Optimal Structures for Voltage 

Controllers in Inverters 

Preprint 

Brian Johnson, Srinivasa Salapaka, 

Blake Lundstrom, and Murti Salapaka 

National Renewable Energy Laboratory 

Presented at the IEEE International Symposium on 

Mathematical Theory of Networks and Systems 

Minneapolis, Minnesota 

July 12–15, 2016 

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Optimal Structures for Voltage Controllers in Inverters

Brian Johnson, Srinivasa Salapaka, Blake Lundstrom, and Murti Salapaka

Abstract— Output voltage regulation is a primary performance objective in power electronics systems which are not

supported by a stiff voltage source. In this paper, we pose

an optimal voltage control problem for ac inverter systems

and study the structure of the resulting feedback laws. Here,

it is demonstrated that the solution to the optimal voltage

regulation control problem exhibits an inner current-controller

structure even though there are no explicit objectives on

tracking current which are targeted. Furthermore, the outerloop voltage control and inner-loop current control structure

is insensitive to the weighting transfer functions used in the

optimal control problem. Although the inner-outer structure

has been used in prior works, the optimal nature of such a

structure was not known. In deriving the optimal controller, we

also present a systematic design framework which is cognizant

of the physical properties of inverters. Case studies are used

to study the optimal controller and its performance.

I. INTRODUCTION

Voltage controllers form an integral component of microgrid systems, uninterruptible power supplies, dc-dc converters, and systems which are not supported by a stiff

voltage source or grid. In so-called master-slave systems,

the system voltage is supported by a single “master” power

converter which typically acts on a fixed voltage reference

and all remaining units regulate their currents [1]. In parallel

converter systems, a centralized controller can be configured

such that the voltage across a common load tracks a single

reference [2], [3]. For decentralized implementations, droopcontrolled inverters [4], [5] and dc-dc converters [6] each

utilize an independent and variable voltage reference which

depends on the output of each unit. Irrespective of how

the voltage reference is generated, a voltage controller is

needed to modulate the power electronics such that the

output voltage tracks the reference. In this paper, we study

the optimal structure of voltage controllers for ac inverter

systems. In deriving the controller, we present a systematic design framework for designing multivariable voltage

controllers with robust and optimal performance. The design framework provides a systematic means of targeting

performance specifications that arise in typical ac power

B. Johnson and B. Lundstrom are with the National Renewable Energy

Laboratory (NREL), Golden, Colorado, USA. NREL is a national laboratory of the U.S. Department of Energy, Office of Energy Efficiency and

Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Email: {brian.johnson,blake.lundstrom}@nrel.gov.

Their work was supported by the Laboratory Directed Research and Development (LDRD) Program at the National Renewable Energy Laboratory

and by the U.S. Department of Energy under Contract No. DE- AC36-08-

GO28308.

S. Salapaka is with Department of Mechanical Engineering

at the University of Illinois at Urbana-Champaign. Email:

salapaka@illinois.edu

M. Salapaka is with the Department of Electrical and Computer Engineering at the University of Minnesota. Email: murtis@umn.edu

systems. As our main result, we show that the resulting

optimal controller has an outer voltage controller and an

inner current controller embedded in it. This result confirms

that the inner-outer structure is optimal and substantiates the

longstanding observation in conventional designs that innerouter structures have superior performance [7]–[9].

Problems related to robust and optimal design of inverter

controllers have received recent attention. In [10], [11],

inverter controller parameters are obtained after solving

a servomechanism robust optimization problem. The H∞

framework and internal model principle were recently applied with the objective of rejecting periodic disturbances

in microgrids [12], [13]. Along similar lines, optimal controllers for static VAR compensators have been investigated

[14]. Here, our focus is on studying the structure of the optimal controller itself and outlining an unambiguous design

procedure.

The outer-voltage inner-current control structure has a

rich history in the power community and has been utilized

extensively in single- and three-phase [8], [15] inverters

as well as dc-dc converters [7], [9]. Going back several

decades, some of the earliest uses of the inner-outer control

structure are found in dc-dc converter applications [16]

and was first employed due to its superior performance in

comparison to single-loop controllers. Subsequent analysis

and engineering judgment have indicated an inherent robustness to load variations and its application has become

well-established in inverters for ac systems [8]. Despite its

advantages over single-loop controllers, traditional innerouter design methods are iterative in nature, are not tailored

for multiple input systems, and do not offer performance

guarantees. In contrast, the proposed method is well-suited

for multivariable settings and guarantees optimality.

