A national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

National Renewable Energy Laboratory 

Innovation for Our Energy Future 

Electrical Model Development 

and Validation for 

Distributed Resources 

M.G. Simões, B. Palle, S. Chakraborty, 

and C. Uriarte 

Colorado School of Mines 

Golden, Colorado 

Subcontract Report 

NREL/SR-581-41109 

April 2007 

NREL is operated by Midwest Research Institute ● Battelle Contract No. DE-AC36-99-GO10337 
Electrical Model Development 

and Validation for 

Distributed Resources 

M.G. Simões, B. Palle, S. Chakraborty, 

and C. Uriarte 

Colorado School of Mines 

Golden, Colorado 

NREL Technical Monitor: Benjamin Kroposki 

Prepared under Subcontract No(s). XAT-5-55150-01 

Subcontract Report 

NREL/SR-581-41109 

April 2007 

National Renewable Energy Laboratory

1617 Cole Boulevard, Golden, Colorado 80401-3393 

303-275-3000 • www.nrel.gov 

Operated for the U.S. Department of Energy 

Office of Energy Efficiency and Renewable Energy 

by Midwest Research Institute • Battelle 

Contract No. DE-AC36-99-GO10337
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Table of Contents 

1 Introduction....................................................................................................................................1

2 Methodology ..................................................................................................................................2

3 Model of the Backbone .................................................................................................................4

4 Diesel Generator............................................................................................................................5

4.1 Mathematical Analysis...........................................................................................5 

4.2 Simulation Results.................................................................................................6

5 Wind Turbine..................................................................................................................................9

5.1 Mathematical Analysis...........................................................................................9 

5.2 Simulation Results.................................................................................................9

6 Inverter .........................................................................................................................................13

6.1 Mathematical Analysis.........................................................................................13 

6.1.1 d-q Axis Theory .......................................................................................13 

6.1.2 Anti-Islanding Schemes...........................................................................15 

6.1.3 Constant Current Control.........................................................................16 

6.1.4 Constant Power Control...........................................................................18 

6.2 Simulation Results...............................................................................................18

7 Photovoltaic Cell .........................................................................................................................32

7.1 Mathematical Analysis.........................................................................................32 

7.2 Simulation Results...............................................................................................33

8 Microturbine.................................................................................................................................34

8.1 Mathematical Analysis.........................................................................................34 

8.2 Simulation Results...............................................................................................36 

8.2.1 Microturbine ............................................................................................36 

8.2.2 DC Link ...................................................................................................38 

8.2.3 PI Controller.............................................................................................40 

8.2.4 Grid Connection.......................................................................................40

9 Backbone Integration..................................................................................................................43

9.1 Diesel Generator ..................................................................................................45 

9.2 PV Cell Grid Connection.....................................................................................46 

9.3 Microturbine Grid Connection.............................................................................48

10 Conclusion and Future Work .....................................................................................................49

11 References ...................................................................................................................................50

iii 
List of Figures 

Figure 1. Microgrid backbone...............................................................................................4 

Figure 2. Diagram of the three-phase and d-q windings.......................................................5 

Figure 3. D-axis equivalent circuit........................................................................................5 

Figure 4. 125-kW diesel generator........................................................................................6 

Figure 5. Woodward governor ..............................................................................................6 

Figure 6. Output of diesel generator without mechanical dynamics.....................................7 

Figure 7. Output of diesel generator with mechanical dynamics..........................................8 

Figure 8. Maximum value of ...........................................................................................9 Cp

Figure 9. Wind turbine generator ........................................................................................10 

Figure 10. Wind turbine and components.............................................................................10 

Figure 11. Wind turbine output.............................................................................................12 

Figure 12. Schematic of the GE inverter system...................................................................13 

Figure 13. Physical distribution of three stationary axes and a quadrature stationary axis ..14 

Figure 14. Average model of the inverter .............................................................................15 

Figure 15. Constant current control with voltage feedback ..................................................17 

Figure 16. Constant current control with frequency feedback ..............................................17 

Figure 17. Constant power control method...........................................................................18 

Figure 18. PLL and d-q axis transformation .........................................................................18 

Figure 19. PSCAD model of the GE inverter........................................................................19 

Figure 20. PSCAD model of AC feed and diode bridge rectifier .........................................20 

Figure 21. VSC averaged model ...........................................................................................21 

Figure 22. PI controllers........................................................................................................21 

Figure 23. Compensation term for inverter voltage and frequency control ..........................22 

Figure 24. Commands for the VSC averaged inverter..........................................................22 

Figure 25. Grid, load models, and breakers ..........................................................................23 

Figure 26. RMS inverter voltage...........................................................................................24 

Figure 27. Inverter frequency................................................................................................25 

Figure 28. d-axis inverter current (voltage)...........................................................................25

Figure 29. q-axis inverter current (frequency) ......................................................................26 

Figure 30. DC link voltage (rectifier output) ........................................................................26 

Figure 31. Power measurement points..................................................................................27 

Figure 32. Active power injected by the AC feed.................................................................27 

Figure 33. Reactive power injected by the AC feed..............................................................28 

Figure 34. Active power delivered by the inverter................................................................28 

Figure 35. Reactive power delivered by the inverter ............................................................29 

Figure 36. Active power injected into the grid......................................................................29 

Figure 37. Reactive power injected into the grid ..................................................................30 

Figure 38. Active power absorbed by the load......................................................................30 

Figure 39. Reactive power absorbed by the load ..................................................................31 

Figure 40. PV cell equivalent circuit.....................................................................................32 

Figure 41. PV cell current .....................................................................................................33 

Figure 42. PV cell voltage.....................................................................................................33 

Figure 43. Block diagram of a microturbine .........................................................................34 

Figure 44. Total system.........................................................................................................36 

 iv
Figure 45. Microturbine ........................................................................................................37 

Figure 46. Microturbine output voltage ................................................................................37 

Figure 47. Microturbine output current.................................................................................38 

Figure 48. Microturbine output power ..................................................................................38 

Figure 49. Capacitor block ....................................................................................................39 

Figure 50. Capacitor voltage block .......................................................................................39 

Figure 51. DC link voltage ....................................................................................................39 

Figure 52. PI control block....................................................................................................40 

Figure 53. Modulation factor.................................................................................................40 

Figure 54. Grid connection....................................................................................................41 

Figure 55. Power profile........................................................................................................41 

Figure 56. Output power .......................................................................................................42 

Figure 57. Microgrid backbone – PSCAD............................................................................43 

Figure 58. Bus voltage...........................................................................................................44 

Figure 59. Grid simulator......................................................................................................44 

Figure 60. Load simulator .....................................................................................................45 

Figure 61. Active power – diesel generator ..........................................................................46 

Figure 62. Reactive power – diesel generator.......................................................................46 

Figure 63. PV output voltage ................................................................................................47 

Figure 64. Inverter output frequency.....................................................................................47 

Figure 65. Inverter output voltage.........................................................................................47 

Figure 66. Power injected from the inverter..........................................................................48 

Figure 67. Microturbine grid connection ..............................................................................48 

List of Tables 

Table 1. Performance Parameters ......................................................................................11 

Table 2. Rotor Parameters..................................................................................................11 

Table 3. Microturbine parameters......................................................................................35 

 v
 vi
1 Introduction 

This project focuses on the development of electrical models for small (1-MW) distributed 

resources (DR) at the National Renewable Energy Laboratory’s (NREL’s) Distributed Energy 

Resources Test Facility (DERTF). DR include all generation (e.g., photovoltaics, wind 

turbines, fuel cells, microturbines, and engines) and storage (e.g., batteries and flywheels) 

located at or near loads. The DERTF includes a Hybrid Power Test Bed, to which multiple 

DR are connected. These devices include wind turbines, a diesel generator, a microturbine, 

two village simulator load banks, and one alternating current-source (AC-source) grid 

simulator. 

The purpose of this project is to provide NREL with the ability to conduct electrical power 

system modeling and analysis on the distribution system. This capability will allow NREL to 

examine the effects of multiple distributed generation units at various penetration levels on 

the electrical power system. 

