Dunstano del PUERTO-FLORES 1 , Romeo ORTEGA 2 , Jacquelien M.A. SCHERPEN 1 University of Groningen (1), Sup´ elec (2) On Power Factor Improvement by Lossless Linear Filters under Nonlinear Nonsinusoidal Conditions Abstract. Recently, it has been established that the problem of power factor compensation for nonlinear loads with nonsinusoidal source voltage can be recast in terms of the property of cyclodissipativity. The purpose of this brief note is to review and to illustrate the application of this framework to the practical of passive reactive compensation of linear loads with a nonsinusoidal source voltage. We give criteria for determining the optimal values of the compensator parameters that depend of the spectral line of load susceptance and voltage source. Streszczenie. Stwierdzono ostatnio, ˙ ze problem poprawy wsp´ olczynnika mocy odbiornik ´ ow nieliniowych zasilanych napie ¸ ciem niesinusoidalnym mo˙ ze by´ c rozpatrywany z punktu widzenia cyklo-rozpraszania (cyclodissipativity). Celem tego artykulu jest przegl  ad i pokazanie zastosowania takiego podej´ scia przy bezstratnej kompensacji odbiornik ´ ow liniowych zsilanych napie ¸ciem niesinusoidalnym. Podane zostaly kryteria do okre´ slenia optymalnych warto´ sci parametr ´ ow kompensatora, kt ´ ore zale˙ z  a od charakterystyki widmowej susceptancji obci  a˙ zenie i ´ zr´ odla napie ¸ cia. (Poprawa wsp ´ olczynnika mocy filtrami bezstratnymi w warunkach nieliniowych i niesinusoidalnych) Keywords: Power factor compensation; nonlinear network; nonsinusoidal conditions. Slowa kluczowe: Kompensacji wsp´ olczynnika mocy; sie´ c nieliniowych; warunkach niesinusoidalnych. Introduction Optimizing energy transfer from an alternating current (ac) source to a load is a classical problem in electrical engi- neering. In practice, the efficiency of this transfer is typically reduced due to the phase shift between voltage and current at the fundamental frequency. The power factor captures the energy-transmission efficiency for a given load. The standard approach to improving the power factor is to place a compen- sator between the source and the load. The effectiveness of capacitive compensation in systems with nonsinusoidal voltages and currents has been widely studied by [1, 2] and [3]. Unfortunately, in [4] it has been illustrated that the capacitive compensation may not be effec- tive for non-sinusoidal voltages. Therefore, a more complex compensator than only a capacitor is required for the reactive power minimization in such situations. Recently, in [5] it has been established that the classical problem is equivalent to imposing the property of cyclodis- sipativity to the source terminals. Since this framework is based on the cyclodissipativity property, see [6, 7], the im- provement of the power factor (PF) is done independent of the reactive power definition, which is a matter of discus- sions in the power community, see for instance [8] and its references. Most of the approaches used to improve the PF are based on different power definitions and a lack of a uni- fied definition of reactive power produces misunderstanding of power phenomena in circuits with nonsinusoidal voltages and currents, [9]. Similarly, the task of designing compen- sators that aim at improving the PF for nonlinear time-varying loads operating in non-sinusoidal regimes is far from clear. Using the cyclodissipativity framework the classical ca- pacitor and inductor compensators were interpreted in terms of energy equalization, see [5] for more details. And we have presented an extension of this result in [10] where we consid- ered arbitrary lossless linear time invariant (LTI) filters, and proved that for general lossless LTI filters the PF is reduced if and only if a certain equalization condition between the weighted powers of inductors and capacitors of the load is ensured. Here we illustrate the application of this framework to the passive compensation of linear loads with a non-sinusoidal source voltage. We give criteria for improvement of PF with linear capacitors, LC filters, which determine the optimal val- ues for the compensator parameters that depend of the spec- tral line of load susceptance and voltage source. Cyclodissipativity of RLC nonlinear networks In order to make this paper self-contained, the purpose of this section is to briefly review the meaning of cyclodissi- pativity, [6, 7], and some of its connections with the nonlinear circuit theory. Although the dissipativity theory applies to a wider classes of systems, let us consider dynamical systems modeled by ordinary differential equations: Definition 1 (Input-State-Output Representation). The input- state-output representation of the dynamical systems H : U→Y , is of the form (1) ˙ x = f (x, u) , x ∈X⊂ R m y = g (x, u) , u ∈U⊂ R n ,y ∈Y⊂ R n , where f : X×U→ R n and g : X×U→Y are vector functions of class C k , with 0 <k< ∞. Let X be the set of reachable and controllable points. Furthermore, assume that system (1), for u =0, has an equilibrium point in x =0. That is, f (0, 0) = 0 and g (0, 0) = 0. The definition of cyclodissipativity involves a function called supply rate w : U×Y→ R, which is locally inte- grable for every u ∈U , [6]. Definition 2 (Cyclodissipativity). We say that the system H is cyclodissipative on X with the supply rate w (u, y) if there exists a function S, called storage function, such that (2) S (x (t 0 )) +  t1 t0 w (u (t) ,y (t)) dt ≥ S (x (t 1 )) , is satisfied for all u ∈U and all t 0 <t 1 , such that x (t) ∈X for all t ∈ [t 0 ,t 1 ]. The inequality (2) is similar to the usual dissipation in- equality where additionally it is required that S ≥ 0, and which expresses that the increase in energy stored cannot be larger than the energy supplied from the outside. Fur- thermore, if the system H does not produce energy at any time, namely that the energy stored from the system is finite, S(x) ≥ 0 for all x ∈X , then the system is called dissipative. Typical examples of dissipative systems are: passive elec- trical networks, mechanical systems, viscoelastic materials, etc. However, if we consider an electrical network with ac- tive elements, namely negative resistors, tunnel diodes, etc., then the interpretation of (2) leads to some difficulties be- cause the storage energy function needs in general not be bounded from below or above. This merely involved a gen- eralization of the concept of a dissipative system to that of 112 PRZEGL  AD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R.87 NR 1/2011