The main contributions of this paper are as follows: i)

we demonstrate that that the optimal voltage controller has

an inner-outer structure, and ii) in deriving the optimal

controller, we describe a systematic design process which

incorporates practical performance specifications in ac inverter systems.

The manuscript is organized in the following manner.

In Section II we introduce the reader to the inverter system model, formulate the design problem, derive the plant

model, and outline practical considerations for ac power

electronics systems. The classical inner-outer structure is

introduced and its correspondence with the optimal H∞

is established in Section III. Concluding statements are in

Section IV.

1

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C

i

R r

K

L

+

−

+

vinv

u

v

+

vL

iinv

− zL

−

−

∗ v

control

vˆinv

+

− −

+

PWM switching

signals

vinv

+

+ +

id

 ̃

Fig. 1. A system consisting of a single-phase inverter connected across

an impedance load with an LC filter and closed-loop controller. The

unmodelled current source,  ̃id, accounts for unknown sources and loads.

The inverter hardware, shown on the bottom, produces an instantaneous

switched voltage vˆinv. Using averaging, the inverter is modeled as a

controllable voltage source which follows the signal vinv.

II. SYSTEM MODEL, PROBLEM FORMULATION, AND

CONTROL DESIGN

A. Inverter System Description

Consider the system in Fig. 1 which consists of a power

electronics inverter connected across an impedance load, zL.

The inverter draws power from a dc source and generates a

switched ac voltage, vˆinv, which serves as a control variable.

Using pulse width modulation (PWM) techniques and a

high switching frequency, the switch cycle average of vˆinv

follows the command vinv [8]. The inductance, L, and

capacitance, C, filter high-frequency harmonics generated

by the switching action. R accounts for ohmic losses in the

inductor and r is the branch resistance which interfaces the

inverter terminals to the remaining system.  ̃id encapsulates

the behavior of unknown loads and sources in the system

and is treated as a disturbance. Since the switching period

is typically much smaller than the filter time-constants, we

model the switch terminals as a controllable voltage source

which follows the control signal vinv. This averaged model

facilitates design, modeling, and analysis.

We assume that the measured signals include the inductor

current iinv, capacitor voltage v, and output branch current

i. The objective is to design a feedback law that generates

a voltage command, vinv, which ensures, i) v tracks v

∗

,

ii) control performance is robust to parametric uncertainty,

iii) the recovery time after disturbances and transients is

small, and iv) the control signal, vinv, respects bandwidth

limitations.

H∞ methods provide a framework for addressing multiple objectives such as those in the previous paragraph. With

this approach, a linear stabilizing control law is obtained

by posing and solving an optimization problem. In the

remainder of the paper, we apply the H∞ framework to

derive a controller K(s) and subsequently show that the

optimal design exhibits an inner-outer structure.

K rLC

−1 Y η

∗ v

vL i

id

u iinv v

W1 Wd

W2

z1

z2

+

+

+

+ +

− − η

+

+

+

r

Yinv YC

plant model

id

 ̃

Fig. 2. Block-diagram of the inverter circuit. The regulated variables, z1

and z2, are shown in red and the measurements are shown in green.

B. Problem Formalization

We denote i) the external input as w(s)

1 where

w(s) = id(s) for the system in Fig. 1, ii) the control input as u(s) = vinv(s) − v(s) for the single inverter system, iii) the regulated output as z(s) and we

pick z(s) = [W1(s)(v

∗

(s) − v(s)), W2(s)vinv(s)]> where

W1(s) and W2(s) are user-defined weighting transfer functions, and iv) the measured output as y(s) where y(s) =

[v

∗

(s) − v(s), i(s), iinv(s)]>for the inverter circuit. The

physical configuration for this system is shown in Fig. 1.

