Multiparametric Linear Programming Research Articles

Impact of short-term and long-term energy storage units on power system operation economy: A parametric linear programming method

This paper evaluates the economic impact of short-term and long-term energy storage capacity on power system operation cost. First, the unit commitment (UC) model with short-term and long-term energy storage comprising a year-round hourly operation simulation is established to minimize the optimal operation cost. By introducing the convex hull of the original feasible region of decision variables, the integer variables in UC model are relaxed as continuous variables, transforming the UC problem as a linear programming (LP). Then, by regarding the power and energy capacities as parameters, the LP is cast as a multiparametric linear programming (mp-LP), where the optimal value function (OVF) is explicitly expressed as a piecewise linear function of the capacity parameters. An approximation method is used to calculate the analytical optimal value function and critical regions. The analytical expression of OVF contains rich sensitivity information. Finally, case studies validate our proposed method and provide visualization results. The sensitivity information of analytical OVF is exploited. As a potential application, the economic evaluation results could be applied in energy storage capacity sizing.

A Data-Driven Pool Strategy for Price-Makers Under Imperfect Information

This paper studies the pool strategy for price-makers under imperfect information. In this occasion, market participants cannot obtain essential transmission parameters of the power system. Thus, price-makers should estimate the market results with respect to their offer curves using available historical information. The linear programming model of economic dispatch is analyzed with the theory of rim multi-parametric linear programming (rim-MPLP). The characteristics of system patterns (combinations of status flags for generating units and transmission lines) are revealed. A multi-class classification model based on support vector machine (SVM) is trained to map the offer curves to system patterns, which is then integrated into the decision framework of the price-maker. The performance of the proposed method is validated on the IEEE 30-bus system, Illinois synthetic 200-bus system, and South Carolina synthetic 500-bus system.

Flexibility Requirement When Tracking Renewable Power Fluctuation With Peer-to-Peer Energy Sharing

Flexible load at the demand-side has been regarded as an effective measure to cope with volatile distributed renewable generations. To unlock the demand-side flexibility, this paper proposes a peer-to-peer energy sharing mechanism that facilitates energy exchange among users while preserving privacy. We prove the existence and partial uniqueness of the energy sharing market equilibrium and provide a centralized optimization to compute the equilibrium. The centralized optimization is further linearized by a convex combination approach, turning into a multi-parametric linear program (MP-LP) with renewable power output deviations being the parameters. The flexibility requirement of individual users is calculated based on this MP-LP. To be specific, an adaptive vertex generation algorithm is proposed to construct a piecewise linear estimator of the optimal total cost subject to a given error tolerance. Critical regions and optimal strategies are retrieved from the obtained approximate cost function to evaluate the flexibility requirement. The proposed algorithm does not rely on exact characterization of optimal basis invariant sets and thus is not influenced by model degeneracy, a common difficulty faced by existing approaches. Case studies validate the theoretical results and show that the proposed method is scalable.

Set Membership Estimation with Dynamic Flux Balance Models

Dynamic flux balance models (DFBM) are used in this study to infer metabolite concentrations that are difficult to measure online. The concentrations are estimated based on few available measurements. To account for uncertainty in initial conditions the DFBM is converted into a variable structure system based on a multiparametric linear programming (mpLP) where different regions of the state space are described by correspondingly different state space models. Using this variable structure system, a special set membership-based estimation approach is proposed to estimate unmeasured concentrations from few available measurements. For unobservable concentrations, upper and lower bounds are estimated. The proposed set membership estimation was applied to batch fermentation of E. coli based on DFBM.

Reliability and Vulnerability Assessment of Multi-Energy Systems: An Energy Hub Based Method

Multi-energy systems (MESs) make it possible to satisfy consumer's energy demand using multiple coupled energy infrastructures, thus increasing the reliability of the energy supply compared to separate energy systems (SESs). To accurately and efficiently assess and improve the reliability of MESs, this paper proposes a MES reliability and vulnerability assessment method using energy hub (EH) model. The energy conversion, transmission and storage in MESs are compactly and linearly described by EH model, making reliability and vulnerability assessment of MESs tractable. Indices for MES vulnerability assessment are proposed to find the key components for improving MES reliability. Multi-parametric linear programming (MPLP) with a self-adaptive critical region set is proposed to reduce the computational burden caused by iteratively solving LP problems for a large number of samples during the assessment process. The results of a case study show that the proposed reliability and vulnerability assessment method is able to effectively evaluate the energy supply reliability of different energy sectors in MES as well as find the critical component of an MES from reliability perspective to support its planning. The proposed algorithm, i.e., MPLP with a self-adaptive critical region set, can improve the computational efficiency by an order of magnitude compared to the traditional LP method.

