Reduced-Order Averaged Model

Abstract

This chapter deals with modeling methodologies used for obtaining simplified – in the sense of reduced order – power electronic converter models, which are able to represent their low-frequency average behavior and are more easily employed in simulation or control law design.

Problems

Problem 6.1

Linearized ROAM of buck-boost converter in dcm

Let us consider the buck-boost circuit in Fig. 6.5. It is required to obtain the small-signal ROAM.

Solution

In order to derive the small-signal ROAM, Eq. (6.19) – representing the nonlinear ROAM of the buck-boost converter in discontinuous conduction – should be linearized around an equilibrium point:

$$ C\overset{\cdotp }{{\left\langle {v}_C\right\rangle}_0}=\frac{E^2T}{2L{\left\langle {v}_C\right\rangle}_0}{\upalpha}^2-\frac{{\left\langle {v}_C\right\rangle}_0}{R}, $$

(6.37)

where notations have the meanings as introduced in the example detailed in Sect. 6.3.2. Equation (6.37) describes the first-order averaged dynamic of the capacitor voltage in relation to the input represented by duty ratio α. Supposing that voltage E is constant, the equilibrium point results from zeroing the time derivative in Eq. (6.37) for a fixed value α _e ; hence

$$ 2L{\left\langle {v}_C\right\rangle}_{0e}^2={E}^2 RT{\upalpha}_e^2, $$

which gives

$$ {\left\langle {v}_C\right\rangle}_{0e}=E{\upalpha}_e\sqrt{(RT)/(2L)}. $$

(6.38)

Equation (6.37) can be written as the dynamic equation of the variable 〈v _C 〉 ²₀ depending linearly on α²:

$$ C\overset{\cdotp }{\left({\left\langle {v}_C\right\rangle}_0^2\right)}=\frac{E^2T}{L}{\upalpha}^2-\frac{2{\left\langle {v}_C\right\rangle}_0^2}{R}. $$

(6.39)

Let $\tilde{\cdotp}$ be the notation dedicated to denoting small variations around a given equilibrium point. Notations ${\left\langle {v}_C\right\rangle}_0={\left\langle {v}_C\right\rangle}_{0e}+\tilde{{\left\langle {v}_C\right\rangle}_0}$ and $\upalpha ={\upalpha}_e+\tilde{\upalpha}$ are adopted; hence, $\overset{\cdotp }{{\left\langle {v}_C\right\rangle}_0}=\overset{\cdotp }{\tilde{{\left\langle {v}_C\right\rangle}_0}}$ and $\overset{\cdotp }{\upalpha}=\overset{\cdotp }{\tilde{\upalpha}}$. Linearization uses the first-order Taylor series approximation of function x ², namely x ² ≈ x ²₀  + 2x ₀(x − x ₀). Thus, Eq. (6.39) gives by linearization around the equilibrium operating point (6.38)

$$ \overset{\cdotp }{\tilde{{\left\langle {v}_C\right\rangle}_0}}=\frac{E^2T{\upalpha}_e^2}{2 LC{\left\langle {v}_C\right\rangle}_{0e}}+\frac{E^2T{\upalpha}_e}{ LC{\left\langle {v}_C\right\rangle}_{0e}}\cdot \tilde{\upalpha}-\frac{{\left\langle {v}_C\right\rangle}_{0e}}{ RC}-\frac{2}{ RC}\cdot \tilde{{\left\langle {v}_C\right\rangle}_0}, $$

where $\frac{E^2T{\upalpha}_e^2}{2 LC{\left\langle {v}_C\right\rangle}_{0e}}-\frac{{\left\langle {v}_C\right\rangle}_{0e}}{ RC}=0$ according to Eq. (6.38). Therefore:

$$ \overset{\cdotp }{\tilde{{\left\langle {v}_C\right\rangle}_0}}=\frac{E^2T{\upalpha}_e}{ LC{\left\langle {v}_C\right\rangle}_{0e}}\cdot \tilde{\upalpha}-\frac{2}{ RC}\cdot \tilde{{\left\langle {v}_C\right\rangle}_0}. $$

(6.40)

Equation (6.40) corresponds to a first-order linear dynamic system having as input small variations of duty ratio $\tilde{\upalpha}$ and as output small variations of averaged capacitor voltage $\tilde{{\left\langle {v}_C\right\rangle}_0}$. That is, the ROAM dynamic of a buck-boost converter around a given equilibrium point is of first order, linear, having $\frac{E^2 RT{\upalpha}_e}{2L{\left\langle {v}_C\right\rangle}_{0e}}$ as gain and $\frac{ RC}{2}$ as time constant.

