Power Flow Control of a Single DistributedGeneration Unit with Nonlinear Local Load

Min Dai, Mohammad N. Marwali, Jin-Woo Jung, and Ali KeyhaniDepartment of Electrical and Computer Engineering

The Ohio State UniversityColumbus, OH 43210

Email: keyhani@ece.osu.edu

Abstract— Distributed generation units with small energysources, such as fuel cells, micro-turbines, and photovoltaicdevices, can be connected to utility grid as alternative energysources besides providing power to their local loads. The dis-tributed generation units are interfaced with utility grid usingthree phase inverters. With inverter control, both active andreactive power pumped into the utility grid from the distributedgeneration units can be controlled. Reactive power flow controlallows the distributed generation units to be used as staticvar compensation units besides energy sources. This paperpresents a distributed generation unit control technique whichprovides robust voltage regulation with harmonic eliminationunder island running mode and decoupled active and reactivepower flow control under grid-connected mode. The controltechnique, which combines discrete-time sliding mode currentcontrol, robust servomechanism voltage control, and integralpower control, enables seamless switching between island modeand grid-connected mode and guarantees sinusoidal line currentwaveform under nonlinear local load. The P Q coupling issue isaddressed and the stability of the power control loop is provedusing Lyapunov direct method.

I. INTRODUCTION

Distributed generation (DG) units interfaced with utility gridwith static inverters are being applied and focused increasinglydue to the fact that conventional electric power systems arebeing more and more stressed by expanding power demand,limit of power delivery capability, complications in buildingnew transmission lines, and blackouts. Power quality, safetyand environmental concerns and commercial incentives aremaking alternative energy sources, e.g., fuel cells, photovoltaicdevices, wind power, and gas-fired micro-turbines, more desir-able. Especially, developments in power electronics and digitalcontrol technology are providing more possibility and betterflexibility of using these new sources in conventional electricpower systems, to not only increase electric energy productionbut also help to enhance the power system stability via powerflow control.

Previous research results have shown that DG could havesignificant impacts on transmission system stability at heavypenetration levels [1], where penetration is defined as thepercentage of DG power in total load power in the system.A DG unit affects the system stability by generating orconsuming active and reactive power. Therefore, power controlperformance of the DG unit determines its impact on theutility grid it connects to. If the power control performance

is good, the DG unit can be used as means to enhance thesystem stability and improve power quality; otherwise it couldundermine the system stability.

Control issues of grid-connected inverter systems with lo-cal load have been addressed by a number of researches.Chandorkar et al. [2] and Kwon et al. [3] have studiedline interactive inverters to be used as uninterruptible powersupplies (UPS), where the inverters do not power the localload and the utility mains simultaneously, and therefore donot contribute to transmission system stability. Some pastresearches on line interactive inverters [4]–[9] only concentrateon current waveform control where power control is notconsidered. Although current waveform control is one of thegoals of the inverter control, power control performance mustbe addressed before it can be practically connected to anypower system.

Liang et al. [10] have presented a power control methodfor a grid-connected voltage source inverter which achievesgood P Q decoupling and fast power response. However, thisapproach requires knowledge of the value of power systemequivalent impedance, which is impractical. Even though thepossible error in the power factor angle of the impedancehas been considered, it is the magnitude of the impedancethat truly causes the sensitivity of P and Q responses, whichis however not addressed. Therefore, the effectiveness of thecontrol approach claimed in [10] is undermined.

Illindala et al. [11] have presented a different power controlstrategy based on frequency and voltage droop characteristicsof power transmission, which allows decoupling of P and Q atsteady state. However, the presented simulation result does notshow a satisfactory performance in response time and controlmagnitude which could be caused by low feedback gain andimplies that higher gain may cause problems.

In this paper, a new solution to the power flow controlproblem of a grid-connected single DG unit will be proposed.The proposed approach combines voltage regulation plusharmonic minimization under island mode and decoupled Pand Q control under grid-connected mode with a nonlinearlocal load. The proposed control does not require knowledgeof the equivalent impedance of the utility grid the DG unit isconnected to and yields seamless switching transients betweenisland mode and grid-connected mode, strong P and Q regu-lation capability, fast enough response, and purely sinusoidal

line current. The P Q coupling issue will be addressed and aproof of the power loop stability based on Lyapunov’s methodwill be presented.

II. THE CONTROL SYSTEM

The proposed control solution is developed for a grid-connected DG unit shown in Fig. 1. The DG unit consistsof a DC bus powered by any DC source or AC source with arectifier, a voltage source inverter, an LC filter stage, a ∆/Ygtype isolation transformer with secondary side filtering. TheDG unit has a local load, linear or crest, and is connectedto the utility grid through a three-phase triac. The utility gridis modelled as an equivalent three-phase AC source with anequivalent internal impedance.

Under island mode, the inverter conducts voltage control,where the load voltage VoutABC should track the given ref-erence. The voltage control goal is strong voltage regulation,low static error in RMS, fast transient response, and low totalharmonic distortion (THD). If the voltage of the utility mainEABC is measured and used as the reference, VoutABC willbe controlled to match EABC in magnitude and synchronizedin phase angle.