The first step of the project was to discuss which DR would be integrated into the microgrid. 

After that was defined, a mathematical analysis was performed for each component. Most of 

the modeling was based on the information contained in [1]. 

Before modeling the microgrid, it was necessary to determine which simulation software 

would be most appropriate. Of the software packages tested and analyzed, it was determined 

that PSCAD would be most suitable for this task. 

With the mathematical analysis done and the appropriate software chosen, the modeling of the 

backbone of the microgrid began. The purpose was to analyze the system and the interactions 

of the DR with one another. Each machine was modeled individually and then inserted onto 

the grid. Interactions between models and the microgrid could then be monitored and adjusted 

to match their physical counterparts at the NREL testing facility. 

The next step of this research is to model a fuel cell and integrate it into the microgrid. In 

addition, validation of the developed models through actual testing of multiple types of DR 

must be done to examine the electrical impacts of multiple DR on the electrical power system. 

1 
2 Methodology 

NREL, in collaboration with the Colorado School of Mines, developed advanced models for 

multiple DR. The work developed in this project can be used to understand and model the 

electrical effects of multiple DR on the distribution system and to determine the effects of 

large penetrations of DR. 

The following methodology was adopted for this project: 

1. DR model literature search 

A literature search for available DR electrical models was conducted for inverters, 

synchronous generators, microturbines, photovoltaic (PV) cells, and wind turbines and 

induction generators [1, 2, 4, 6, 7]. This task helped bring together in a central location 

the current electrical models for the various types of DR. Most of the models came 

from [1], but the PV model was based on [2]. 

2. Electrical system modeling software evaluation 

This part of the project involved selecting the best simulation software for analysis of 

electrical power systems. The software chosen was PSCAD. 

PSCAD is a power-system simulator for the design and verification of power quality 

studies, power electronic design, distributed generation, and transmission planning. It 

was developed by the Manitoba HVDC Research Center and has been in use since 

1975. PSCAD is a graphical front end to EMTDC for creating models and analyzing 

results. In PSCAD, one combines blocks to form a power network. These blocks are 

actually FORTRAN code, which call for an EMTDC code library to combine them 

into executable files. Running these files runs the simulations, and the results can be 

picked up by PSCAD on the run. 

PSCAD is suitable for the NREL DERTF for several reasons. PSCAD has fully 

developed models of various devices used in the NREL facility. The library includes 

models of synchronous and induction machines, transformers, three-phase converters, 

relays, breakers, cables, wind turbines, and transmission lines. Saturation, magnetizing, 

and leakage inductances can be disabled or enabled in the rotating machine models. If 

enabled, up to 10 saturation curve points can be given. PSCAD also offers tools to 

simulate various faults on the power system. Turbine models—including wind turbines, 

governor, and other miscellaneous models—are also in the PSCAD library. In addition, 

PSCAD has models that generate wind speed for the wind turbine. 

In PSCAD, systems may consist of electrical and control-type components, which may 

be interconnected to allow for an all-inclusive simulation study. The control systems 

modeling function section of the PSCAD library provides a complete set of basic 

linear and nonlinear control components. These components can be combined into 

larger, more elaborate systems. Outputs from control components can be used to 

 2
control voltage and current sources, switching signals, and firing pulses for thyristors, 

gate turn-off thyristors, and insulate gate bipolar transistors. Control components can 

also be used for signal analysis, and outputs may be directed to online plots or meters. 

In addition, PSCAD has an inbuilt model for a six-pulse SCR converter. Converter 

models with other switching devices and firing circuits can be built using the control 

systems modeling function library. Models of fuel cells or other devices not available 

in PSCAD can be built using this library and be added to the master library. Users can 

write subroutines for any unavailable components in FORTRAN or C (C++). 

As in Simulink, multiple modules can be built inside a single project, and each module 

can contain other modules. This provides a hierarchical modeling capability. PSCAD 

has an interface to Simulink, but it is not suitable for continuous and simultaneous 

simulations. PSCAD calls Simulink, which runs a whole simulation and then returns 

the result to PSCAD. 

Modeling in the PSCAD environment is not as intuitive as it is in circuit simulators 

such as PSIM. However, once one gets used to wiring and other simulation controls, 

building models in PSCAD is relatively easy. 

PSCAD has complete models of most of the devices used in the NREL facility. 

Models of other devices [such as fuel cells and pulse-width modulated (PWM) 

inverters] that are not available in PSCAD can be built by the user. PSCAD has the 

tools to implement control algorithms such as peak-power tracking for PV cells or 

wind turbines. Graphing, plotting, and exporting results are easy. Therefore, PSCAD 

is suitable for this simulation study. 

3. DR model development 

Under this task, models of DR such as microturbines, PV cells, and diesel generators 

were developed. Electrical models of inverters were also developed. Each model was 

developed and tested individually and then connected to a three-phase distribution 

system. 

4. Validation testing of developed models 

This step is yet to be implemented. It consists of using the models developed in Step 3 

to simulate the electrical configurations of DR at the NREL DERTF. Testing will 

verify the model transient and steady-state models of electrical power flow. 

5. Final report and presentation 

This report details the research conducted under this project. It includes information 

about the availability of models and software to conduct analysis of the electrical 

impacts of distributed generation on the electrical power system and the selection of a 

specific modeling program. It also reports on the models that were developed. 

Validation under actual test conditions is yet to be conducted.

 3
3 Model of the Backbone 

Figure 1 shows the microgrid at the NREL DERTF. The main components of the system are a 

250-kVA grid simulator; a 120:480-V, 250-kVA transformer; a 480-V bus; a 165-kW–404-

kVAR variable R-L-C load bank; a 125-kW diesel generator; a 60-kW wind turbine; a 33-kW 

microturbine; and a few three-phase and single-phase PWM inverters to inject power into the 

bus from direct current (DC) sources. 

Figure 1. Microgrid backbone 

The following sections discuss each DR. 

 4
4 Diesel Generator 

4.1 Mathematical Analysis 

The diesel generator used a synchronous machine model available in the PSCAD library. 

Equation 1 shows the transformation of the stator windings into equivalent windings, using 

the dq0 transformation, as follows: 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−θθ −θ

−θθ −θ

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

c

b

a

o o

o o

0

q

d

V

V

V

2

1

2

1

2

1

)240sin()120sin()sin(

)240cos()120cos()cos(

U

U

U

(1) 

 

The three-phase rotor winding may be transformed into a two-phase equivalent winding, with 

additional windings added to each axis, as shown in Figure 2. 

Figure 2. Diagram of the three-phase and d-q windings [3] 

The d-axis equivalent circuit for the generalized machine is shown in Figure 3. 

Figure 3. D-axis equivalent circuit [3] 

 5
The mechanical torque is calculated by Equation 2 [3, 11]. 

)III(LIL

)III(LIL

IIT

3D2D1DMD1D1d

3Q2Q1QMQ1Q1q

1Qd1Dq

+=Ψ ++

+=Ψ ++

Ψ−Ψ=

(2) 

 

4.2 Simulation Results 

Figure 4 is the 125-kW synchronous generator driven by a diesel engine. Included in this 

schematic is the synchronous generator with exciter and timers with an AC source on the right 

side representing an infinite bus to maintain voltage and a 50-μF capacitor for power 

correction. L2N and S2M send a signal to the generator at 10 and 9.5 s, respectively, to allow 

it to operate as a machine with mechanical dynamics rather than as a simple AC source. This 

model is a better representation of a real-time system. The signal BRK is used to close the 

breaker at 0.5 s to allow power flow through the system. 

3 Phase

RMS

TIN

BRK

*

1

BRK TIME

Close BRK

at 0.5 Second

1

S

Te

3

A

V

Tm

Ef0

w Tm

Ef If

If

v

0.48

Timer

Enab Mech Dynamics

0->1 @ 10.0 Sec

L2N

Timer

S2M

Source to Machine

0->1 @ 9.5 Sec

TM

VT

IT 3

Ef If

Ef0

Vref

Exciter_(AC1A)

W

RRL

50.0

Figure 4. 125-kW diesel generator 

Below is the control system, also known as the Woodward governor, for the diesel engine. It 

consists of a set of controls—whose input is the offset of the actual speed and the reference 

speed—that control the actual diesel engine (labeled IC engine) and result in a controlled 

output torque that is sent to the input torque of the synchronous generator. 