The inductor current is given by

iinv(s) = Yinv(s) (vinv(s) − v(s)), (1)

where Yinv(s) := (sL + R)

−1

. The load voltage can be

expressed as

vL(s) = η(s)iinv(s) + ̃id(s)



Y

−1

rLC(s), (2)

where we define

η(s) := zC(s)

zC(s) + r

, (3)

YrCL(s) := YC(s)η(s) + YL(s), (4)

and YC(s) := z

−1

C

(s) = sC. Along similar lines, the

capacitor voltage can be written as

v(s) = (riinv(s) + vL(s)) η(s). (5)

Lastly, the capacitor state-equation can be rearranged to

yield

i(s) = iinv(s) − YC(s)v(s). (6)

Using (1)–(6), recalling u(s) = vinv(s) − v(s) and z(s) =

[W1(s)(v

∗

(s) − v(s)), W2(s)vinv(s)]>, the block-diagram

of the inverter circuit with the regulated variables is given

in Fig. 2. Note that the transfer function, Wd(s) is a

user-defined transfer function which defines the expected

frequency range of disturbances.

1From here forward, s = σ + jω is a complex variable and proper

transfer functions are assumed throughout.

2

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The mapping from inputs w(s) and u(s) to outputs z(s)

and y(s) is given as:













W1(s)(v

∗

(s) − v(s))

W2(s)vinv(s)

v

∗

(s) − v(s)

i(s)

iinv(s)













= G(s)



id(s)

vinv(s) − v(s)



,

(7)

where, the generalized plant transfer matrix is

















−

W1(s)Wd(s)η(s)

YrLC(s) −

W1(s)η(s)(1+rYL(s))Yinv(s)

YrLC(s)

W2(s)Wd(s)η(s)

YrLC(s) W2(s)



1+ η(s)(1+rYL(s))Yinv(s)

YrLC(s)



−

η(s)

YrLC(s) −

η(s)(1+rYL(s))Yinv(s)

YrLC(s)

−

η(s)Yc(s)

YrLC(s)

η(s)YL(s)Yinv(s)

YrLC(s)

0 Yinv(s)

















| {z }

=G(s)

. (8)

The derivation of (8) is summarized in Appendix-A. Having

obtained G(s), the controller K(s) can be found after

specifying the user-defined transfer functions W1(s), W2(s),

and Wd(s) and solving the H∞ design problem.

Remark 1: Although the load is generally unknown and

does not have static parameters, an estimate of the load

admittance can be utilized in the plant model. Subsequently,

we show robust performance to load variations.

C. Design of Weighting Transfer Functions

Here, we outline a set of guidelines for designing the cost

transfer functions which encapsulate practical performance

objectives and system properties.

1) Voltage regulation function W1(s): Referring to the

first row in (7), it is apparent that W1(s) has a direct

relationship on the voltage tracking error. Accordingly,

W1(s) is designed such that tracking error at the nominal

ac frequency, ωo, is minimized, the resonant behavior of

the inverter LC filter is damped by the controller, and

performance within the inverter bandwidth is emphasized.

Accordingly, W1(s) can be ascribed the following general

form:

W1(s) = κ1Gωo(s)Grlc(s)Gbw(s), (9)

where κ1 is an overall gain factor, Gωo(s) is tuned such

that it has a large amplitude at ωo, Grlc(s) damps out

the resonant behavior of the RLC elements in the filter,

and Gbw(s) emphasizes performance within the inverter

bandwidth.2 Gωo(s) can be written as the superposition of

a band-pass and notch filter with the following expression

Gωo(s) = κNs

2 + 2κBPζωos + κBPω2

o

s

2 + 2ζωos + ω2

o

, (10)

where κN and κBP are the gains of the notch and bandpass

components, respectively, and ζ is the damping factor. Since

we wish to emphasize performance at the rated frequency,

ωo, we must pick κBP > κN. For variable frequency ac

2

In practice, the inverter bandwidth is limited by the switching frequency,

sampling frequency, and time-step size of the digital controller.

systems, it may be necessary to increase ζ to broaden the

frequency range where Gωo(s) is large. This approach of

emphasizing the controller response at the rated system frequency is similar to the widely used proportional-resonant

controller for ac systems [20], [21].

