Operacije nad politopskim skupovima kod optimalnog upravljanja sustava s ograničenjima

In the last decade a lot of research has focused on the explicit solution of optimal and robust control problems for the class of constrained discrete-time systems. Many newly developed control algorithms for such control problems internally use operations on polytopic sets. We review basic polytopic manipulations and analyze them in the context of the computational effort. We especially consider the so-called regiondiff problem where the set difference between a polyhedron and union of polyhedra needs to be computed. Regiondiff problem and related polycover problem – checking if a polytope is covered by the union of other polytopes – are utilized very often in derivation of the explicit solutions to the constrained finite time optimal control problems for piecewise affine systems. Similar observation holds for the computation of the (positive) controlled invariant sets, infinite time optimal control solution and/or controllers with reduced complexity for piecewise affine systems. We describe an in-place depth-first exploration algorithm that solves the regiondiff problem in an efficient manner. We derive strict upper bound for the computational complexity of the described algorithm. In extensive testing we show that our algorithm is superior to the mixed integer linear programming approach when solving the polycover problem.U posljednjih desetak godina znatna istraživačka aktivnost usmjerena je na pronalaženje eksplicitnih rješenja optimalnog i robusnog upravljanja za klasu vremenski diskretnih sustava s ograničenjima. Brojni razvijeni algoritmi interno koriste operacije nad politopskim skupovima. U ovom radu analiziramo osnovne operacije nad politopskim skupovima sa stajališta njihove računske kompleksnosti. Naročita pozornost dana je takozvanom regiondiff problemu, odnosno problemu proračuna razlike poliedarskog skupa i unije poliedara. Isto tako je analiziran i srodni polycover problem – provjera je li poliedarski skup u potpunosti prekriven unijom poliedara. Oba ova problema često se sre´cu pri konstruiranju ekplicitnih rješenja optimalnog upravljanja po dijelovima afinih sustava uz konačan horizont predikcije, kao i pri proračunu pozitivnih invarijantnih skupova, optimalnog upravljanja uz beskonačan horizont predikcije i/ili proračunu regulatora smanjene kompleksnosti za po dijelovima afine sustave. Razvijen je efikasan algoritam za rješenje regiondiff problema zasnovan na dubinskom pretraživanju stablaste strukture problema. Izvedena je teoretska gornja ograda za kompleksnost dobivenog algoritma, i pokazano je zašto je takva ograda konzervartivna u praksi. Na nizu simulacija pokazana je računsku superiornost razvijenog algoritam za polycover problem u odnosu na pristup zasnovan na rješavanju mješovitog cjelobrojnog programa

Operacije nad politopskim skupovima kod optimalnog upravljanja sustava s ograničenjima

In the last decade a lot of research has focused on the explicit solution of optimal and robust control problems for the class of constrained discrete-time systems. Many newly developed control algorithms for such control problems internally use operations on polytopic sets. We review basic polytopic manipulations and analyze them in the context of the computational effort. We especially consider the so-called regiondiff problem where the set difference between a polyhedron and union of polyhedra needs to be computed. Regiondiff problem and related polycover problem – checking if a polytope is covered by the union of other polytopes – are utilized very often in derivation of the explicit solutions to the constrained finite time optimal control problems for piecewise affine systems. Similar observation holds for the computation of the (positive) controlled invariant sets, infinite time optimal control solution and/or controllers with reduced complexity for piecewise affine systems. We describe an in-place depth-first exploration algorithm that solves the regiondiff problem in an efficient manner. We derive strict upper bound for the computational complexity of the described algorithm. In extensive testing we show that our algorithm is superior to the mixed integer linear programming approach when solving the polycover problem.U posljednjih desetak godina znatna istraživačka aktivnost usmjerena je na pronalaženje eksplicitnih rješenja optimalnog i robusnog upravljanja za klasu vremenski diskretnih sustava s ograničenjima. Brojni razvijeni algoritmi interno koriste operacije nad politopskim skupovima. U ovom radu analiziramo osnovne operacije nad politopskim skupovima sa stajališta njihove računske kompleksnosti. Naročita pozornost dana je takozvanom regiondiff problemu, odnosno problemu proračuna razlike poliedarskog skupa i unije poliedara. Isto tako je analiziran i srodni polycover problem – provjera je li poliedarski skup u potpunosti prekriven unijom poliedara. Oba ova problema često se sre´cu pri konstruiranju ekplicitnih rješenja optimalnog upravljanja po dijelovima afinih sustava uz konačan horizont predikcije, kao i pri proračunu pozitivnih invarijantnih skupova, optimalnog upravljanja uz beskonačan horizont predikcije i/ili proračunu regulatora smanjene kompleksnosti za po dijelovima afine sustave. Razvijen je efikasan algoritam za rješenje regiondiff problema zasnovan na dubinskom pretraživanju stablaste strukture problema. Izvedena je teoretska gornja ograda za kompleksnost dobivenog algoritma, i pokazano je zašto je takva ograda konzervartivna u praksi. Na nizu simulacija pokazana je računsku superiornost razvijenog algoritam za polycover problem u odnosu na pristup zasnovan na rješavanju mješovitog cjelobrojnog programa