Under grid-connected mode, the DG unit conducts powercontrol, where the output active power P and reactive powerQ from the DG unit to the utility grid should be regulatedto desired values Pref and Qref . Both Pref and Qref can bepositive or negative, which provides possibility for the DG unitto help with the energy production and stability enhancementof the power system or sustain power supply to local loadwhen it exceeds the capacity of the DG. The control goal ofpower regulation is stability, low static error, and fast response.

In the switching transient from island mode to grid-connected mode, even though VoutABC matches EABC beforethe triac is turned on, transient current will still flow due to thechange of the circuit topology. The duration of the transientis determined by the power control performance. Due to thematching between VoutABC and EABC and the low equivalentimpedance Z, the VoutABC transient is mostly negligible andregarded seamless.

A. Voltage and Current Control

For high quality of VoutABC with strong regulation, lowTHD, and overload protection, a dual-loop voltage and currentcontrol structure is used as shown in Fig. 2, where the innerloop is for current control and outer loop is for voltage control.A robust servomechanism controller (RSC) is used for voltagecontrol and a discrete-time sliding mode controller (DSMC)is used for current control. The three-phase quantities aretransformed from ABC reference frame into a stationary αβreference frame. The details of this approach have been givenin [12], [13]. Some key points are summarized below forreaders convenience.

The DSMC is used in the current loop to limit the invertercurrent under overload condition because of the fast and no-overshoot response it provides. The RSC is adopted for voltagecontrol due to its capability to perform zero steady state

tracking error under unknown load and eliminate harmonicsof any specified frequencies with guaranteed system stability.The theory behind the RSC is based on the solution ofrobust servomechanism problem (RSP) [14] where the internalmodel principle [15] and the optimal control theory for linearsystems are combined. The internal model principle is appliedin the DG voltages control by including the fundamentalfrequency mode and the frequency modes of the harmonics tobe eliminated into the controller. The optimal control theoryof linear systems is combined in the RSC in order to guaranteethe stability of the closed loop system and providing arbitrarygood transient response based on desired control priorities.

In a DG unit shown in Fig. 1, most of the voltage harmonics,like the triplen (3rd, 9th, 15th, . . . ) and even harmonics, areeither non-existing or uncontrollable, or negligible in values.Therefore, only the fundamental and the 5th and 7th harmonicsare left for the control to handle. Since the overload protectionis a strongly desired feature for inverter systems, a DSMC isincluded in the inner loop to limit the current under overloadconditions. With the existence of the DSMC in the inner loop,the RSC in the outer loop are designed taking the dynamicsof the DSMC into account, so that the stability and robustnessof the overall control system is guaranteed.

1) The discrete-time sliding mode current control: The cir-cuit shown in Fig. 1 is a linear system and can be modelled instate space. The current controller controls the inverter current,i.e., the inductor current. The LC filter can be represented indiscrete time Xi(k + 1) = AiXi(k) + Bivinv(k) + Eiiout(k)

yi(k) = CiXi(k) (1)

where Xi(k) =[

vTxfm(k) iTinv(k)

]T, vinv , and load

disturbance iout are all represented in αβ reference frame.Given inverter current command iref (k), the DSMC equiva-lent control law is

vinv,eq(k) = (CiBi)−1 [iref (k)

−CiAiX(k) − CiEiiout(k)]. (2)

With inverter control voltage limit vmax determined by the DCbus voltage and PWM technique, the actual control voltagebecomes

vinv = vinv,eq if ‖vinv,eq‖ vmax,

vmax

‖vinv,eq‖vinv,eq otherwise. (3)

2) The robust servomechanism voltage control: The voltagecontroller generates current command iref for the currentcontroller. For the system shown in Fig. 1, considering thedynamics of the DSMC current controller, the overall statespace model is

X(k + 1) = AX(k) + Biref (k) + Eiout(k), (4)

where X(k) =[

vTxfm(k) iTinv(k) vT

out(k) iTxfm(k)]T

,with all element vectors represented under αβ reference frame,and ixfm is the transformer secondary current and will bereplaced by iout feedback in the control due to the negligibledifference.

LfCs

vDC

Inverter Controller

iinvA

iinvB

iinvC

3-ph Load

Z

iLineA

iLineB

iLineCeB

eA

eC

ioutA

ioutB

ioutC

voutA

voutB

voutC

Cf

vxfmAB

vxfmBC

vxfmCAvinvAB

vinvBC

vinvCA

Fig. 1. A grid-connected DG unit with local load.

vref,αβ

+ ev,αβ

RSC i∗ref,αβ

|i| Limiter iref,αβ

+ ei,αβ

DSMC vinv,αβ

SVPWM vαβ

Plant X

X

iout,αβ

−vout,αβ

−

iout

Fig. 2. Control structure for island mode.