Undesired inverter filter resonance can be mitigated by

including a model of the filter within Grlc(s). Referring

to Fig. 1, denote the parallel impedance of the RL branch

and capacitor as zf(s) := (R + sL) k(sC)

−1

. We ascribe

Grlc(s) the following form:

Grlc(s) = (zf(s) + 1) den(zf(s))

α(s)

. (11)

In practice, the output filter is designed with minimal

resistive losses such that high efficiency is maintained. Consequently, zf(s) usually exhibits a tight peak at 1/

√

LC and

small gain at all other frequencies. To provide robustness

to parameter variations (typical filter component tolerances

are ±10%) we wish to damp frequencies around 1/

√

LC

by tuning Grlc(s). Towards that end, den(zf(s))/α(s) is

selected to attain the desired response around 1/

√

LC and

the addition of 1 in the first factor of (11) flattens the gain

to unity at asymptotically high and low frequencies.

Reference tracking at frequencies within the inverter

bandwidth can be further enhanced by including the transfer

function, Gbw(s). Specifically, to ensure W1(s) does not

emphasize high frequencies which cannot be realized by

the hardware, Gbw(s) can be selected as a low-pass filter.

Here, we pick

Gbw(s) = (ωlpf/ (s + ωlpf))2. (12)

Lastly, the overall gain factor, κ1, in (9) is proportional to

the voltage regulation aggressiveness of the controller and

can be tuned accordingly.

2) Inverter voltage control effort weighting function

W2(s): Referring to (7), it follows that W2(s) corresponds

to shaping the performance of vinv. Since there are no corresponding reference signals to track, W2(s) can be designed

to suppress high-frequency control effort. Accordingly, a

high-pass filter or a related variation can be used as given

below

W2(s) = W3(s) = c1s + ωw

s + c2ωw

, (13)

where c1 and c

−1

2

are the asymptotic gains at high frequencies and dc, respectively, and c1ωw and c

−1

2

are the

frequency breakpoints.

3) Disturbance current transfer function Wd(s): The

function, Wd(s), characterizes the response of expected

disturbances. For systems with linear loads, we can assume

that disturbance currents have components primarily at ωo.

However, if nonlinear disturbances, such as diode rectifier

loads and switching power supplies, are anticipated, Wd(s)

can be chosen to emphasize higher order harmonics. If the

frequency content of unmodelled loads is anticipated, Wd(s)

can be chosen accordingly. For instance, the harmonics of

rectifier loads are well characterized in [18], [19]. Here,

we choose this function as a low-pass filter (i.e., Wd(s) =

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+

−

+

+

+

+ + +

−

+

−

voltage control current control −

LC filter plant

∗ v Kv Ki v

inv

∗

i

iinv

vinv sL+R

1

sC

1

i

Fig. 3. Conventional voltage controller with an inner-outer structure.

1/



ω

−1

d

s + 1) such that low-order harmonic rejection is

emphasized.

III. OPTIMAL CONTROL STRUCTURE AND

PERFORMANCE

In this section, we first introduce the classical voltage

controller and subsequently demonstrate its correspondence

with the proposed H∞ design.

A. The Outer-Voltage Inner-Current Control Structure

Consider the well-known voltage controller in Fig. 3

which exhibits an inner-outer structure [7], [8], [15]. In

this classical system, an outer voltage controller, Kv(s),

generates a reference, i

∗

inv(s), for an inner current controller,

Ki(s). The feedforward terms, i(s) and v(s), are used to

cancel signals which act as disturbances in the LC filter

plant. The intuition behind this design is that the inner

current controller allows for direct control of the energy

delivered by the inductor without introducing delays from

the capacitor. The architecture in Fig. 3 dates back several

decades [16]. Referring to Fig. 3, the current controller

output can be written as

vinv(s) − v(s)

= Ki(s) (i

∗

inv(s) − iinv(s))

= Kv(s)Ki(s) (v

∗

(s) − v(s)) + Ki(s) (i(s) − iinv(s))(14) .

Next, let’s consider the H∞ controller, K(s), that

we designed in Section II. This controller has inputs

[v

∗

(s) − v(s), i(s), iinv(s)]>and output signal is u(s) =

vinv(s)−v(s).

3 Denote the controller transfer matrix entries

3The decision to pick u(s) = vinv(s) − v(s) becomes clear since this

provides an intrinsic voltage feedforward.