1

sT e G -sT 1 + sT11 + sT2

G

1 + sT

N(s)

D(s)

Order = 2

*

D 17 -

F

+

WREF

SPEED

w

Tm

FL IC Engine

W

TS1

TS2

Diesel engine model to represent

cylinder torque ripple and misfire

Figure 5. Woodward governor 

 6
Output real power, reactive power, and terminal voltage are measured at the generator’s 

output. Below are output graphs of the synchronous machine running solely as a source 

without transitioning to machine with the rotor unlocked and the mechanical dynamics 

enabled. Figure 6 provides the output graphs of the generator. The desired output of 125 kW 

is achieved as is the desired terminal voltage, which at 1 p.u. is equal to 480 V. The blue

output represents the diesel generator; the green line represents the AC source. The negative 

power output of the AC source represents power supplied to the source from the diesel 

generator. 

0.0 5.0 10.0 15.0 20.0 

-75 

-50 

-25 

0 

25 

50 

75 

100 

125 

150 

kW

REA L POWER GEN REA L POWER SRC

-40 

-30 

-20 

-10 

0 

10 

20 

kVAR

REA CTIV E POWER GEN REA CTIV E POWER SRC

0.9990 

0.9992 

0.9995 

0.9997 

1.0000 

1.0002 

1.0005 

1.0008 

p.u.

TERMINA L V OLTA GE

Figure 6. Output of diesel generator without mechanical dynamics 

Figure 7 displays the output when the machine transitions from a source to a machine at t = 

9.5 s and the rotor is unlocked at t = 10 s, enabling machine dynamics. In the following 

graphs, it is clear the machine is experiencing transient problems in the large power output 

sine wave. 

 7
0.0 5.0 10.0 15.0 20.0 

-100 

-50 

0 

50 

100 

150 

200 

250 

300 

kW

REAL POWER GEN REAL POWER SRC

-250 

-200 

-150 

-100 

-50 

0 

50 

100 

kVAR

REACTIVE POWER GEN REACTIVE POWER SRC

0.900 

0.920 

0.940 

0.960 

0.980 

1.000 

1.020 

1.040 

p.u.

TERMINAL VOLTAGE

Figure 7. Output of diesel generator with mechanical dynamics 

 8
5 Wind Turbine 

5.1 Mathematical Analysis 

The wind turbine also used a model in the PSCAD library. One way to control the active 

power of a wind turbine is to regulate the power coefficient, Cp, of the rotor turbine. Cp can be 

calculated by [4]: 

1

035.0

08.0

11

e54.0 116 22.0),(C

3

i

5.12

i

p

i

+β − β+λ = λ

⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −β− λ =λβ λ

−

(3) 

Where: 

λ = tip speed ratio 

β = pitch angle. 

The maximum power coefficient can be determined graphically. An example is given in 

Figure 8. With the maximum Cp value, it is possible to determine the optimum tip speed, 

where there is maximum power capture. With this, the optimum turbine speed curve is 

obtained and used as a reference in active power control. 

0

0,1

0,2

0,3

0,4

0,5

0,6

02468

TSR

Cp

10

Figure 8. Maximum value of [1] Cp

5.2 Simulation Results 

The 65-kW Northern Power wind turbine also used a model available in PSCAD. Figure 9 is 

the synchronous generator that closely resembles the actual wind turbine generator. Included 

in the schematic is the exciter, the generator, an infinite bus with an AC source, and a 50-μF 

capacitor for power factor correction. The input, Tm, is at the bottom of the synchronous 

machine. The input torque—which usually is driven by a turbine, engine, or in this case, a 

windmill—enters here. 

 9
TIME TIME S2M L2N

w

Ef

3 Phase

RMS

0.48

50.0

VT

IT 3

Ef If

Ef0

Vref

Exciter_(AC1A)

Testdy

0.001

Sync1

VT

IT

3

w

Te

A

B

C

EfEf0 If

Tm0Tm

Tm

Figure 9. Wind turbine generator 

Figure 10 is a schematic of the wind turbine and its components, including an input wind 

speed that may be adjusted to ramp, gust, or create wind noise and a blade angle pitch 

dynamics control. 

ES Vw

Vw Tm

Beta

W P

Wind Turbine

MOD 2 Type

A six Pole Machine

Mechanical speed = 

W(pu)*2*pi*f/(pole paris)

ES Vw

Wind Source

Mean

1.0

A

B

Ctrl

Ctrl = 1 w

CNT

*

376.99

N

D

N/D

14.0

Pole pairs

BETA

TIME

Wind Turbine

Governor

Wm

Beta

Pg

MOD 2 Type

P

A

B

Ctrl

Ctrl = 1

1.0

A

B

Ctrl

Ctrl = 1

CNT

0.58333

Signal CNT enables the 

pitch angle dynamics

w

*

376.99

Figure 10. Wind turbine and components 

 10
To make the simulation as close as possible to the real turbine, actual values were taken from 

the Northern Power wind turbine Operation and Maintenance Manual. The extracted values 

are provided in tables 1 and 2. 

Table 1. Performance Parameters 

Performance Parameters

Nominal power rating 100 kW 

Rated wind speed 14 m/s 

Cut-in wind speed 4 m/s 

Cut-out wind speed 25 m/s 

Survival wind speed 70 m/s 

Table 2. Rotor Parameters 

Rotor 

Rotor diameter 19.1 m 

Swept area 287 m2

Rotational speed 63.5 rpm 

Figure 11 depicts the output of the Northern Power wind turbine. For simple analysis, the 

wind speed was set at a constant 14 m/s, and pitch angle dynamics were disabled to allow 

study of the effects of a constant wind speed without change in wind direction. The 

fluctuation of real power and machine speed (ω) is due to the synchronous machine enabling 

mechanical dynamics and unlocking the rotor, which results in a large out flux of output 

power. After about 10 s, the machine settles to a constant speed of 377 rad/s (60Hz). 

 11
Untitled

0.0 5.0 10.0 15.0 20.0 25.0 

-15.0 

-10.0 

-5.0 

0.0 

5.0 

10.0 

15.0 

20.0 

25.0 

Power

P (kW)

-60 

-50 

-40 

-30 

-20 

-10 

0 

10 

20 

30 

40 

50 

Reactive Power

Q (MVar)

376.00 

376.50 

377.00 

377.50 

378.00 

378.50 

Machine Speed (elec)

w (rad/s)

Figure 11. Wind turbine output 

 12
6 Inverter 

6.1 Mathematical Analysis 

Figure 12 shows the main schematic of the inverter model. The main components of the 

system are a 480-V AC feed, a 489:575 star-delta transformer, a three-phase diode bridge 

rectifier, a three-phase PWM inverter modeled using controlled current sources, a three-phase 

balanced R-L-C load, and breakers for connecting the system to load and utility. The 

modeling of each of these components in PSCAD is explained in later sections. 

A d-q axis reference frame is used to control the inverter voltage and frequency by controlling 

the d-axis and q-axis currents, respectively. Initial results from the original simulation model 

are shown in the results section. Later in the results section, power measurements taken at 

various points in the schematic are presented. The p.u. system of units was used to display 

some of the results. Base value for voltage is 0.2771 kV (phase), and base current is 0.1208 

kA (phase). 

Figure 12. Schematic of the GE inverter system 

6.1.1 d-q Axis Theory 

Figure 13 shows a physical distribution of the three stationary axes as, bs, and c

s

 , which 

are120° apart from one another, and of the Cartesian axes, where qs

 is a horizontal axis 

aligned with phase as

 and a vertical axis rotated by –90° is indicated by ds. Three-phase 

voltages varying in time along the axes as, bs, and c

s can be algebraically transformed onto 

two-phase voltages varying in time along the axes ds and qs [10]. 

