Derivation of the VOC method

Sketch of an idea Catalin Arghir November 4, 2014 Suppose we start with the following model of a 3-phase inverter in α-βframe: ˙x1=u1−w1 ˙x2=u2−w2(1) Where xrepresents the output capacitor voltages, uthe converter side currents charging the two 1 Farad capacitors and wthe grid side currents. We plan to stabilize the set: γ=x:px2 1+x2 2=ρ0 We use the following coordinate transformation: r=px2 1+x2 2 φ= arctan(x2 x1)(2) By taking the time derivatives of (2) we get: ˙r=1 2 2x1˙x1+2x2˙x2 √x2 1+x2 2 =x1(u1−w1)+x2(u2−w2) r=1 rx1x2u1 u2−1 rp ˙ φ=1 1+ x2 2 x2 1 x1˙x2−x2˙x1 x2 1 =x1(u2−w2)−x2(u1−w1) r2=1 r2−x2x1u1 u2−1 r2q (3) Where p and q are the instantaneous active and reactive powers: p=x1w1+x2w2 q=x1w2−x2w1(4) We rewrite eq. (3) as: ˙r ˙ φ=1 r2x1r x2r −x2x1u1 u2−1 rp 1 r2q(5) We use the following feedback transformation (invoking a transverse feedback linearization argument or simply to resemble a harmonic oscillator): u1 u2=x1−x2 x2x1b ω(6) Eq. (5) becomes: 1 ˙r ˙ φ=1 r2x1r x2r −x2x1x1−x2 x2x1b ω−1 rp 1 r2q=1 r2(x2 1+x2 2)r0 0 (x2 1+x2 2)b ω−1 rp 1 r2q We end up with the following system: ˙r ˙ φ=r0 0 1b ω−1 rp 1 r2q(7) By letting ˙ φ=ω+˙ θand setting ω=const., we get the following phase dynamics: ˙ θ=− 1 r2q By letting b=b01−r2 ρ2 0, where ρ0is the radius of the circle-set to be stabilized, we ge the following amplitude dynamics: ˙r=b0r1−r2 ρ2 0− 1 rp In fact one could pick any function b(r) that is zero at ρ0, strictly positive for r < ρ0and strictly negative for r > ρ0as b is the real part of the complex poles of the harmonic oscillator: ˙x1 ˙x2=b−ω ω b x1 x2(8) The sketch shows that if instead of starting with a Lienard oscillator, one starts with a 2 state integrator, by applying a certain coordinate and feedback transformation one is able to highlight feedback that turns a given set into an attractive limit cycle. Moreover, for a particular choice of feedback, the closed loop system may take the shape of a Van der Pol oscillator. 2