Simone Paoletti

Identification of Hybrid Systems A Tutorial

This tutorial paper is concerned with the identification of hybrid models, i.e. dynamical models whose behavior is determined by interacting continuous and discrete dynamics. Methods specifically aimed at the identification of models with a hybrid structure are of very recent date. After discussing the main issues and difficulties connected with hybrid system identification, and giving an overview of the related literature, this paper focuses on four different approaches for the identification of switched affine and piecewise affine models, namely an algebraic procedure, a Bayesian procedure, a clustering-based procedure, and a bounded-error procedure. The main features of the selected procedures are presented, and possible interactions to still enhance their effectiveness are suggested.

A bounded-error approach to piecewise affine system identification

This paper proposes a three-stage procedure for parametric identification of piecewise affine autoregressive exogenous (PWARX) models. The first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems (MIN PFS) problem for a suitable set of linear complementary inequalities derived from data. Second, a refinement procedure reduces misclassifications and improves parameter estimates. The third stage determines a polyhedral partition of the regressor set via two-class or multiclass linear separation techniques. As a main feature, the algorithm imposes that the identification error is bounded by a quantity /spl delta/. Such a bound is a useful tuning parameter to trade off between quality of fit and model complexity. The performance of the proposed PWA system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick-and-place machine.

Comparison of four procedures for the identification of hybrid systems

In this paper we compare four recently proposed procedures for the identification of PieceWise AutoRegressive eXogenous (PWARX) and switched ARX models. We consider the clustering-based procedure, the bounded-error procedure, and the Bayesian procedure which all identify PWARX models. We also study the algebraic procedure, which identifies switched linear models. We introduce quantitative measures for assessing the quality of the obtained models. Specific behaviors of the procedures are pointed out, using suitably constructed one dimensional examples. The methods are also applied to the experimental identification of the electronic component placement process in pick-and-place machines.

A greedy approach to identification of piecewise affine models

This paper addresses the problem of identification of piece-wise affine (PWA) models. This problem involves the estimation from data of both the parameters of the affine submodels and the partition of the PWA map. The procedure that we propose for PWA identification exploits a greedy strategy for partitioning an infeasible system of linear inequalities into a minimum number of feasible subsystems: this provides an initial clustering of the datapoints. Then a refinement procedure is applied repeatedly to the estimated clusters in order to improve both the data classification and the parameter estimation. The partition of the PWA map is finally estimated by considering pairwise the clusters of regression vectors, and by finding a separating hyperplane for each of such pairs. We show that our procedure does not require to fix a priori the number of affine submodels, which is instead automatically estimated from the data.

A Survey on Switched and Piecewise Affine System Identification

Abstract Recent years have witnessed a growing interest on system identification techniques for switched and piecewise affine models. These model classes have become popular not only due to the universal approximation properties of piecewise affine functions, but also because the proposed identification procedures have proven to be effective in problems involving complex nonlinear systems with large data sets. This paper presents a review of recent advances in this research field, including theoretical results, algorithms and applications.

Data classification and parameter estimation for the identification of piecewise affine models

This paper proposes a three-stage procedure for parametric identification of piece wise affine auto regressive exogenous (PWARX) models. The first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the MIN PFS problem (partition into a minimum number of feasible subsystems) for a set of linear complementary inequalities derived from input-output data. Then, a refinement procedure reduces misclassifications and improves parameter estimates. The last stage determines a polyhedral partition of the regressor set via two-class or multi-class linear separation techniques. As a main feature, the algorithm imposes that the identification error is bounded by a fixed quantity /spl delta/. Such a bound is a useful tuning parameter to trade off between quality of fit and model complexity. Ideas for efficiently addressing the MIN PFS problem, and for improving data classification are also discussed in the paper. The performance of the proposed identification procedure is demonstrated on experimental data from an electronic component placement process in a pick-and-place machine.

Load forecasting for active distribution networks

This paper addresses the problem of electric load forecasting for distribution networks with Active Demand (AD), a new concept in smart-grids introduced within the EU project ADDRESS. By changing the typical consumption pattern of the consumers, AD adds a new dimension to the problem of load forecasting, and therefore makes currently available load forecasting techniques no more suitable. A new approach to load forecasting in the presence of AD is therefore proposed. The approach is based on a decomposition of the load into its components, namely the base load (representing different seasonal patterns), and a residual term depending both on stochastic fluctuations and AD effects. The performance of the proposed approach is illustrated through a numerical example. Since data sets including AD are not yet available, in the numerical example AD effects are simulated and added to real measurements representing the aggregated load of about 60 consumers from an Italian LV network.

Recent techniques for the identification of piecewise affine and hybrid systems

The problem of piecewise affine identification is addressed by studying four recently proposed techniques for the identification of PWARX/HHARX models, namely a Bayesian procedure, a bounded-error procedure, a clustering-based procedure and a mixed-integer programming procedure. The four techniques are compared on suitably defined one-dimensional examples, which help to highlight the features of the different approaches with respect to classification, noise and tuning parameters. The procedures are also tested on the experimental identification of the electronic component placement process in pick-and-place machines.

Models and Techniques for Electric Load Forecasting in the Presence of Demand Response

Demand-side management has been recently recognized as a strategic concept in smart electricity grids. In this context, active demand (AD) represents a demand response scenario in which households and small commercial consumers participate in grid management through appropriate modifications of their consumption patterns during certain time periods in return of a monetary reward. The participation is mediated by a new player, called aggregator, who designs the consumption pattern modifications to make up standardized products to be sold on the energy market. The presence of this new input to consumers generated by aggregators modifies the load behavior, asking for load forecasting algorithms that explicitly consider the AD effect. In this paper, we propose an approach to load forecasting in the presence of AD, based on gray-box models where the seasonal component of the load is extracted by a suitable preprocessing and AD is considered as an exogenous input to a linear transfer function model. The approach is thought for a distribution system operator that performs technical validation of AD products, and therefore possesses full information about the AD schedule in the network. A comparison of the performance of the proposed approach with techniques not using the information on AD and with approaches based on nonlinear black-box models is performed on a real load time series recorded in an area of the Italian low voltage network.

Optimal Allocation of Energy Storage Systems for Voltage Control in LV Distribution Networks

This paper addresses the problem of finding the optimal configuration (number, locations, and sizes) of energy storage systems (ESSs) in a radial low voltage distribution network with the aim of preventing over- and undervoltages. A heuristic strategy based on voltage sensitivity analysis is proposed to select the most effective locations in the network where to install a given number of ESSs, while circumventing the combinatorial nature of the problem. For fixed ESS locations, the multi-period optimal power flow framework is adopted to formulate the sizing problem, for whose solution convex relaxations based on semidefinite programming are exploited. Uncertainties in the storage sizing decision problem due to stochastic generation and demand, are accounted for carrying out the optimal sizing over different realizations of the demand and generation profiles, and then taking a worst-case approach to select the ESS sizes. The final choice of the most suitable ESS configuration is done by minimizing a total cost, which takes into account the number of storage devices, their total installed capacity and average network losses. The proposed algorithm is extensively tested on 200 randomly generated radial networks, and successfully applied to a real Italian low voltage network and a modified version of the IEEE 34-bus test feeder.