 13
Figure 13. Physical distribution of three stationary axes and a quadrature stationary axis 

The transformations are shown in Equation 4. 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−=

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−−=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

cs

bs

as

s

s0

s

ds

s

qs

s

s0

s

ds

s

qs

cs

bs

as

V

V

V

2

1

2

1

2

1

3

3

3

3 0

001

V

V

V

V

V

V

1

2

3

2

1

1

2

3

2

1

101

V

V

V

(4) 

 

Three methods commonly used to control a grid-connected inverter feeding power from a DG 

source are [12]: 

• Constant current control 

• Constant power control 

• Constant DC bus voltage control. 

This report presents the first two inverter control methods with anti-islanding schemes. The 

control block diagrams presented used the average model of the PWM inverter. The average 

model is usually preferred over the switching model for simulation purposes because the 

average model needs less computation time. In addition, small signal analysis can be 

performed on the average model. The average model of the inverter with controlled voltage 

and current sources is shown in Figure 14. 

 14
Figure 14. Average model of the inverter 

6.1.2 Anti-Islanding Schemes 

Various methods to detect islanding of a DG inverter connected to a grid are discussed in [5]. 

However, the most commonly used methods are passive inverter-resident methods, active 

inverter-resident methods, active methods not resident in the inverter, and the use of 

communications between the utility and the inverter. 

A report published by GE [6] showed that active inverter-resident methods have advantages 

over other methods. Active inverter-resident methods use a variety of schemes to cause an 

abnormal condition in the inverter voltage and/or frequency that can be detected to prevent 

islanding. The anti-islanding scheme proposed by [6] is based on d-q implementation of the 

positive feedback concept. The following equations explain the basics of the positive 

feedback concept with d-q axis modeling. 

Considering an R-L-C load, active and reactive power can be defined as: 

⎟

⎠

⎞ ⎜

⎝

⎛ = −= ωL

1

ωCVQand R

V P 2

2

 (5) 

 

The proposed islanding scheme creates abnormal voltage and frequency conditions (force 

voltage and frequency of the nominal ranges) to detect islanding. In the voltage feedback 

method, when the controller detects a change in the d-axis output voltage, it commands an 

increase in active power, which results in a further increase in d-axis voltage and eventually 

forces the voltage out of the nominal range. Thus, islanding can be detected. 

 15
The same analysis can be applied to frequency feedback methods. When the controller detects 

a change in frequency, it forces the reactive power and results in a further increase or decrease 

of inverter frequency in the same direction. These two islanding detection schemes can be 

implemented in the two control methods, as described in the next section. 

6.1.3 Constant Current Control 

In this method, the inverter output currents are controlled to the given reference currents. The 

error between the measured and reference currents is processed by PI controllers to generate 

commands for the PWM converter. 

Block diagrams of the control method are shown in figures 15 and 16. In the first figure, the 

voltage feedback loop is implemented. In Figure 16, the frequency feedback loop is 

implemented. These are added as feed-forward terms to the d-axis and q-axis reference 

currents, respectively. The measured voltage and frequency are passed through a band pass 

filter operating at a very narrow frequency band, a gain, and a limiter to become a current 

variation, Δi, that is added to Idref or Iqref. 

 16
Figure 15. Constant current control with voltage feedback 

Figure 16. Constant current control with frequency feedback 

 17
 18

6.1.4 Constant Power Control 

In this method, the active and reactive powers supplied by the inverter are controlled to the 

given reference values. The error between the measured and reference powers are processed 

to generate reference current commands. Feed-forward terms for islanding detection similarly 

can be added to the control scheme. 

Figure 17. Constant power control method 

6.2 Simulation Results 

The “abc to dqo” component in the PSCAD library performs the transformation of threephase stationary frame components to synchronously rotating two-phase reference frame 

components. The d-q transformation in PSCAD is modified so that q axis leads d axis by 90°, 

as shown in Figure 18. A vector control scheme with the d-axis reference frame oriented 

along the “Phase a” voltage vector is used. Direct axis current is used to control the 

magnitude of the inverter voltage, and quadrature axis current is used to control the 

frequency. Voltage angle is derived from the “PLL” block in the PSCAD library. This angle is 

modified so the waveform described by the cosine (angle) is in phase with “Phase a” voltage. 

Figure 19 shows the complete PSCAD model. 

Egridb

Eg

Egrida

ridc

theta_grid *

Pi by 180

Va

Vb

Vc

PLL theta

B

-

D +Shift

90.0

AngPLL

Figure 18. PLL and d-q axis transformation

 

ia id

ib iq

ic

theta

abc to dq EDGb

EDGc

EDGa

*

2.5516

*

2.5516

*

2.5516

AngPLL

EDGd(p.u.)

*

-1.0 EDGq(p.u.)
B C

Ioutc

IoutbIouta

Ic

Ia

EDGa

EDGb

EDGc

EgridaEgridb

Egridc

A

Ib

C

B

A

DG_BRK

0.1048 [MW]

0.04016 [MVAR]

display

Egridb

Egridc

Egrida

theta_grid

*

Pi by 180

VaVb

Vc

PLL theta

IcomC

IcomB

IcomA

id iaiq ib

theta ic

dq to abc

Vpn

IcomC

IcomB

IcomA

IoutaIoutb

Ioutc

*

*

dVpna

dlma

*

*

dVpnb

dlmb

*

*

dVpnc

dlmc

dlma dlmb dlmc 1.0e-005

VF Ph

0.0

0.0

0.0

1.0e-005

VF Ph

1.0e-005

VF Ph

90.0

90.0

90.0

dVpna

dVpnb

dVpnc

B

-

D +

Shift

90.0

ia idib iq

ic

theta

abc to dq Egridb

Egridc

Egrida Egrid_d(p.u.)

Egrid_q(p.u.)

ia idib iq

ic

theta

abc to dq EDGb

EDGc

EDGa EDGd(p.u.)

EDGq(p.u.)

*

2.5516

Id

Iq

Phase lock loopThe d-axis voltage is aligning with the phase a voltage (in lock) and the q-axis voltage is 90 degree leading the d-axis voltage.The waveform descriped by cos(angle) should be in phase with phase a voltage. 

*

-1.0

*

-1.0

*

-1.0

The relationship between the input signal and output voltageEDGa=sqrt(2)*dVpnaEDGb=sqrt(2)*dVpnbEDGc=sqrt(2)*dVpnc

Vbase=0.2771kV(Phase RMS) V=0.3919kV(Phase peak) 1/V=2.5516IBase=0.12028 kA (Phase RMS)I=0.1701kA (Phase peak)1/I=5.8788

*

2.5516

*

2.5516

*

2.5516

*

2.5516

*

2.5516

IdIq

ErrDErrQ

B

+

D

-

F

+

B

+

D

-

F

+

0.0

IqREF

A

B

Ctrl

Ctrl = 1

CTROL1

0 A

B

Ctrl

Ctrl = 1

CTROL1

0 CTROL1 TIME 1

Disable Compensator before 0.2 seconds

ErrD Enb_ErrD ErrQ Enb_ErrQ

ia idib iq

ic

theta

abc to dq

IaIb

Ic

Id

Iq

1.0

IdREF

*

-1.0

*

5.8788

*

5.8788

*

5.8788

AngPLL

AngPLL AngPLLAngPLL

AngPLL

Igrida

Igridb

Igridc

ABC

Util_BRK

1.327e-007 [MW]

2.087e-007 [MVAR]

Iloadb

Iloadc

Iloada

C

B

A

LOAD_BRK

0.1 [MW]