Electrical Power Distribution System Reconfiguration: Case Study of a Real-life Grid in Croatia

This paper describes the application of a nonlinear model predictive control algorithm to the problem of dynamic reconfiguration of an electrical power distribution system with distributed generation and storage. Power distribution systems usually operate in a radial topology despite being physically built as interconnected meshed networks. The meshed structure of the network allows one to modify the network topology by changing the status of the line switches (open/closed). The goal of the control algorithm is to find an optimal radial network topology and optimal power references for controllable generators and energy storage units that will minimize cumulative active power losses while satisfying all system constraints. The validation of the developed algorithm is conducted in a case study of a reallife distribution grid in Croatia. Realistic simulations show that large loss reductions are feasible (more than 13%), i.e., the developed control algorithm can contribute to significant savings for the grid operato

Electrical Power Distribution System Reconfiguration: Case Study of a Real-life Grid in Croatia

This paper describes the application of a nonlinear model predictive control algorithm to the problem of dynamic reconfiguration of an electrical power distribution system with distributed generation and storage. Power distribution systems usually operate in a radial topology despite being physically built as interconnected meshed networks. The meshed structure of the network allows one to modify the network topology by changing the status of the line switches (open/closed). The goal of the control algorithm is to find an optimal radial network topology and optimal power references for controllable generators and energy storage units that will minimize cumulative active power losses while satisfying all system constraints. The validation of the developed algorithm is conducted in a case study of a reallife distribution grid in Croatia. Realistic simulations show that large loss reductions are feasible (more than 13%), i.e., the developed control algorithm can contribute to significant savings for the grid operato

Reference snage vjetroagregata u koordiniranom upravljanju vjetroelektranama

The new grid regulations require that a grid-connected wind farm acts as a single controllable power producer. To meet this requirement a traditional wind farm control structure, which allowed individual wind turbines to internally define their power production, has to be modified. This paper investigates the opportunity for wind turbine load reduction that arises from dynamic power control of wind turbines. The wind farm controller design is proposed that utilizes coordinated power control of all wind turbines to achieve the wind farm regulation requirements and to minimize the wind turbine loads.Nova mrežna pravila zahtijevaju da vjetroelektrane spojene na električnu mrežu djeluju kao jedinstveni upravljivi proizvođač električne energije. Da bi se zadovoljio takav zahtjev, tradicionalni način upravljanja vjetroelektranama, koji dozvoljava da vjetroagregati interno definiraju svoju referencu snage, treba biti modificiran. U ovom radu proučavaju se mogućnosti smanjenja opterećenja vjetroagregata korištenjem dinamičkog upravljanja snage vjetroagregata. Predložen je koncept regulatora vjetroelektrane koji koristi koordinirano upravljanje snagom vjetroagregata u svrhu zadovoljenja mrežnih pravila i smanjenja opterećenja vjetroagregata

Konveksna optimizacija u učenju CMAC neuronskih mreža

Simplicity of structure and learning algorithm play an important role in the real-time application of neural networks. The Cerebellar Model Articulation Controller (CMAC) neural network, with associative memory type of organization and Hebbian learning rule, satisfies these two conditions. But, Hebbian rule gives poor performance during off-line identification, which is used as a preparation phase for on-line implementation. In this paper we show that optimal CMAC network parameters can be found via convex optimization technique. For standard l2 approximation this is equivalent to the solution of Quadratic Program (QP), while for l1 or l‡ approximation solving Linear Program (LP) suffices. In both cases physical constraints on parameter values can be included in an easy and straightforward way.Jednostavnost građe i algoritama učenja od iznimne su važnosti u primjenama neuronskih mreža u stvarnom vremenu. CMAC neuronska mreža s asocijativnom memorijskom organizacijom i Hebbianovim algoritmom učenja udovoljava ovim zahtjevima. Međutim, Hebbianov algoritam učenja ne daje dobre rezultate pri off-line identifikaciji, koja se koristi kao pripremna faza za on-line identifikaciju. U ovom se članku pokazuje da se optimalne vrijednosti parametara CMAC neuronske mreže mogu dobiti primjenom tehnika konveksne optimizacije. Za standardnu l2 aproksimaciju koristi se kvadratno programiranje (QP), a za l1 i l‡ aproksimacije linearno programiranje (LP). U oba je slučaja jednostavno uključiti fizikalna ograničenja na vrijednosti parametara u algoritam optimizacije