Magnitude [dB]

ω, [rad/s]

0 10 1 10 2 10 3 10 4 10 5

10 6 10 7 10

0

20

-20

-40

-60

-80

40

60

80

W1

W2

Fig. 4. The user-defined weights, as defined in Section II-C, are illustrated

for the case L = 1 mH, C = 22 μF, and R = 10 mΩ. The peaks in

W1(s) correspond to the rated system frequency and resonant frequency

of the output filter. Here we illustrate the case when the performance

coefficient is κ1 = 102

ω (rad/s)

Kui

Phase (deg) Magnitude (dB)

ω (rad/s)

Kue

Phase (deg) Magnitude (dB)

101102103104105106 101102103104105106

−90

0

90

180

15

20

25

30

35

90

180

270

0

20

40

60

80 Kuinv

(a)

ω (rad/s)

Phase (deg) Magnitude (dB)

ω (rad/s)

Kue

Phase (deg) Magnitude (dB)

101102103104105106 101102103104105106

−90

0

90

180

30

35

40

45

45

90

135

180

225

30

40

50

60

70

Kui Kuinv

(b)

Fig. 5. Responses of the controller for filter parameters L = 1 mH,

C = 22 μF, and R = 10 mΩ when (a) κ1 = 102, and (b) κ1 = 104.

The voltage tracking error to control signal transfer function is shown on

the left. The responses from the output and inductor currents to control

signal are given on the right. Note that the magnitudes of the currents-tocontrol transfer functions coincide over most of the frequency range and the

phases are separated by 180o. Here, the optimality of a control architecture

with an “outer” voltage controller and “inner” current controller is implied.

as K(s) = [Kue(s), Kui(s), Kuinv(s)], where Kue(s) is

the transfer function between the voltage error and control

signal, Kui(s) processes the output current measurement,

and Kuinv(s) corresponds to the inverter inductor current.

If Kui(s) ≈ −Kuinv(s), it follows that

u(s) := vinv(s) − v(s)

= Kue(s) (v

∗

(s) − v(s)) + Kui(s)i(s) + Kuinv(s)iinv(s)

≈ Kue(s) (v

∗

(s) − v(s)) + Kui(s) (i(s) − iinv(s)). (15)

Comparing (15) and (14), it is apparent that if Kui(s) ≈

−Kuinv(s) holds, then the H∞ controller, K(s), has the

same structure as the classical controller in Fig. 3. This leads

us to the main result of the paper.

B. Optimality of inner current control and outer voltage

control configuration

Given the generalized plant matrix in (8), the weighting

transfer functions in (9)–(13), and the parameters in Table

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ω (rad/s)

Phase (deg) Magnitude (dB)

ω (rad/s)

Kue

Phase (deg) Magnitude (dB)

101102103104105106 101102103104105106

−90

0

90

180

25

30

35

40

90

180

270

0

20

40

60

80 Kui Kuinv

(a)

ω (rad/s)

Phase (deg) Magnitude (dB)

ω (rad/s)

Kue

Phase (deg) Magnitude (dB)

101102103104105106 101102103104105106

−90

0

90

180

35

40

45

50

45

90

135

180

225

40

60

80

Kui Kuinv

(b)

Fig. 6. Responses of the controller for filter parameters L = 3 mH,

C = 66 μF, and R = 30 mΩ when (a) κ1 = 102, and (b) κ1 = 104.

As shown, the inner-outer structure remains despite variations in the filter

parameters and control gains.

1, the controller, K(s), is obtained after solving the H∞

design problem. As illustrated in Fig. 4, the voltage tracking

weighting transfer function, W1(s), has peaks at 2π60 rad/s

and 1/

√

LC. The transfer function, W2(s), exhibits a highpass characteristic. Having obtained the controller, we now

study the responses of Kui(s) and Kuinv(s).

As shown in, Figs. 5 and 6, the control response to both

current measurements is similar in magnitude over a broad

frequency range and they exhibit a 180o phase shift between

each other. This result confirms Kui(s) ≈ −Kuinv(s)

which implies the relationship in (15) and establishes a

correspondence with the conventional inner-outer response

in (14). This observation holds for parametric variations in

the plant output filter and the controller gains. In particular,

the L, C, and R filter values in Figs. 5 and 6 differ by a

factor of 3 and the respective subplots utilize a weighting

coefficient κ1 that varies by a factor of 100. This provides

strong evidence that Kui(s) ≈ −Kuinv(s) holds over a

broad set of conditions. It is worth noting and especially

interesting that the inner-current controller appears despite

the exclusion of current as a regulated variable.