-0.0008449 [MVAR

grid impedance0.012 [ohm]0.0003056 [H] Timed Breaker

Logic Util_BRK

Timed

Breaker

Logic LOAD_BRK

dDdQ

omiga omiga

60.0

Normalized

fre fre

N

D

N/D

3 Phase

RMS Vt Vt

A

B

C

0.01 0.00250.01 0.00250.01 0.0025

dQdD

I

P

I

P

Enb_ErrDEnb_ErrQ D +

F

+

B

-

D +

*

Fre

Iq

*

0.40885

Id

*

*

Fre 0.40885

dDdQ

Re

+

Le

+ Re+ Re+

Le

+ Le+

0.012

Re

Egrid_d

Egrid_q

*

2 G sT 1 + sT G Egrid_d 1 + sT

Fre ai_frequency

a

*

80 G sT 1 + sT G 1 + sT

0.0

EnVoltaegAI

*

*

0.0

EnFrequencyAI

TIME

*

a

ai_voltage

ai_frequency

Util_BRK

Timed

Breaker

Logic

Open@t0

DG_BRK

DG_BRK

10000.0

D D DD D D

A

A

B

C

A

B

C 0.48

#2 #1

0.575

0.15 [MVA]

Vpn Vpn

B

C

1.1e-006

A

B

C

Vbase=0.2771kV(Phase RMS) V=0.3919kV(Phase peak) IBase=0.12028 kA (Phase RMS)I=0.1701kA (Phase peak)

scale factor

1/Sqrt(2)

Vpn

*

Vpn_corr

step-up transformerprimary: 0.48KVsecondary: 0.575KV three-phased rectifier

three-phased VSC averaged model

Vpn_corrVpn_corrVpn_corr

Figure 19. PSCAD model of the GE inverter 

 19
Inverter modeling included: 

• 480-V feed, transformer, and diode bridge rectifier 

Figure 20 shows the PSCAD model of a 480-V AC source with a transformer feeding 

the diode bridge rectifier. The transformer is a grounded star-delta rated at 150 kVA 

with a turn ratio of 480:575. Diode models in PSCAD are arranged to form a threephase diode bridge rectifier. A large (1 mF) capacitor is connected across the DC side 

of the rectifier. 

 

10000.0

D D D

D D D

A

A

B

C

A

B

C 0.48

#2 #1

0.575

0.15 [MVA]

B

C

1.1e-006

A

B

C

Figure 20. PSCAD model of AC feed and diode bridge rectifier 

• Inverter model 

A three-phase voltage source converter (VSC) averaged model was used to model a 

PWM converter, as shown in Figure 21. The commands for the PWM converter are 

generated by two PI control loops that control the voltage magnitude and frequency at 

the converter output. Ia, Ib, and Ic and Egrida, Egridb, and Egridc are transformed to 

d-q quantities for the PI control loops, as shown in Figure 22. IdREF is the reference 

value for the d-axis inverter current, and IqREF is the reference value for the q-axis 

inverter current. These are compared with the actual currents, and the error signals are 

passed into two PI controllers. Figure 18 shows that a feed-forward term is being 

added to the error signals. The two feed-forward terms, ai_voltage and ai_frequency, 

are generated from the actual inverter voltage and frequency. However, these signals 

are currently disabled in the model, and the simulation would not run if these signals 

were enabled. Compensation terms are added to the PI controller outputs for crosscoupling effect. The controllers are disabled for the first 0.2 s. 

 20
Ioutc

Ioutb

Iouta

EDGa

EDGb

EDGc Vpn

dlma dlmb dlmc

1.0e-005

V F Ph

0.0

0.0

0.0

1.0e-005

V F Ph

1.0e-005

V F Ph

90.0

90.0

90.0

dVpna

dVpnb

dVpnc

Ioutc

Ioutb

Iouta

Ic

Ia

Egrida

Egridb

Egridc

Ib

C

B

A

DG_BRK

0.1048 [MW]

0.04016 [MVAR]

3 Phase

RMS Vt Vt

A

B

C

0.01 0.0025

0.01 0.0025

0.01 0.0025

Figure 21. VSC averaged model 

A

B

Ctrl

Ctrl = 1

CTROL1

0 A

B

Ctrl

Ctrl = 1

CTROL1

0 CTROL1 TIME 1

Disable Compensator before 0.2 seconds

ErrD Enb_ErrD ErrQ Enb_ErrQ

IcomC

IcomB

IcomA

id ia

iq ib

theta ic

dq to abc *

-1.0

AngPLL

dD

dQ

dQ

dD

I

P

I

P

Enb_ErrD

Enb_ErrQ D +

F

+

B

-

D +

*

Fre

Iq

*

0.40885

Id

*

*

Fre 0.40885

dD

dQ

Figure 22. PI controllers 

 21
Figure 23 shows how the compensation terms ai_voltage and ai_frequency are 

generated. The terms are supposedly being used to provide better tracking of inverter 

voltage magnitude and frequency. However, as these signals are multiplied by zero, 

the voltage and frequency are not tracked directly. The results section shows that the 

PI controller does not keep the inverter voltage and frequency constant when the grid 

or load is disconnected from the inverter system. 

*

2 G sT

1 + sT

G

Egrid_d 1 + sT

Fre ai_frequency

ai_voltage

 K=2

 Tw=1/(2*pi*1)=0.159

 T0=1/(2*pi*10)=0.0159

*

80 G sT

1 + sT

G

1 + sT

island

K=80

Tw=1/(2*pi*1)

T0=1/(2*pi*10)

0.0

EnVoltaegAI

*

* 0.0

EnFrequencyAI

TIME * ai_voltage

ai_frequency

Figure 23. Compensation term for inverter voltage and frequency control 

The d-q outputs from the PI controllers are transformer back to a-b-c to generate the 

commands for the PWM converter. The inverter commands dlma, dlmb, and dlmc and 

dVpna, dVpnb, and dVpnc shown in Figure 24 are obtained as follows. 

IcomC

IcomB

IcomA

id ia

iq ib

theta ic

dq to abc *

-1.0

AngPLL

dD

dQ

scale factor

1/Sqrt(2)

Vpn * Vpn_corr

IcomA

Iouta

*

*

dVpna

dlma

Vpn_corr

IcomB

Ioutb

*

*

dVpnb

dlmb

Vpn_corr

IcomB

Ioutb

*

*

dVpnb

dlmb

Vpn_corr

Figure 24. Commands for the VSC averaged inverter 

 22
• Grid and load connections 

The three-phase inverter outputs are connected to the grid and a variable R-L-C load 

through two breakers, as shown in Figure 25. These breakers are operated using the 

“timed breaker logic” block in PSCAD. The behavior of the inverter simulation model 

when these breakers are described is shown in the results section. 

A B C Igrida Igridb Igridc

AB

C

Util_BRK

1.327e-007 [MW]

2.087e-007 [MVAR]

Iloadb

Iloadc

Iloada

C

B

A

LOAD_BRK

0.1 [MW]

-0.0008449 [MVAR]

grid impedance

0.012 [ohm]

0.0003056 [H]

3 Phase

RMS

A

B

C

R

+

R

+

R

+

L

+

L

+

L

+

Cap

+

Cap

+

Cap

+

2.3

R

0.003395

L

2072.0

C

R

L

Cap

Re

+

Le

+ Re

+ Re+

Le

+ Le+

0.012

Re

0.0003056

L Re Le * 1.0 factor 1.0 factor

*

Vt Vt

Timed

Breaker

Logic Util_BRK

Timed

Breaker

Logic LOAD_BRK

Util_BRK

Timed

Breaker

Logic

Open@t0

DG_BRK

DG_BRK

Figure 25. Grid, load models, and breakers 

 23
The three breakers shown in Figure 25 are operated at the following times: 

o DG_BRK is closed at 0.2 s to connect the inverter to the grid and load. 

o Util_BRK is opened at 4 s to disconnect the inverter from the grid. 

o LOAD_BRK is opened is 10 s to disconnect the R-L-C load from the inverter. 

Figure 26 and Figure 27 show the inverter RMS output voltage and frequency during 

the breaker operations. Inverter voltage is expressed in the p.u. system with a base 

value of 0.2771 kV (phase). Figure 27 shows that the inverter frequency gradually 

increases when the system is disconnected from the grid. Voltage remains pretty 

constant during this period. However, when load is disconnected from the system at 10 

s, both frequency and voltage change rapidly. This is probably due to the averaged 

model of the inverter, which has three controlled current sources in the three inverter 

legs. 

