The numerical results corroborate the effectiveness of the proposed structure and highlight that modeling POZ and MFU is an essential task in the TEP problems.