## Description

Problem 1. Consider a balanced, three-phase capacitive circuit, where each leg has Capacitance C

with effective series resistance, R. Denote the voltages and currents across the RC combination as va,

vb, vc and ia, ib, ic. Derive corresponding expressions for the d and q-axis components.

Problem 2. The dynamics of a linear balanced-three phase inductive circuit in the αβ domain are

given by:

vα = (L − M)

diα

dt

vβ = (L − M)

diβ

dt . (1)

What are the corresponding dynamics in the abc domain?

Problem 3. In this problem, we will examine some dynamic attributes of a three-phase grid-forming

(GFM) voltage-source inverter (VSI). The inverter includes a dc voltage source, vdc, a hex-bridge

converter, and an output inductive filter with inductance Lf and resistance Rf (these may also include

line impedances, if any). We assume droop control is employed for voltage and frequency regulation.

The system architecture with this control type is depicted in Fig. 2. The three-phase voltages corresponding to the external network and the inverter terminals are denoted by eabc and vabc, respectively.

In the αβ reference frame, we can represent the network and inverter-terminal voltages as:

(

eα =

√

2E cos ωet

eβ =

√

2E sin ωet, (

vα =

√

2V cos ωit

vβ =

√

2V sin ωit.

(2)

Implicit in the definitions above is that the amplitudes of the two voltages are denoted by √

√

2E,

2V ; and frequencies are denoted by ωe, ωi

. In what follows, we will also find it useful to define

corresponding voltage phase angles by θe = ωet, θi = ωit, respectively. To facilitate analysis, we will

reference the angle difference δ = θi − θe in subsequent developments. The inverter output currents

(referred interchangeably as line currents) in the αβ reference frame are denoted by iα, iβ, respectively,

and in the local dq reference frame by id, iq, respectively. Figure 1 illustrates several of the quantities

referenced above.

Now consider the implementation of droop control depicted in Fig. 2. At the core are the following

linear trade-offs:

V = Vnom − mq(Q − Q

?

), (3a)

ωi = ωnom − mp(P − P

?

), (3b)

where P , Q are filtered active- and reactive-power values measured at the inverter terminals (we

discuss this shortly), and mq, mp are determined by the voltage- and frequency-droop specifications.

For instance, if we assume a 5% voltage droop and 0.5 Hz frequency droop while the inverters are

running at rated power Srated, then it follows that mq = 0.05Vnom/Srated and mp = 2π0.5 Hz/Srated.

To reject double-frequency pulsating components that arise from imbalances and switching ripple

(which are inescapable in practical systems) from the power calculations, a low-pass (LP) filter is

required. In this problem, we will assume a first-order LP filter. The dynamics of the filtered activeand reactive-power, denoted by P and Q, respectively, can be written as

˙P = ωc(P − P),

˙Q = ωc(Q − Q), (4)

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EE 8744: Modeling, Analysis, and Control of Renewable Energy Systems Spring 2021

where ωc is the cut-off frequency which typically ranges from several Hz to tens of Hz. In general,

the LP filter will hinder control responsiveness as the filtered quantities are leveraged inside the droop

controller. Therefore, the selection of the cut-off frequency, ωc, is an important design choice and it

presents an important trade-off between power-filtering performance and system-transient response.

With these definitions in place, we see that the dynamics of the terminal voltage amplitude V and

angle θi

in (3) are given by:

V˙ = −mq

˙Q = mqωc(Q − Q

?

), (5a)

˙θi = ωnom − mp(P − P

?

). (5b)

In subsequent developments, we will express the phase dynamics with the power angle δ as follows

˙δ = ωnom − ωe − mp(P − P

?

). (6)

Part (i). Let ed, eq correspond to the dq reference-frame representations of the external network

voltage eabc. Using an appropriate reference transformation, express ed, eq as a function of δ and E.

Part (ii) Derive the state-space model for the dynamics of id, iq with inputs ed, eq.

Part (iii) Show that the instantaneous active and reactive power at the inverter terminals P, Q are

given by

P =

3

2

√

2V id, Q = −

3

2

√

2V iq. (7)

Part (iv) List down the dynamical equations for id, iq, V , δ, and P clearly indicating all parameters

and inputs. This fifth-order model captures all dynamics of the GFM inverter with droop control.

Note that the filtered reactive power is linearly related to the terminal voltage, and therefore, while

we can recover it’s dynamics, we will not carry it forward in the analysis.

Figure 1: Illustrating frequently referenced voltage and current signals in pertinent reference frames.

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EE 8744: Modeling, Analysis, and Control of Renewable Energy Systems Spring 2021

Part (v) Show that the equilibria of the dynamics of the GFM inverter with droop control are

obtained from the solution of the following nonlinear equations:

ωnom − ωe − mp

Peq − P

?

= 0, (8a)

Veq − Vnom − mq

3

2

√

2Veqiq,eq + Q

?

= 0, (8b)

−

Rf

Lf

id,eq + ωiiq,eq +

√

2

Lf

(Veq − E cos δeq) = 0, (8c)

−

Rf

Lf

iq,eq − ωiid,eq +

√

2

Lf

(E sin δeq) = 0, (8d)

Peq −

3

2

√

2Veqid,eq = 0, (8e)

where δeq, Veq are the terminal voltage amplitude and phase-angle equilibria corresponding to the voltage and phase dynamics; id,eq, iq,eq are the equilibrium values of the d and q axis current dynamics;

and Peq is the equilibrium value of the filtered active power.

Part (vi) We will now derive a small-signal model for the dynamical model. The state vector of

the small-signal model is defined as ∆x = [∆δ, ∆V, ∆id, ∆iq, ∆P]

>. The dynamics of the small-signal

model are obtained by linearizing the dynamical model you derived in Part (iv). The small-signal

model is compactly represented as ∆ ˙x = A∆x, where A is the Jacobian matrix of the nonlinear dynamical model evaluated for the equilibria referenced above. Write out the A matrix (25 entries) for

this small-signal model.

Figure 2: Voltage source inverter with droop controller. The “power calculation” block calculates the

active- and reactive-power using (7), the “low-pass filter” block comprises two low-pass filters with

cutoff frequency of ωc, the “droop control” block denotes the droop relationships given in (3), and

the “voltage generator” block generates the PWM modulation signals in the αβ reference frame given

corresponding polar-coordinate inputs.

EE 8744: Modeling, Analysis, and Control of Renewable Energy Systems Spring 2021

Problem 4. Consider the half-bridge circuit with current control illustrated in Fig. 3. Suppose we

employ the following PI compensator:

Gc(s) = kp + ki

1

s

. (9)

Part (i) With appropriate feed-forward cancellation, show that you obtain the circuit in Fig. 4 in a

local dq reference frame.

Part (ii) Show that the closed-loop response from the d and q-axis current references to currents are

given by

H(s) = kp + ki/s

(kp + ki/s) + (R + sL)

. (10)

Part (iii) Prove that the closed-loop transfer function, H(s), is first-order with time constant, τ ,

H(s) = 1

1 + τs

, (11)

if and only if the time constant τc of the PI compensator, τc =

kp

ki

, matches the time constant τf of

the inductive output filter τf =

L

R

.

Figure 3: Voltage source half-bridge circuit with output RL filter.

Figure 4: Voltage source half-bridge circuit with output RL filter in dq reference frame with feedforward cancellation.

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