DG : Graphs

 T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.00 

0.25 

0.50 

0.75 

1.00 

1.25 

1.50 

1.75 

2.00 

2.25 

Inverter RMS voltage (P.U.)

Vt

Figure 26. RMS inverter voltage 

 24
DG : Graphs

 T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

58.00 

58.50 

59.00 

59.50 

60.00 

60.50 

61.00 

Frequency (Hz)

omiga

Figure 27. Inverter frequency 

Figures 28 and 29 show the d-axis and q-axis inverter currents, respectively. The 

reference value for d-axis current is 1.0 (P.U), and the reference value for q-axis 

current is 0.0 (p.u.). The PI controllers do a good job tracking these reference values 

until the instant when load is disconnected from the system. The magnitude of the DC 

link voltage is shown in Figure 30. 

DG : Graphs

 T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.20 

0.00 

0.20 

0.40 

0.60 

0.80 

1.00 

1.20 

d - axis current (P.U)

Id

Figure 28. d-axis inverter current (voltage) 

 25
DG : Graphs

T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.100 

-0.050 

0.000 

0.050 

0.100 

0.150 

0.200 

0.250 

0.300 

0.350 

q - axis current (P.U.)

Iq

Figure 29. q-axis inverter current (frequency) 

DG : Graphs

T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.00 

0.20 

0.40 

0.60 

0.80 

1.00 

1.20 

DC bus voltage (kV)

Vpn

Figure 30. DC link voltage (rectifier output) 

Active and reactive power measurement points in the schematic are shown in Figure 

31. Figure 32 and Figure 33 show the active and reactive power injected by the 480-V 

AC feed into the system. It delivers about 105 kW of active power and 32 kVAR of 

reactive power when the R-L-C (23 Ω–3.395 mH–2072 μF) load is connected at the 

inverter output. Figure 34 and Figure 35 show the active and reactive power delivered 

by the inverter. The magnitudes are slightly lower than the power delivered by the AC 

feed, which can be attributed to the switching losses in the diode bridge rectifier and 

the PWM converter. (The AC sources used in the average model have a small series 

resistance). Figure 36 and Figure 37 show the active and reactive powers injected into 

the grid. Both of these are close to zero, which means that no real amount of power is 

being injected into the grid. Active and reactive power absorbed by the R-L-C load are 

shown in Figure 38 and Figure 39. The magnitudes of these powers are very close to 

the power measured at the inverter output. 

 26
Figure 31. Power measurement points 

DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.000 

0.050 

0.100 

0.150 

0.200 

0.250 

Active power, Feed (MW)

Pf

Figure 32. Active power injected by the AC feed 

 27
DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.025 

0.000 

0.025 

0.050 

0.075 

0.100 

0.125 

0.150 

0.175 

0.200 

0.225 

Reactive power, Feed (MVAR)

Qf

Figure 33. Reactive power injected by the AC feed 

DG : Graphs

 T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.020 

0.000 

0.020 

0.040 

0.060 

0.080 

0.100 

Active power, Inverter (MW)

Pin

Figure 34. Active power delivered by the inverter 

 28
DG : Graphs

 T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.0050 

0.0000 

0.0050 

0.0100 

0.0150 

0.0200 

0.0250 

0.0300 

0.0350 

0.0400 

0.0450 

Reactive power, Inverter (MVAR)

Qin

Figure 35. Reactive power delivered by the inverter 

DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.020 

0.000 

0.020 

0.040 

0.060 

0.080 

0.100 

Active power, Grid (MW)

Pg

Figure 36. Active power injected into the grid 

 29
DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.060 

-0.050 

-0.040 

-0.030 

-0.020 

-0.010 

0.000 

0.010 

0.020 

Reactive power, Grid (MVAR)

Qg

Figure 37. Reactive power injected into the grid 

DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.000 

0.020 

0.040 

0.060 

0.080 

0.100 

Active power, Load (MW)

PL

Figure 38. Active power absorbed by the load 

 30
DG : Graphs

T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.060 

-0.050 

-0.040 

-0.030 

-0.020 

-0.010 

0.000 

0.010 

0.020 

Reactive power, Load (MVAR)

QL

Figure 39. Reactive power absorbed by the load 

 31
7 Photovoltaic Cell 

7.1 Mathematical Analysis 

The simulation model of the PV cell is based on the circuit-based model developed in [2]. The 

schematic of the circuit-based model is shown in Figure 40. The schematic is based on the 

double exponential equation, which models a PV cell, derived from the physics of the p-n 

junction. 

( 1 s e V IR

AkT s

ph s

p

V IR I I Ie

R

+ ⎛ ⎞ − + =− − ⎜ ⎟ ⎝ ⎠

(6)

D1

Rp

+

0.137

Rs Rs

D2

Rs

+

10.0

BRK

Timed

Breaker

Logic

Open@t0 BRK

150.43

Iph

528.45

Rp Rp

IL

IL

Vo

Vo

Figure 40. PV cell equivalent circuit 

The parameters in the equivalent circuit are derived from the environmental parameters of 

temperature and irradiance. They are related as shown below. 

( )

12

0 1

9

8 10

( )

11

1 ph

s

K T

p

I K E KT

K R K K

E

R Ke

= +

= +

=

T

(7)

Where: 

E is the irradiance in W/m2

T is temperature in Kelvin 

K0-12 are constants specific to a given cell. 

 32
These constants are derived from the I-V curve of the PV cell. 

7.2 Simulation Results 

The following are the simulation results, with the parameters shown in Figure 1. The 

temperature used was 25° C. All other parameters were taken from [2]. Figure 41 and Figure 

42 show the response of the PV cell to load changes. 

Time ... 0.0 1.0 2.0 3.0 4.0 5.0 

0.010 

0.020 

0.030 

0.040 

0.050 

0.060 

0.070 

0.080 

0.090 

Current (kA)

Current

Figure 41. PV cell current 

 Time ... 0.0 1.0 2.0 3.0 4.0 5.0 

0.7380 

0.7400 

0.7420 

0.7440 

0.7460 

0.7480 

0.7500 

0.7520 

Voltage (KV)

PV Voltage

Figure 42. PV cell voltage 

 33
8 Microturbine 

The microturbine model is connected to the grid through an inverter and a DC link capacitor. 

A PI controller is used to set the inverter output voltage to match the desired power profile 

from the microturbine. 

8.1 Mathematical Analysis 

The microturbine model was based on the system of Figure 42 from [1]. 

PI

-

+ KT

1 + st t

throttle/torque

transfer function

Te

idc

+

Bw

1

sJ

KE

+

w Eg Vdc

-

KEw

Ki Ki E dc X dc - 2

Vdc

Pout

-

-

w

power

estimation

map

P

^

w*

P

^

w*

Figure 43. Block diagram of a microturbine 

This model assumes that the torque response is a first-order system. The model was modified 

to accept a constant input shaft speed. Also, the Ke*ω*idc feedback was eliminated because it 

does not give a voltage value. In this model, the desired power output and shaft speed are set. 

The system then stabilizes, and the output power follows the desired power. The parameters 

Ke and Kx are easily obtained through experimentation. Ke is the ratio between no-load 

voltage and no-load shaft speed. Kx can be determined using Ke and one other operation point. 

(8)

1DC

1

e

1

1DC

x

loadno

loadDCno e

V

K V

K

V K

ω−

− ω =

ω =

−

−

From these parameters, voltage, power and torque can be found. These equations are used in 

the model of the microturbine. 

(9)

2

dcxdce

2

edc xdc dc

edc x dc

IKIK

IKIKP

IKKV

⋅−⋅=τ

⋅ω⋅−⋅ω⋅=

⋅ω⋅−ω⋅=

 34
The other parameters, such as the shaft inertia, J; the throttle gain; and the time constant can 

be obtained through calculation or experimentation. This simulation used constants from a 

Honeywell microturbine from a University of Wisconsin paper [2]. The values are shown 

below. 

