Acta Montanistica Slovaca Ročník 13 (2008), číslo 1, 136-145 Ultracapacitor Modelling and Control Using Discrete Fractional Order State-Space Model Andrzej Dzielinski 1 and Dominik Sierociuk Ultrakapacitné modelovanie a riadenie použitím diskrétneho čiastočného riadenia modelu s rožloženými parametrami. In this paper the modelling of ultracapacitor system using discrete fractional order state-space system is presented. The obtained model is used for design and testing of state feedback controller with observer. Key words: Ultracapacitor Modelling, State-Space Model, Discrete Fractional Order Introduction Ultracapacitors (aka supercapacitors) are electrical energy storage devices which offer high power density which was not possible to achieve in traditional capacitors. There are many aproaches to ultracapacitors modelling. Many of the autors use more or less complicated RC models (Buller, Karden, Kok, Doncker, 2002), which are accurate especially for low frequencies. Some of the authors describe ultracapacitors by RC transmision line (Belhachemi, Rael, Davat, 2000). In the papers (Quintana, Ramos, Nuez, 2006; Westerlund, Ekstam; 1994) very efficient approach using fractional order calculus was presented. In this paper the fractional order calculus approach to ultracapacitor modelling is used, especially discrete fractional order state-space model (DFOSS). Foundations of ultracapacitors technology The capacity of typical ultracapacitors is many times bigger than typical capacitors and has values from fractional parts of Farad to thousands of Farads. Its energy density has values about 2…9 Wh.kg -1 and power density is 3…8kW.kg -1 . These values locate ultracapacitors between electrolytic capacitors and batteries. Thanks to their huge capacity, and small dimensions the ultracapacitors have found numerous applications in modern technology. One of the most prominent is the ultracapacitor application in hybrid cars (Burke, 2001; Wight, 2002). When a hybrid car is decelerating the electric motor acts as a generator producing a short, but high value energy impulse. This is used to charge the ultracapacitor. Charging the conventional batteries with such a short impulse would be extremely ineffective. Similarly, during start-up of the electric motor a short-time but substantial in value increase of the source power is needed. This is achieved by using the ultracapacitor. Other areas of ultracapacitor applications is in power electronics converters (mainly inverters) with DC circuit (Wodecki ,Koczara, 2004; Rufer, Barrade, 2002). Ultracapacitors can be found in wind power stations (Abbey, Joos, 2007), where they stabilise the power supplied to the grid. They are charged during the period of strong wind and discharge during calm periods. They can also be applied as energy saving subsystems in underground energy supply system. They are placed along the tracks and they collect the energy during braking and give it back during start-up. Also some back- up systems in electronics and IT use ultra capacitors (e.g. computer memory back-up). In most of the applications mentioned it is essential to have a fairly detailed model of ultracapacitor. This model makes the design of control systems possible. The more accurate model we have, the more advanced control schema can be achieved. Control systems are needed e.g. to stabilise the ultracapacitor voltage which tends to fluctuate significantly. Ultracapacitor is build from two activated carbon plates mixed with electrolyte and separator (Zorpette, 2005). Its working rule is based on Helmholtz effect, who found out that there exists some level of voltage below which the electrolysis does not take place. Below this level the electrolyte behaves as an insulator. When to the ultracapacitor electrodes the voltage below this level is connected the electrolysis does not start and the current does not flow through the electrolyte. However, in this case, the motion of ions contained in electrolyte (positive to negative electrode and negative to positive electrode) takes place. Because 1 Assoc. Prof. Andrzej Dzielinski, PhD., mgr inż. Dominik Sierociuk, Institute of Control and Industrial Electronics, University of Technology, Koszykowa 75, 00-662 Warsaw, Poland, adziel @isep.pw.edu.pl , dsieroci @isep.pw.edu.pl (Recenzovaná a revidovaná verzia dodaná 28. 11. 2007) 136