Problem 6.2

Consider the system in Fig. 5.30 from Problem 5.1 in Chap. 5. It is about a circuit used to convey the energy between a variable DC source and a single-phase strong AC grid via a DC link, the reactive power being zero.

Given the system’s large-signal model developed in the solution to Problem 5.1, obtain a small-signal reduced-order averaged model used for control purpose by taking into account the following considerations.

The system must have the possibility of varying the input power; this is done by controlling the DC-source current by means of control input u ₁. The primary control purpose is to ensure the power flux between the primary DC source and the AC grid while preserving the rated operating conditions in terms of voltage and current; this is done by means of control input u ₂. To this end, a widely employed cascaded control structure can be used (Aström and Hägglund 1995), in which the DC-link voltage controller imposes the set-point of the grid inductor current. Therefore, the control structure is composed of two loops: the inner loop used for controlling the grid-side inductor current and the outer loop for controlling the DC-link voltage.

Solution

As shown in the solution of (b) in Problem 5.1, computation of the small-signal model and of its associated transfer functions may be a difficult way to obtain necessary information for designing control laws. Depending on each particular control objective, some remarks may be useful in order to introduce simplifying assumptions before starting control design procedures. In our case, let us focus on the regulation of the average value of the DC-link voltage, v _DC . As AC voltage is stiff, the (variable) output power can be controlled by means of the inductor L ₂ current (the output current loop). This supposes a variable DC current drain from the DC-link. In order to maintain the value of v _DC within operating limits, a voltage loop should also be used. In fact, the invariance of v _DC guarantees the input–output power balance. Figure 6.18 presents briefly this control approach.

Fig. 6.18

Simplified diagram of cascaded control structure for the DC-AC converter given in Fig. 5.30 (Chap. 5). Auxiliary elements – filters, i _d computation using PLL, etc. – have been omitted

Let us review the result from Problem 5.1 (Eq. 5.83), preserving the significance of notations introduced there:

$$ \left\{\begin{array}{c}\kern-5.8em \frac{d{\left\langle {i}_{L1}\right\rangle}_0}{ dt}=\frac{1}{L_1}\left[E-{v}_{DC}\left(1-\upalpha \right)\right]\\ {}\kern-7em \frac{d{i}_d}{ dt}=\frac{V}{L_2}-\frac{v_{DC}}{L_2}\cdot {\upbeta}_d-\frac{i_d\cdot r}{L_2}\\ {}\frac{d{v}_{DC}}{ dt}=\frac{1}{C}{\left\langle {i}_{L1}\right\rangle}_0\left(1-\upalpha \right)-\frac{1}{ RC}{v}_{DC}-\frac{i_d}{2C}\cdot {\upbeta}_d,\end{array}\right. $$

(6.41)

describing the averaged large-signal model of the circuit, where v _DC  = 〈v ₀〉₀, β _d  = 〈u ₂〉₁ and i _d  = 〈i _L2〉₁.

First note that variable i _L1 is controlled independently; it represents a perturbation for the remainder of the system and its dynamic is not of interest here.

The dynamic of current i _d is rendered significantly faster than the dynamic of v _DC by means of the inner control loop, i.e., i _d equals i ^* _d practically instantaneously in relation to the dynamic of v _DC . Therefore, one can assume that variable i _d influences v _DC by means of its steady-state regime, characterized by zeroing the derivative of the corresponding dynamic equation in relations (6.41) and taking into account neglecting the resistance of inductor L ₂, r = 0. Thus, the steady-state value of β _d is reached, which depends on the value of v _DC , namely β _{d _ st}  = V/v _DC . At its turn, the latter relation is used in the dynamic equation of v _DC to obtain

$$ \overset{\cdotp }{v_{DC}}=\frac{1}{C}{\left\langle {i}_{L1}\right\rangle}_0\left(1-\upalpha \right)-\frac{1}{ RC}{v}_{DC}-\frac{i_dV}{2C{v}_{DC}} $$

or, equivalently,

$$ \overset{\cdotp }{v_{DC}^2}=\frac{2{P}_{in\backslash \_ DC}}{C}-\frac{2}{ RC}{v}_{DC}^2-\frac{V}{C}{i}_d, $$

(6.42)

where P _{in _ DC}  = 〈i _L1〉₀ ⋅ v _DC (1 − α) signifies the power supplying the DC link.

Relation (6.42) represents the linear dynamical equation of the DC-link voltage squared. This relation also represents the nonlinear ROAM of the circuit because the resulting system order has been decreased by one.