Table 3. Microturbine parameters 

Ke 0.092 

Kx 8.594e-5 

J 0.085 

t 10s 

The DC link capacitor current is the difference between the microturbine current (I1) and the 

output current (I2). I2 is the ratio between the output power and the DC link voltage. The 

voltage in the capacitor can then be calculated with the capacitor differential equation. 

)II( C

1

t

V

V

P I

21

DCLink

DCLink

out

2

−= ∂

∂

= (10)

 The modulation factor that controls the inverter is given by: 

∫ p i ⋅+⋅= errKerrKm (11)

The inverter converts the DC voltage from the microturbine into AC voltage to be connected 

to the AC grid. The output AC voltages are calculated using the equations: 

)240t602sin( 2

V mV

)120t602sin( 2

V mV

)t602sin( 2

V mV

dc c

dc

b

dc a

= °−π

= °−π

= π (12)

The phase currents are calculated using the electrical equation for voltage across a resistor 

inductor load. 

 35
( ) 1

aga a

a iRVV

t L

i ⋅−−= ∂∂ (13)

Finally, the three-phase output power is calculated by: 

ccbbaaout ⋅+⋅+⋅= iViViVP (14)

8.2 Simulation Results 

The microturbine model is transformed from a dynamic model into an electrical model using 

a voltage-controlled voltage source. 

Figure 44 shows the entire system. 

DClink

I1

Pm

Vd

C

Vdc_lin

5000

desired shaft speed

1

D -

F

+

C

B

A

BRK_m A

B

C

V

0.1

N

D

N/D1000.0

Pm

BRK_m

Timed

Breaker

Logic

Open@t0

*

*

0.5

V

N

D

N/D127.0

N

D

N/D Pm 1.0e-006

N

D

N/D Pm 1.0e-006

PI In1 Out1

Microturbine

w

Pin

Pr

V_dc

I_dc

P_out

Power Profilepower

Am

Bm

Cm

3 Phase

RMS Vt_m Vt_m

A

B

C

A

B

C

A

B

C 0.264

#2 #1

0.99

0.15 [MVA]

Figure 44. Total system 

8.2.1 Microturbine 

Figure 45 shows the microturbine. The model is based on equations instead of circuit 

elements. The output voltage is calculated with Equation 9. 

 36
D +

F

-

G

1 + sT w

Pin

I_dc

V_dc

P_out

*

B

-

D +

F

-

*

0.1

1

sT

D -

F

+ X2 const 0.092 *0.008459 const

*

0.096 D +

F

+

1.0

const

1.0

reciprocal

N

D

N/D *

*

Pr

Figure 45. Microturbine 

Figure 46, Figure 47, and Figure 48 show the output voltage, current, and power, respectively. 

Figure 46. Microturbine output voltage 

 37
Figure 47. Microturbine output current 

Figure 48. Microturbine output power 

8.2.2 DC Link 

The DC link capacitor connects the microturbine to the inverter. The voltage level is used to 

drive the inverter and the error in the capacitor voltage, and the microturbine voltage is used 

as the input to the PI controller. 

The output current and voltage are calculated with Equation 10. Because I2 is in terms of the 

DC link voltage, a feedback has to be created. It used a capacitance of 1 F. 

Figure 49 shows the capacitor block, and Figure 50 shows the capacitor voltage block. 

 38
Vdc_link

*

Vd

Pm

N

D

1 N/D

C

I1

I1

I2

C

Vdc_lin

Figure 49. Capacitor block 

1

D sT +

F

- *

I2

C

Vdc_link

N

D

1 N/D

I1

Figure 50. Capacitor voltage block 

Figure 51 shows the DC link voltage. It has a constant value of 435 V. 

Figure 51. DC link voltage 

 39
8.2.3 PI Controller 

The PI controller (Figure 52) helps the output power follow the desired power profile. It uses 

an error from the difference in DC link voltage and the microturbine voltage to create a 

modulation factor, “m” (Figure 53), that will then drive the inverter. The m value is put 

through a saturation limit between 0 and 1, but under normal operations, it should not get to 

these boundary conditions. They are in place to keep the inverter model from creating 

voltages it physically could not create. 

The proportional and integral gains Ki and Kp were determined through a trial-and-error 

method. 

D +

F

+

1

sT

*

0.02

*

0.0002

in

Out1

Figure 52. PI control block 

Figure 53. Modulation factor 

8.2.4 Grid Connection 

Figure 54 shows the system connected to the grid through a voltage-controlled voltage source. 

The mVdc/2 controls the voltage source. The grid is considered to be 60 Hz, and the 120o

offset produces a balanced three-phase system. The breaker measures the output power. 

 40
A

B

C

C

B

A

BRK A

B

C

V

N

D

N/D1000.0

50.0 [ohm]50.0 [ohm]50.0 [ohm]

V

Ia

Ea

Figure 54. Grid connection 

Figure 55 shows the power profile fed into the microturbine to drive the system. This profile 

can be changed. 

Figure 56 shows that the output power follows the reference. (Note that the output power 

graph is in megawatts but the power reference graph is in watts.) The PI controls the 

transients of the systems. Changing the proportional and integral gains will change how fast 

the transients occur and how much the output oscillates. This case uses both gains at 0.002. 

Figure 55. Power profile 

 41
Figure 56. Output power 

 42
9 Backbone Integration 

After the proper functioning of each component was verified, the components were integrated 

into the backbone. 

Figure 57 shows the PSCAD model of the system that has been simulated. The model 

includes a 62.5-kVA AC grid simulator; a 250-kVA, Y-connected 216.5-V/480-V 

transformer; variable R-L-C load; a 125-kW diesel generator (represented as a synchronous 

generator); a 60-kW wind turbine; a PV cell; a 33-kW microturbine; and three 3.5-kW PWM 

inverters. Each PWM inverter is controlled using the control approaches discussed in earlier 

sections. Also included in the figure are circuit breakers, at which current, voltage, and power 

are measured. Other devices, such as a fuel cell, may be added to the grid. 

Graphs

GridPout

GridRPout

theta

GridcurrentA

GridcurrentB

GridcurrentC

.150 [MW.173 ] [MVAR .173 ][MVAR]

AC Device 1 - village load simulator

(Resistive 100 kW, Inductive 173 kVAR and Capacitive 173 kVAR)

LoadBreak

loadpower

loadRpower

loadcurrentA

loadcurrentB

loadcurrentC

3 Phase

RMS

Timed

Breaker

Logic

Open@t0 LoadBreak

BRKR

BRKL

BRKC

Timed

Breaker

Logic

Open@t0 BRKR

Timed

Breaker

Logic

Closed@t0 BRKL

Timed

Breaker

Logic

Closed@t0 BRKC

Ea

Ea

SourceRMS

A

B

C

Diesel Gen BRK_Diesel

BRK_Source

BRK_Source

BRK_Diesel

Timed

Breaker

Logic Open@t0

Timed

Breaker

Logic Closed@t0

BRK_Inverter

Timed

Breaker

Logic Open@t0

INV_CC_F

A

B

C

INV_CC_V

A

B

C

BRK_Inverter_V

BRK_Inverter_V

Timed

Breaker

Logic

Open@t0

BRK_Inverter

INV_CP

A

B

C

BRK_Inverter_V

.250 [MVA]

.480 [kV] / .277 [kV]

RRL

P1

Q1

P3

Q3

Inv_Power

Pd

Qd

Other_Power

Pgrid

Qgrid

BRK_Inverter_m

PV_Inverter

A

B

C

BRK_PV

BRK_PV

Timed

Breaker

Logic

Open@t0

PV_Plots

BRK_Inverter_m

Timed

Breaker

Logic

Open@t0

Microturbine

Am

Bm

Cm

Figure 57. Microgrid backbone – PSCAD 

 43
Figure 58 shows the three-phase bus voltage. 

Figure 58. Bus voltage 

Figure 59 shows the grid simulator active and reactive powers, current, and voltage. 