Optimal power and energy sizing of compressed air energy storage in distribution network using multiparametric programming

As a new type of mechanical energy storage, compressed air energy storage (CAES) has attracted wide attention in recent years. This paper studies the optimal sizing problem of CAES in power distribution network (PDN). CAES plays a role in shaving the peak and filling the valley at demand side and thus reduces the operation cost of PDN to purchase electricity from the main grid. A multi-parametric linear programming (MP-LP) model is formulated where the power and energy capacities of CAES are regarded as parameters; the parameterized optimal value function (OVF) provides a graphical tool to describe the impact on operation cost of CAES configuration, which helps determine the planning strategy in a visual manner. The visualized results reveal not only the optimal solution, but also some useful information, like the sensitivity of operation cost to parameters. Case studies conducted on the IEEE 33-bus distribution system verifies the effectiveness of the proposed method.

Impact of electricity price forecasting errors on bidding: a price‐taker's perspective

Electricity price forecasting is very important for market participants in a deregulated market. However, only a few papers investigated the impact of forecasting errors on the market participants' behaviours and revenues. In this study, a general formulation of bidding in the electricity market is considered and the participant is assumed to be a price-taker which is general for most of the participants in power markets. A numerical method for quantifying the impact of forecasting errors on the bidding curves and revenues based on multiparametric linear programming is proposed. The forecasted prices are regarded as exogenous parameters for both deterministic and stochastic bidding models. Compared with the existing method, the proposed method can calculate how much improvement will be achieved in the cost or revenue of the bidder if he reduces the price forecasting error level, and such calculation does not require any predefined forecasting results. Numerical results and discussions based on real-market price data are conducted to show the application of the proposed method.

Design of reduced complexity controllers for linear systems under constraints using data cluster analysis

ABSTRACT A numerical method is proposed to reduce the complexity and computational effort involved in the application of the multiparametric linear programming technique in the design of offline controllers for linear systems subject to constraints. For this purpose, the concept of controlled invariant sets and the K q-flat data cluster analysis algorithm are applied. Specifically, we show how the K q-flat algorithm can be used to establish a smaller number of polyhedral regions associated with a piecewise affine explicit state feedback control law. We also propose a new approach in the design of sub-optimal controllers that further reduce the number of regions. Numerical examples show that a significant reduction in the complexity of the control law can be achieved by the proposed approach.

Impact of Energy Storage on Economic Dispatch of Distribution Systems: A Multi-Parametric Linear Programming Approach and its Implications

As a main flexible resource, energy storage helps smooth the volatility of renewable generation and reshape the load profile. This paper aims to characterize the impact of energy storage unit on the economic operation of distribution systems in a geometric manner that is convenient for visualization. Posed as a multi-parametric linear programming problem, the optimal operation cost is explicitly expressed as a convex piecewise linear function in the MW/MWh parameter of the energy storage unit. Based on duality theory, a dual linear programming based algorithm is proposed to calculate an approximate optimal value function (OVF) and critical regions, circumventing the difficulty of degeneracy, a common challenge in the existing multi-parametric linear programming solvers. When the uncertainty of renewable generation is considered, the expected OVF can be readily established based on OVFs in the individual scenarios, which is scalable in the number of scenarios. The OVF delivers abundant sensitivity information that is useful in energy storage sizing. Leveraging the OVFs, a robust stochastic optimization model is proposed to determine the optimal MW-MWh size of the storage unit subject to a given budget, which gives rise to a simple linear program. Case study provides a clear sketch of the outcome of the proposed method, and suggests that the optimal energy-power ratio of an energy storage unit is between 5 and 6 from the economical perspective.

Minimax strategy in approximate model predictive control

It is known that a model predictive control law for a linear dynamical system with a linear or quadratic cost function can be explicitly computed as a piece-wise affine function. However, the number of regions required grows rapidly with the horizon length, the number of states and constraints limiting the deployment of explicit solutions to relatively small MPC problems, and motivating approximate solutions requiring less storage for online implementation. Unfortunately, the offline computation required to generate the approximate solution can be very high using many existing algorithms. In this paper, we propose a new procedure to generate the approximate solution based on barycentric interpolation whilst retaining a (less conservative) certification of the controller. This novel certification procedure requires solving a number of small-scale multiparametric linear programs together with convex optimization problems. During the online implementation of the approximate MPC, only a small linear program has to be solved to evaluate the control law. The efficacy of the proposed approach is demonstrated through both simulation and also in application to a canonical cart–pole stabilization problem.