One can use relation (6.42) to design linear control laws to regulate v ² _DC (instead of v _DC ) by using i _d as control input, knowing that P _{in _ DC} is a perturbation. The plant transfer function used to this end is

$$ {H}_{i_d\to {v}_{DC}^2}(s)=\frac{v_{DC}^2}{i_d}=-\frac{V\cdot R}{s\frac{ RC}{2}+1}. $$

(6.43)

If the transfer from i _d to v _DC is sought, then linearization of Eq. (6.42) around an equilibrium operating point v _DCe is necessary. This operating point obeys the following relation:

$$ \frac{2{P}_{in\backslash \_ DCe}}{C}-\frac{2}{ RC}{v}_{DCe}^2-\frac{V}{C}{i}_{de}=0. $$

(6.44)

As in the previous solved problem, the linearization makes use of the first-order Taylor series approximation of function x ², namely x ² ≈ x ²₀  + 2x ₀(x − x ₀). Variables of interest can respectively be written as the sum of their equilibrium values and their small variations:

$$ \left\{\begin{array}{l}{v}_{DC}={v}_{DC e}+\tilde{v_{DC}}\\ {}{i}_d={i}_{de}+\tilde{i_d}.\end{array}\right. $$

Therefore, linearization of Eq. (6.42) yields

$$ 2{v}_{DC e}\cdot \overset{\cdotp }{\tilde{v_{DC}}}=\frac{2{P}_{in\backslash \_ DCe}}{C}-\frac{4{v}_{DC e}}{ RC}\tilde{v_{DC}}-\frac{2{v}_{DC e}^2}{ RC}-\frac{V}{C}{i}_{de}-\frac{V}{C}\tilde{i_d}, $$

(6.45)

where the power supplying the DC link has been supposed a sufficiently slowly variable, i.e., constant: $\tilde{P_{in\backslash \_ DCe}}=0$. By substituting relation (6.44) into (6.45) one obtains

$$ 2{v}_{DC e}\cdot \overset{\cdotp }{\tilde{v_{DC}}}=-\frac{4{v}_{DC e}}{ RC}\tilde{v_{DC}}-\frac{V}{C}\tilde{i_d}, $$

from which the transfer function results as

$$ {H}_{\tilde{i_d}\to \tilde{v_{DC}}}(s)=\frac{\tilde{v_{DC}}}{\tilde{i_d}}=-\frac{V\cdot R}{4{v}_{DC e}}\cdot \frac{1}{s\frac{ RC}{2}+1}. $$

(6.46)

Equation (6.46) gives the small-signal ROAM that expresses the linear first-order dynamic of the DC voltage as depending on current i _d . Its utility consists in providing the basis for the design of a DC-link voltage regulator that imposes the i _d value in response to v _DC evolution (see Fig. 6.18).

Note that transfer functions given by relations (6.43) and (6.46) have the same time constant – meaning that they correspond to the same dynamic – but different gains (in the latter case this depends on the chosen equilibrium point).

The following problems are proposed to the reader to solve.

Problem 6.3

Modeling a buck power stage operating in dcm

Let us consider the buck converter with capacitive output filter having the circuit diagram presented in Fig. 2.10 of Chap. 2. The converter operates in discontinuous-inductor-current mode. The following points should be addressed.

• (a) Deduce the converter ROAM by taking the duty ratio as control input and the output voltage as controlled variable.

• (b) Determine the conversion ratio between the input and the output voltages.

• (c) Deduce the small-signal ROAM.

Problem 6.4

Modeling a flyback converter operating in dcm

Answer the same questions as in Problem 6.3 for the case of a flyback converter having the circuit diagram given in Fig. 4.24a of Chap. 4.

Problem 6.5

ROAM of a series-resonance supply of a diode rectifier

Let us consider the circuit composed of a voltage inverter, a resonant tank and a diode rectifier given in Fig. 6.19.

Fig. 6.19

Series-resonance supply of a diode rectifier

The voltage inverter is assumed to operate in full wave at a frequency close to the resonance frequency of the alternative circuit. Address the following points.

• (a) Deduce the switched model.

• (b) Explain why the classical first-order harmonic approach is suitable for studying the resonant circuit.

• (c) Write the ROAM of this converter by taking the output voltage v _C0 as a controlled variable and the inverter frequency as a control input variable.

• (d) Deduce the small-signal model of the previously obtained ROAM.

• (e) Give the equivalent diagram corresponding to the ROAM of the converter.