Quantum optimization of energy systems: The problem of commitment to units in electricity generation

Problem opredjeljenja jedinice ili unit commitment problem - UC je nelinearni problem mješovitog cjelobrojnog programiranja koji se koristi za minimiziranje operativnih troškova proizvodnih jedinica u elektroenergetskom sustavu. Može se opisati kao procedura uključivanja i isključivanja generatora kako bi se povećala izlazna snaga i zadovoljila tržišna potražnja. Naravno, prilikom odlučivanja koji od generatora će biti uključeni, a koji ne, niz ograničenja elektroenergetskog sustava mora biti zadovoljeno. Na početku rada, upoznajemo se s radom i osnovnim pojmovima kvantnog računanja kao što su kvantni bit ili kubit, te superpozicija. Također se upoznajemo s definicijom Problema opredjeljenja jedinica i najpoznatijim algoritmom za njegovo rješavanje - algoritmom optimizacije rojem čestica (eng. particle swarm optimization, PSO). PSO algoritam koristi se za pronalaženje maksimuma ili minimuma funkcije definirane u višedimenzionalnom vremenskom prostoru. Sastoji od jata pojedinaca (čestica) koji se kreću u prostoru pretra živanja kako bi pronašli rješenje za dani problem. U obzir se uzima kretanje svake čestice prema brzini i prošlom iskustvu čestice, kao i iskustvo susjednih čestica. Nadalje, predlaže se binarni algoritam optimizacije rojem čestica ili BPSO. Zašto binarna inačica? Pa upravo je ona najjednostavnija za reprezentaciju stanja generatora: 0 - isklju čeno i 1 - uključeno Razlika izmedu PSO i BPSO pristupa je u definiranju vektora položaja čestice, koji je u BPSO binaran. Kako bismo uspješno opisali rad LAQPSO algoritma, uvodimo pojam kvantno inspiriranog evolucijskog algortima - QEA. Temelji se na konceptima i principima kvantnog računarstva kao što su kvantni bit i superpozicija stanja. Okarakteriziran je odabirom (reprezentacijom) pojedinca, evaluacijskom funkcijom i dinamikom populacije. Uvodi se nova, vjerojatnosna oznaka za prikaz kubita, te pojam kubit pojedinca kao niza kubita. Iako su pristupi temeljeni na BPSO algoritmu uspješno primijenjeni na kombinatoriku problema optimizacije u raznim područjima, BPSO algoritam ima nekoliko nedostataka od kojih je jedan preuranjena konvergencija pri rješavanju težih problema. Kako bi se prevladali spomenuti nedostatci, uveden je novi binarni PSO algoritam inspiriran kvantnim računarstvom - QBPSO algoritam. Predloženi QBPSO kombinira konvencionalni BPSO s konceptima kvantnog računarstva. QBPSO koristi pojam kubit pojedinca za probabilistički prikaz, koji zamjenjuje postupak ažuriranja brzine u optimizaciji roja čestica. Predlažu se i novu rotacijska vrata, odnosno vrata za rotaciju koordinata za ažuriranje kubit pojedinaca u kombinaciji s dinamičkim kutom rotacije za određivanje veličine kuta. Kako bi se performanse poboljšale i ukupni minimalni operativni trošak smanjio, algoritam roja kvantnih čestica lokalnog privlačenja - LAQPSO za rješavanje UC problema nastaje na temelju kvantnog evolucijskog algoritma (QEA) i QBPSO algoritma. Novi kvantni mehanizam prikaza bitova nazvan kvantni kut koristi se za kodiranje rješenja, te se uvodi pojam lokalnog atraktora koji je zadužen za automatsko odredivanje rotacijskog kuta kvantnih rotacijskih vrata. Tijekom procesa traženja globalnog rješenja, veličina kuta rotacije prilagodava se važnom parametru zvanom koeficijent kontrakcije, koji može kvantitativno odrediti kompromis izmedu sposobnosti istraživanja i sposobnosti eksploatacije. Predloženi algoritam primjenjuje se za rješavanje UC problema za energetski sustav od 26 jedinica (generatora). Uspoređuje se s binarnim PSO (BPSO), poboljšanim kvantnim BPSO (IQBPSO) i drugim tehnikama kako bi se pokazala učinkovitost i točnost predloženog algoritma. Rezultati pokazuju vrhunske performanse LAQPSO algoritma za minimiziranje ukupnih troškova u usporedbi s drugim navedenim algoritmima.Unit commitment problem (UC) is a nonlinear problem of mixed integer programming used to minimize the operating costs of generating units in the power system. It can be described as a procedure to turn the generator on and off to increase output power and meet market demand. When deciding which of the generators will be included and which will not, a number of limitations of the power system must be met. At the beginning of the paper, we get acquainted with the work and basic concepts of quantum computing such as quantum bit or qubit, and superposition. We are also introduced to the definition of the Unit Commitment Problem and the most well-known algorithm for its solution - the particle swarm optimization (PSO) algorithm. The PSO algorithm is used to find the maximum or minimum of a function defined in multidimensional vector space. It consists of a swarm of individuals (particles) moving in the search space to find a solution to a given problem. The movement of each particle according to the velocity and past experience of the particle is taken into account, as well as the experience of neighboring particles. Furthermore, a binary particle swarm optimization algorithm or BPSO is proposed. The difference between the PSO and BPSO approaches is in defining the particle position vector, which is binary in BPSO. In order to successfully describe the work of the LAQPSO algorithm, we introduce the concept of a quantum-inspired evolutionary algorithm - QEA. It is based on the concepts and principles of quantum computing such as quantum bit and state superposition. It is characterized by the selection (representation) of the individual, the evaluation function and the dynamics of the population. A new, probabilistic notation for representation is introduced, which is based on the concept of qubits and the notion of the qubit of an individual as a series of qubits. Although BPSO-based approaches have been successfully applied to the combinatorics of optimization problems in various fields, the BPSO algorithm has several shortcomings, one of which is premature convergence in solving more difficult problems. In order to overcome the mentioned shortcomings, a new binary PSO algorithm inspired by quantum computing - QBPSO algorithm - was introduced. The proposed QBPSO combines conventional BPSO with concepts of quantum computing. QBPSO uses the term qubit individual for probabilistic representation, which replaces the speed update process in particle swarm optimization. A new rotary gate is also proposed, ie a gate for the rotation of coordinates to update the qubit of individuals in combination with a dynamic rotation angle to determine the size of the angle. In order to improve performance and reduce the overall minimum operating cost, the local attraction quantum particle swarm algorithm - LAQPSO for UC problem solving is based on the quantum evolution algorithm (QEA) and the PSO algorithm. A new quantum bit expression mechanism called quantum angle is used to encode the solution to the particle, and the notion of a local attractor is introduced which is in charge of automatically determining the rotational angle of a quantum rotating gate. During the process of finding a global solution, the magnitude of the rotation angle is adjusted to an important parameter called the contraction coefficient, which can quantify the trade-off between exploration capability and exploitation capability. The proposed algorithm is applied to solve UC problems for a 26-unit (generator) power system. It is compared with binary PSO (BPSO), improved quantum BPSO (IQBPSO) and other techniques to demonstrate the efficiency and accuracy of the proposed algorithm. The results show the superior performance of the LAQPSO algorithm to minimize total costs compared to the other listed algorithms