Memory-Efficient Mixed-Precision Implementations for Robust Explicit Model Predictive Control

We propose an optimization for space-efficient implementations of explicit model-predictive controllers (MPC) for robust control of linear time-invariant (LTI) systems on embedded platforms. We obtain an explicit-form robust model-predictive controller as a solution to a multi-parametric linear programming problem. The structure of the controller is a polyhedral decomposition of the control domain, with an affine map for each domain. While explicit MPC is suited for embedded devices with low computational power, the memory requirements for such controllers can be high. We provide an optimization algorithm for a mixed-precision implementation of the controller, where the deviation of the implemented controller from the original one is within the robustness margin of the robust control problem. The core of the mixed-precision optimization is an iterative static analysis that co-designs a robust controller and a low-bitwidth approximation that is statically guaranteed to always be within the robustness margin of the original controller. We have implemented our algorithm and show on a set of benchmarks that our optimization can reduce space requirements by up to 20.9% and on average by 12.6% compared to a minimal uniform precision implementation of the original controller.

Probabilistic Optimal Energy Flow of District Multienergy Systems: An MPLP-Based Online Dictionary-Learning Approach

The integration of multienergy systems (MES) provides an important opportunity to increase the technical, economic, and environmental performance of the overall system compared with separately operated energy systems. The operation of district MES encounters numerous uncertainties such as fluctuations in energy prices as well as loads in different energy forms. In this paper, we formulate a probabilistic optimal energy flow (P-OEF) model of a district MES into a linear program (LP) problem based on the linear energy hub model. We propose an improved multiparametric LP (MPLP) based online dictionary-learning approach to efficiently calculate the P-OEF problem. We then derive a generalized critical region theory of the MPLP problem to jointly consider the change of parameters in terms of both the objective function and the constraints of P-OEF in MES. Finally, we explore the physical significance of the MPLP's critical region in the P-OEF problem. We perform a case study to verify the validity and effectiveness of the proposed approach compared to the traditional Monte Carlo approach for the P-OEF of the MES.

MPLP-Based Fast Power System Reliability Evaluation Using Transmission Line Status Dictionary

Composite power system reliability evaluation is computationally time-consuming because the optimal power flow (OPF) with the least load curtailment is calculated for a large number of samples. Most current studies of probabilistic simulation methods focus on sampling techniques to improve the sampling efficiency and decrease the calculations of the OPF. This paper proposes a fast reliability evaluation method that accelerates the calculation of the OPF using multi-parametric linear programming (MPLP). In this paper, the sampled transmission line statuses are considered by the transmission line status dictionary (TLSD). We match the line status of each sample with the scenarios in the TLSD. For matching samples, the generation status and the sampled load are treated as MPLP parameters of the DC-OPF evaluation model. A dynamic learning algorithm is applied to solve the MPLP problem. Case studies are conducted on both IEEE Reliability Test Systems and a provincial power system in China. The results show that the proposed method improves the evaluation efficiency by 23–30 times compared with the normal DC-OPF-based model.

Static output feedback control design for constrained linear discrete‐time systems using data cluster analysis

Methods for the design of static output feedback controllers for linear discrete-time systems subject to state and control constraints, additive disturbance and measurement noise are proposed in this study. First, the authors show how the design can be performed through the construction of an output feedback controlled invariant set together with the solution of several multiparametric linear programming (mp-LP) problems. This technique is commonly used to characterise the optimal control action as a piecewise affine (PWA) function over polyhedral regions. However, when applied to constrained multiple output systems, it may require high online computational cost and large memory capacity, due to the number of polyhedral regions resulting from the mp-LP solution. Within this context, the authors present a new approach to compute a PWA static output feedback control law over a smaller number of polyhedral regions using the K q-flat data cluster analysis algorithm. Optimisation problems to compute PWA sub-optimal static output feedback control laws over an even smaller number of regions are also proposed, and illustrated by numerical examples.