Figure 59. Grid simulator 

 44
Figure 60 shows the load simulator graphs. 

Figure 60. Load simulator 

9.1 Diesel Generator 

Figure 61 and Figure 62 show the outputs of the diesel generator when connected to the 

backbone. 

 45
Figure 61. Active power – diesel generator 

Figure 62. Reactive power – diesel generator 

9.2 PV Cell Grid Connection 

To integrate the PV cell model into the backbone, the GE inverter model is connected at the 

output of the cell. The PV model is used as the DC source for the current-controlled inverter. 

The following figures show the simulation results with the inverter model. Anti-islanding 

schemes were not implemented. The grid connection was removed at 4 s and connected back 

at 7 s. Figure 63 shows the PV cell output voltage. The magnitude of the voltage is 

approximately 680 V. Figure 64 and Figure 65 show the frequency and voltage waveforms at 

the inverter output. As expected, frequency and voltage vary when the grid is disconnected 

from the inverter. Figure 66 shows the power injected from the inverter into the grid and the 

variable R-L-C load. 

 46
T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.00 

0.10 

0.20 

0.30 

0.40 

0.50 

0.60 

0.70 

0.80 

0.90 

DC bus voltage (kV)

Vpn

Figure 63. PV output voltage 

 T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

56.0 

57.0 

58.0 

59.0 

60.0 

61.0 

62.0 

63.0 

64.0 

65.0 

Frequency (Hz)

omiga

Figure 64. Inverter output frequency 

 T(s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

0.00 

0.20 

0.40 

0.60 

0.80 

1.00 

1.20 

1.40 

1.60 

1.80 

Inverter RMS voltage (P.U.)

Vt

Figure 65. Inverter output voltage 

 47
 T (s) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 

-0.100 

-0.050 

0.000 

0.050 

0.100 

0.150 

0.200 

0.250 

0.300 

0.350 

0.400 

Active power, Inverter (MW)

Pin

Figure 66. Power injected from the inverter 

9.3 Microturbine Grid Connection 

The microturbine model is integrated into the backbone through an inverter. Figure 67 shows 

the integration with the backbone. The transformer steps up the voltage to the bus voltage 480 

V. When connected to the backbone, the microturbine responds very slowly. 

C

B

A

A BRK_m

B

C

V

0.1

N

D

N/D1000.0

BRK_m

Timed

Breaker

Logic

Open@t0

V

Am

Bm

Cm

3 Phase

RMS Vt_m Vt_m

A

B

C

A

B

C

A

B

C 0.264

#2 #1

0.99

0.15 [MVA]

Figure 67. Microturbine grid connection 

 48
10 Conclusion and Future Work 

DR have an important role in the alleviation of transmission bottlenecks because they are 

designed to be located at or near the point of end-use and thus avoid the need to transmit 

electricity through congested wires. Another role of DR is outage response and mitigation. By 

placement at strategic locations in a networked configuration, DR, operating in “intentional 

islanding” mode, can provide power to protect critical loads during power failures. 

In this project, valid models for DR were created to simulate NREL’s DERTF. All of the 

simulations were run in PSCAD. Models for PWM inverters, microturbines, and PV cells 

were developed. The diesel generator and wind turbine used models available in the PSCAD 

library. 

Two inverter control methods with two anti-islanding detection schemes were presented. 

Simulation results showed that the constant current control method works well to track 

desired d-q axis currents and detect islanding. The inverter simulation with constant power 

controller is quite stable, but there are problems controlling the reactive power. 

 

Procedures to develop models for a microturbine and a PV suitable for use in electrical 

engineering were developed. Both models work as expected. The microturbine output power 

follows the reference power as expected. The PV output voltage is constant at 680 V. 

The diesel generator was modeled as a synchronous machine and, therefore, used a model 

available in PSCAD. The desired output and the desired terminal voltage were achieved. The 

power is negative because the synchronous machine operates as a generator. 

The wind turbine also used a model available in PSCAD. It used an input wind speed and a 

blade angle pitch dynamics control. 

After the models were developed and tested individually, they were connected together to the 

backbone. Except for the microturbine, all the components worked properly and injected 

power into the grid. The microturbine worked well by itself, but when connected to the 

backbone, it became slow and took too long to stabilize. 

This project developed electrical models for simulation and analysis of different DR. It can be 

used to understand the effects of DR on the electric power system. A fuel cell model and the 

validation of the system have not yet been implemented. 

 49
11 References 

[1] Simoes, M; Farret, F. Integration of Alternative Sources of Energy. Wiley Press, 

January 2006. 

[2] Gow, J.A.; Manning, C.D. “Development of a Photovoltaic Array Model for Use in 

Power-Electronics Simulation Studies.” Electric Power Applications, IEEE 

Proceedings, Vol.146, Issue 2, March 1999, pp. 193–200. 

[3] PSCAD help file. 

[4] Perdana, A.; Carlson, O.; Persson, J. “Dynamic Response of Grid-Connected Wind 

Turbine With Doubly Fed Induction Generator During Disturbances.” Available at 

http://www.elkraft.ntnu.no/norpie/10956873/Final%20Papers/054%20-

%20Abram_DynamicResponseDFIG.PDF. 

[5] Verhoeven, B. “Probability of Islanding in Utility Networks Due to Grid-Connected 

Photovoltaic Power Systems.” IEA PVPS T5-07: 2002. 

[6] Ye, Z.; Walling, R.; Garces, L.; Zhou, R.; Li, L.; Wang, T. Study and Development of 

Anti-Islanding Control for Grid-Connected Inverters. NREL/SR-560-36243. Work 

performed by the General Electric Global Research Center, Niskayuna, NY. Golden, 

CO: National Renewable Energy Laboratory, May 2004. 

[7] Venkataramanan, G.; Illindala, M.S.; Houle, C.; Lasseter, R.H. Hardware Development 

of a Laboratory-Scale Microgrid Phase 1 – Single Inverter in Island Mode Operation. 

NREL/SR-560-32527. Work performed by the Wisconsin Power Electronics Research 

Center, Madison, WI. Golden, CO: National Renewable Energy Laboratory, November 

2002, pp. 31–66. 

[8] IEEE P1547.4, IEEE Guide for Design, Operation, and Integration of Distributed 

Resource Island Systems with Electric Power Systems, Draft 1. August 2005. 

[9] Davis, M.; Costyk, D. Guide for Design, Operation, and Integration of Distributed 

Resource Island Systems with Electric Power Systems, Draft 1. August 2005 

[10] Chakraborty, S. “D-Q and P-Q Theory: Axis Transformation, Related Mathematics, and 

Applications.” Not published. 

[11] Chapman, S. Electric Machinery Fundamentals. 4th edition. McGraw-Hill, 2005; pp. 

267–342. 

[12] Mohan, N.; Undeland, T.; Robbins, W. Power Electronics. 3rd edition. New York: 

Wiley, 2003, pp. 16–30. 

[13] Flannery, P.; Venkataramanan, G.; Shi, B. Integration of Distributed Technologies – 

Standard Power Electronic Interfaces. P500-2005-119. California Energy Commission 

consultant report, April 2004. 

[14] Simoes, M.; Farret, F. Renewable Energy Systems: Design and Analysis with Induction 

Generators. CRC Press, May 2004. 

 50
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1. REPORT DATE (DD-MM-YYYY)

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5a. CONTRACT NUMBER

DE-AC36-99-GO10337 

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4. TITLE AND SUBTITLE

Electrical Model Development and Validation for Distributed 

Resources 

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

NREL/SR-581-41109 

5e. TASK NUMBER

DPE6.1000 

6. AUTHOR(S)

M.G. Simoes, B. Palle, S. Chakraborty, C. Uriarte 

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Colorado School of Mines 

1500 Illinois St. 

Golden, CO 80401 

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14. ABSTRACT (Maximum 200 Words)

This project focuses on the development of electrical models for small (1-MW) distributed resources at the National 

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15. SUBJECT TERMS

distributed energy; distributed resources; DE; DR; models; modeling; National Renewable Energy Laboratory; NREL 

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