ELEC5206 Chap.8 Dynamic Modelling and Voltage Control of the PVSC
Dynamic Modelling and Voltage Control of the PVSC
This chapter combines the PVSC control lecture with the lecture on relative stability. It argues for regulating PV voltage rather than PV current, places MPPT outside the feedback loop as the source of the voltage reference, and derives the boost PVSC’s averaged and small-signal models.
Linearising the PV curve with its negative dynamic resistance gives a second-order transfer function from duty ratio to PV voltage, whose damping falls as the operating point moves into the current-source region. A controller is then designed by affine parameterisation: choose the closed-loop response, solve for the controller, and map it to PID gains.
Finally the loop is judged by phase margin, gain margin and sensitivity peak, read from Bode and Nyquist plots. Expect derivations, parameter calculations and a robustness judgement.
What this chapter covers
- 01
Why PV voltage, not PV current, is regulated
- 02
Two loops: the PVSC voltage loop and the DC-link loop
- 03
Averaged and small-signal models of the boost PVSC
- 04
Dynamic resistance and the second-order duty-to-voltage plant
- 05
Natural frequency, damping and the current-source region
- 06
Affine parameterisation and the resulting PID gains
- 07
Absolute, internal and relative stability
- 08
Phase margin, gain margin and sensitivity peak
Worked example · free
PID time constant and integral gain from an affine design
- 1ωcl = 4 × 25,797 = 103,187 rad/s.
- 1α1 = 2ζcl/ωcl = 1.4/103,187 = 1.357 × 10−5 s and α2 = 1/ωcl² = 9.39 × 10−11 s².
- 1τd = α2/α1 = 6.92 × 10−6 s.
- 1Ki = ωn²/(K0α1) = 6.655 × 108/(−3.99 × 1010 × 1.357 × 10−5) = −1228.
Key terms
- Dynamic resistance
- The local slope RPV = ΔV/ΔI of the PV curve, negative at every operating point.
- Damping ratio
- The parameter ζ of a second-order system that measures how quickly oscillations decay.
- Affine parameterisation
- A design method that chooses the desired closed loop FQ and solves for the controller C = Q/(1 − QG0).
- Phase margin
- 180° plus the loop phase at the frequency where the loop gain magnitude equals one.
- Gain margin
- The factor by which loop gain could rise before instability, 1/|CG0| at the phase crossover.
- Sensitivity peak
- The maximum of 1/|1 + CG0|, equal to one over the Nyquist curve's closest distance to minus one.
Dynamic Modelling and Voltage Control of the PVSC FAQ
Why is the PVSC plant hard to control near the current-source region?
Damping is inversely proportional to the magnitude of the dynamic resistance. On the flat, current-source part of the I-V curve that magnitude is large, so damping falls and the loop oscillates; the lecture flags this as an oscillation alert and advises avoiding that region.
Why are the controller gains negative in the case study?
The plant gain K0 equals minus the DC-link voltage divided by the product of inductance and PV capacitance, so it is negative: raising the boost duty ratio lowers the PV voltage. The affine design divides by K0, so every PID gain inherits the negative sign, which keeps the overall feedback negative.
Is a stable loop automatically good enough?
No. Affine design guarantees input-output and internal stability, but robustness, the distance from instability, is judged separately by phase margin, gain margin and sensitivity peak. The lecture rejects a stable controller with a 32.5° phase margin and a sensitivity peak of 2.44 as not robust enough.
Why does MPPT sit outside the voltage feedback loop?
MPPT decides where the PV voltage should be and changes that reference slowly, while the voltage loop keeps the PV voltage at the reference against fast disturbances. Separating the two lets each be designed for its own time scale, and the lecture shows that tracking becomes much faster once a dedicated voltage loop is present.
Exam move
Derive the small-signal model once from the switching equations, writing every step, then practise only the final three formulas for ωn, ζ and K0 with your own component values. Memorise the four affine design steps as a sentence each.
For stability, practise reading margins from a Bode plot and computing one margin by hand from magnitudes and angles, and always finish with a verdict on robustness. Explain aloud why a lightly damped plant needs a well-designed loop.
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