On Robust Tie-Line Scheduling in Multi-Area Power Systems

The tie-line scheduling problem in a multi-area power system seeks to optimize tie-line power flows across areas that are independently operated by different system operators (SOs). In this paper, we leverage the theory of multi-parametric linear programming to propose algorithms for optimal tie-line scheduling, respectively, within a deterministic and a robust optimization framework. Aided by a coordinator, the proposed methods are proved to converge to the optimal schedule within a finite number of iterations. A key feature of the proposed algorithms, besides their finite step convergence, is that SOs do not reveal their dispatch cost structures, network constraints, or natures of uncertainty sets to the coordinator. The performance of the algorithms is evaluated using several power system examples.

Sensibility and Uncertainties Analysis method dedicated to home energy management problem

This paper focuses on the energy management problem applied to residential sector. The studied optimization problem is defined as the optimal management of production and consumption activities in buildings. A scheduling problem is identified to adjust the energy consumption to both the energy cost and the users’ comfort. The optimization procedure of energy management is based on available predictions (weather forecast, users habit, etc.). These predictions are not entirely known data of the optimization problem because of uncertainties. Parametric uncertainties are introduced in the home energy management problem in order to provide robust energy allocation. To improve the taking into account of uncertainties of prediction and the energy management problems, a Sensibility and Uncertainties Analysis method based on the procedure Branch & Bound and the multi-parametric linear programming is proposed. After a description of general principles and main steps, this algorithm is applied to various energy flow management problems in a smart platform. Practical application: The problem of energy management in buildings is described as the optimization problem which is defined to adjust the energy consumption of buildings in order to reach equilibrium between production and consumption on the grid. This home energy management problem has been modeled with a lot of parameters that are uncertain data. This is generally the case in modeling. This is particularly true in the energy management problem because of the high sensitivity of the decision (the energy allocation) to the data. The proposed approach permits to analyze the sensitivity of the solutions according to uncertain parameters. Then the impact of an uncertainty can be analyzed. Each family of solutions can be used to quantify locally the stability of a robust solution over the uncertainties.

POP – Parametric Optimization Toolbox

In this paper, we describe POP, a MATLAB toolbox for parametric optimization. It features (a) efficient implementations of multiparametric programming problem solvers for multiparametric linear and quadratic programming problems and their mixed-integer counter-parts, (b) a versatile problem generator capable of creating random multiparametric programming problems of arbitrary size, and (c) a comprehensive library of multiparametric programming test problems featuring benchmark test sets for multiparametric linear, quadratic, mixed-integer linear, and mixed-integer quadratic programming problems. In addition, POP is equipped with a graphical user interface which enables the user-friendly use of all functionalities of POP and a link to the solvers of the Multi-Parametric Toolbox (MPT), as well as the ability to design explicit MPC problems. These features are demonstrated in detailed computational studies providing insights into the versatility and applicability of POP. Additionally, the example of a period...

Fast Real-Time Constrained Predictive Current Control in Permanent Magnet Synchronous Machine-Based Automotive Traction Drives

The current control in permanent magnet synchronous machine (PMSM) drives is a challenging problem which has to deal with physical and real-time constraints. The goal of this paper is to provide a control solution that can deal effectively with these challenges. The objective is to adjust the stator voltages to obtain fast and monotonic transient current response while satisfying the existing constraints. To this end, using the PMSM model in the rotor coordinates, an explicit one-step ahead predictive control law which uses tools from Lyapunov theory is developed. First, a polytopic approximation of the quadratic constraints is used to reduce the complexity. Then, the optimization problem is reduced to a linear program. Furthermore, a piecewise affine explicit control law is obtained via multiparametric linear programming. As a consequence, the online computation of the control law is reduced to a point location problem yielding a fast control method suitable for real-time implementation. Second, the speed-dependent terms of the mathematical model are considered as disturbance. Then, a state and disturbance observer is designed using a quadratic Lyapunov function to guarantee asymptotic stability of the estimation error. Real-time results obtained in an industrial hardware-in-the-loop test-bench are reported and analyzed.

A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.

An efficient explicit algorithm for multi-rate predictive control systems

In this paper, explicit model predictive control (MPC) schemes for discrete-time linear-invariant multi-rate systems with constraints on inputs and states are studied. An efficient multi-rate predictive control algorithm is proposed by solving a multiparametric linear programming (mp-LP), and the explicit state feedback control laws are developed. Further, based on the closed-loop multi-rate piecewise affine (PWA) structure, stability analysis is presented by linear matrix inequality (LMI). The simulation results show the effectiveness of the explicit MPC for multi-rate system.