LARES – Laboratorij za sustave obnovljivih izvora energije

Prikazan je povijesni tijek razvoja i sadašnji trenutak Laboratorija za sustave obnovljivih izvora energije (LARES) na FERu. Laboratorij je nastao adresiranjem tehničkih izazova povezanih s upravljanjem pojedinačnih sustava obnovljivih izvora energije, sustava pohrane ili potrošača, s ciljevima njihove ekonomičnosti i dugovječnosti, a danas se mahom bavi upravljanjem njihovih složeno povezanih kombinacija, na razini zgrada i infrastrukture. Glavna metodološka odrednica LARESa su matematička optimizacija i prediktivno upravljanje

Dinamičko upravljanje distribucijskim elektroenergetskim sustavima

This paper considers the problem of optimal dynamic management of electrical power distribution networks with distributed generation and storage. Initially, analysis is performed of the Optimal Power Flow (OPF) problem -- a paramount optimization problem that needs to be solved to ensure optimal steady-state power network operation. In the rest of the paper we present a hierarchical control structure for solving the considered optimal control problem in a dynamical framework. At the upper level a dynamic OPF solver computes the optimal power references for distributed generators and storages at slow rate. These references are then transmitted to the intermediate level, where a faster Model Predictive Control algorithm computes small deviations from power references given by the OPF solver to take into account the variability of load profiles that was neglected at the upper layer. Finally, the power references are forwarded to the primary level where local controllers track these power reference values. A realistic simulation case study of a Croatian power distribution grid is used for testing purposes and to demonstrate the applicability and usefulness of the proposed control strategy.U ovom članku razmatramo problem optimalnog dinamičkog upravljanja elektroenergetskim distribucijskim sustavom sa distribuiranom proizvodnjom i pohranom energije. Analizirali smo problem optimalnih tokova snage (OPF), koji je od najveće važnosti za razmatrani upravljački problem. U nastavku članka smo opisali hijerarhijsku upravljačku strukturu za rješavanje razmatranog problema optimalnog upravljanja. Na najvišoj razini (dinamički) OPF algoritam izračunava optimalne reference snage za distribuirane izvore i spremnike energije na sporoj vremenskoj skali. Te se reference zatim šalju srednjoj razini, gdje brži algoritam, temeljen na modelskom prediktivnom upravljanju, izračunava male devijacije od referenci koje je dobio od nadređene razine, kako bi uzeo u obzir brže varijacije profila potrošnje koje su zanemarene u nadređenoj razini. Konačno, reference se prosljeđuju najnižoj razini gdje se nalaze lokalni regulatori koji su zaduženi za njihovo praćenje. Realističan simulacijski ispitni primjer hrvatske elektroenergetske distribucijske mreže korišten je za ispitivanje i demonstraciju primjenjivosti i korisnosti predložene upravljačke strategije

Neural network based identification and control of non-linear time variable processes

Opisana je identifikacija nelinearnih vremenski promjenljivih procesa, pri čemu je posebna pažnja posvećena primjeni neuronskih mreža u identifikaciji nelinearnih procesa. Objašnjena je veza između općeg ulazno-izlaznog nelinearnog modela i statičkih neuronskih mreža. Detaljno su opisane korištene strukture neuronskih mreža: višeslojna perceptronska mreža (MLP), neizrazita neuronska mreža (FNN) i CMAC mreža. Opisani su i ispitani nerekurzivni i rekurzivni algoritmi učenja neuronskih mreža. Prezentiran je algoritam trenutačne linearizacije. Predložen je sustav prediktivnog upravljanja koji se sastoji od nelinearnog modela zadanog procesa realiziranog neuronskom mrežom, postupka trenutačne linearizacije i poopćenog prediktivnog regulatora. Svojstva sustava upravljanja ispitana su simulacijom na računalu na matematičkom modelu nelinearnog sustava. Temeljem jednadžbi očuvanja tvari i energije izveden je fizikalni model šaržne destilacijske kolone. Model je uspoređen s eksperimentalnim rezultatima dobivenim na laboratorijskoj destilacijksoj koloni. Simulacijom na računalu provjerena je primjena predloženog sustava prediktivnog upravljanja pri upravljanju šaržnom destilacijksom kolonom.System identification of non-linear time variant processes is presented, with special attention given to the identification of non-linear processes using neural networks. Connection between black box input-output non-linear models and static neural networks is described. Structures of MultiLayer Perceptron (MLP), Fuzzy-Neural Network (FNN) and Cerebellar Model Articulation Controller (CMAC) are discussed in great detail. On-line and off-line learning algorithms for these neural networks are described and tested. Instantaneous linearization method is derived. System of predictive control is given, which is comprised from nonlinear neural model of given process, instantaneous linearization method and generalized predictive controller. Characteristics of given system were proved through computer simulations. Mathematical model of batch distillation column based on heat and material balances is derived. Model is compared with the experimental results obtained on a laboratory distillation column. Predictive control of batch distillation column was tested through computer simulations