+5 mV

−5 mV

i

iL

id

−40

−20

0

20

40

0 0.05 0.1 0.15 0.2

−10

−5

0

5

10

, [V] v −∗ v current, [A]

time, [s]

(a)

−20 μV

+20 μV

−40

−20

0

20

40

i

iL

id

0 0.05 0.1 0.15 0.2 −10

−5

0

5

10

, [V] v −∗ v current, [A]

time, [s]

(b)

Fig. 7. A comparison of disturbance rejection for filter parameters L =

1 mH, C = 22 μF, and R = 10 mΩ when, (a) the weighting coefficient

κ1 = 102, and (b) κ1 = 104. In the simulation, a sinusoidal current

disturbance is abruptly started at t = 0.1 s. As illustrated, the controller

with the larger coefficient in W1(s) reduces the error amplitude during

transients and reaches a lower steady-state error. Note that the error during

the transient exhibits oscillatory behavior at the filter resonant frequency.

C. Time-Domain Performance

Here we analyze the time-domain performance of the

system in Fig. 1 with the optimal controller. In the generalized plant matrix, a nominal parallel RL load is assumed

which consumes 1 kW of real power with a power factor

of 0.8. A sinusoidal voltage reference is utilized where

v

∗

(t) = 120√

2 sin (ωot).

We observe the case when a disturbance current, id, is

injected into the load. Referring to Fig. 7, the inverter is

initially delivering power to the nominal load. At t = 0.1 s,

a sinusoidal disturbance current with amplitude 10 A and

120o phase shift is initiated. To illustrate how controller

performance is easily adjusted by tuning the weighting

transfer functions, we compare voltage regulation when the

performance coefficient κ1 in (9) is adjusted. In Figs. 7(a)

and 7(b), κ1 is assigned a value of 102and 104, respectively.

In comparing the response of the two systems, it is evident

that the controller with the larger value of κ1 yields a smaller

voltage tracking error during both transients and in steadystate.

IV. CONCLUSION

Here, we studied the optimal structure of voltage controllers for inverters. It was shown that the optimal conThis report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 5
TABLE I

PLANT AND CONTROLLER PARAMETERS.

R = 10 mΩ, 30 mΩ r = 1 Ω

C = 22 μF, 66 μF L = 1 mH, 3 mH

Y

−1

L(nom) = 12 Ω + s(42.4 mH) κ1 = 102

, 104

κN = 1 κBP = 102

ζ = 0.01 α(s) = LCs2 + 100RCs + 1

ωlpf = 8ωo c1 = c2 = 0.01

ωw = 4ωo ωd = 3ωo

troller, obtained via H∞ synthesis, contains outer-voltage

and inner-current control loops embedded in it. This result is

obtained despite variations in the design parameters and the

intentional exclusion of current design specifications. This

suggests that the optimality of the inner-outer structure holds

over a wide parameter space and is generally applicable. As

part of future efforts, an analytical proof of the inner-outer

structure is in progress.

APPENDIX

A. Computation of G(s)

v

∗

(s) − v(s) is given by

v

∗

(s) − v(s)

= v

∗

(s) −



η(s)

YrLC(s)



id(s)

−



η(s) (1 + rYL(s)) Yinv(s)

YrLC(s)



(vinv(s) − v(s)), (16)

where the final expression follows from

r + η(s)Y

−1

rLC(s) = Y

−1

rLC(s) (1 + rYL(s)), (17)

and YL(s) := z

−1

L

(s). (16) accounts for the first and third

rows of G(s) in (7).

The inverter voltage can be written in terms of id(s) and

(vinv(s) − v(s)) as given below:

vinv(s)

=

Wd(s)η(s)

YrLC(s)

id(s) +



1 +

η(s) (1 + rYL(s)) Yinv(s)

YrLC(s)



(vinv(s) − v(s)),(18)

The result in (18) corresponds to the second row of (7).

Along similar lines, the inverter output current can be

expressed as

i(s)

= −



Wd(s)η(s)Yc(s)

YrLC(s)



id(s)

+



η(s)YL(s)Yinv(s)

YrLC(s)



(vinv(s) − v(s)), (19)

where we utilized the fact that

1 − rYc(s)η(s) −

η

2

(s)Yc(s)

YrLC(s)

=

η(s)YL(s)

YrLC(s)

. (20)

The fourth row of G(s) is characterized using (19) and the

last row is given by (1).

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This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 6