An RSC is a combination of a stabilizing compensator anda servo compensator. The stabilizing compensator is the statefeedback of (4) times a gain K0. The servo compensator is asecond order filter represented by

η(k + 1) = Asη(k) + Bsev(k), (5)

where η is the state vector, ev(k) is the instantaneous voltageregulation error vector, and matrix As has poles at specifiedfrequencies to be cancelled from ev(k), e.g., the fundamental,5th and 7th harmonics. All quantities are in the αβ referenceframe.

The current command for the current controller can beobtained as

iref (k) = K0X(k) + K1η(k), (6)

where the gains K0 and K1 are obtained by solving the linearquadratic optimization problem of the augmented system of Xand η. Therefore, run-time gain calculation is not necessary.

B. Active and Reactive Power Control

Since the DG unit uses a voltage source inverter with astrong voltage control, its output active and reactive power aredetermined by the unit’s output voltage, including magnitudeand phase angle, as stated in

P =VoutE

Xsin δ, (7)

Q =V 2

out − VoutE cos δ

X, (8)

where E is the equivalent main voltage, X is the equivalentline reactance where the resistance is ignored, and δ is the

power angle. Since the DG unit output voltage control alreadyexists, the task of the power controller is to generate voltagecommand for the voltage controller based on the desired powervalues Pref and Qref and actual values P and Q as illustratedin Fig. 3.

It is apparent that the desired DG output voltage V andthe power angle δ can be calculated from (7) and (8) givendesired P and Q values and system parameter X . If this is true,the power control problem is solved. However, in practicalsystems, the above approach is not feasible. Equations (7) and(8) show that both P and Q are sensitive to variation of Xsince it appears in the denominators and especially when Xis small. A large capacity power system is assumed in thisresearch, where the value of X must be small. Practically,the value of X cannot be precisely known and may changedue to the operation of the power system. Once an inaccuratevalue of X is used in the control, the error of X will leadto poor tracking and even instability. Therefore, power controlsolutions requiring knowledge of X cannot be practically used.

It can be observed from (7) and (8) that both P and Q willbe affected by only adjusting one of V and δ, which is socalled coupling between P and Q. However, variations of Vand δ have different levels of impact on P and Q as describedin the following partial derivatives -

∂P

∂δ=

VoutE

Xcos δ, (9)

∂P

∂Vout=

E

Xsin δ, (10)

∂Q

∂δ=

VoutE

Xsin δ, (11)

Pref , Qref

+ eP , eQ

P, Q control |vref |, φref

v, i loops v

P, Q

P, Q

−

i

Fig. 3. Control structure for grid-connected mode.

0.51

1.5

−200

200.5

1

1.5

V

(A)

δ (deg.)

∂P /

∂δ

0.51

1.5

−200

20−0.5

0

0.5

V

(B)

δ (deg.)

∂P /

∂V

0.51

1.5

−200

20−0.5

0

0.5

V

(C)

δ (deg.)

∂Q /

∂δ

0.51

1.5

−200

200

1

2

V

(D)

δ (deg.)

∂Q /

∂V

Fig. 4. Sensitivity of P and Q to V and δ variations with normalized V ,E, and X: (A) ∂P

∂δ, (B) ∂P

∂V, (C) ∂Q

∂δ, (D) ∂Q

∂V.

∂Q

∂Vout=

2Vout − E cos δ

X. (12)

These partial derivatives are plotted in three-dimensional man-ner as shown in Fig. 4 to illustrate the significance of theimpacts of V and δ variations on P and Q under different Vand δ values. The values of V , E, and X are normalized inFig. 4 for comparison purposes.

It can be observed from Fig. 4 that, when |δ| is small andV is close to 1 which is true for large capacity power systems,∂P∂δ is close to 1 and ∂P

∂Voutis close to 0 and reversely, ∂Q

∂δ isclose to 0 and ∂Q

∂Voutis close to 1. This fact indicates that P is

more sensitive to δ and Q is more sensitive to Vout especiallywhen the DG unit is connected to a large capacity systemwhere the power angle δ is usually small. The different levelsof sensitivity of P and Q to δ and Vout provide a chanceto control P and Q relatively independently, not completelyindependently though.

Based on the above analysis, an integral approach to conductthe power flow control can be developed to control P byadjusting δ and control Q by adjusting Vout. If the phaseangle associated to the system voltage E is assumed to be0, δ = φ holds, where φ is the phase angle associated to Vout.The voltage and phase angle references can be generated as

φref =∫

[KP (Pref − P ) + ωn]dt + φ0, (13)

Vref =∫

KQ(Qref − Q)dt + V0, (14)

where ωn = 2π · 60rad./sec. is the system nominal angularfrequency, φ0 and V0 are the initial voltage and phase angle

Pref

+ ePKP

∆ω+ ω T

z−1∆φ

+ φref

P−

ωn

+

φ0+

Qref

+ eQ

KQT

z−1∆V

+ |Vref |

Q− |V0|

+

Fig. 5. The power regulator for P and Q.

at the moment that the DG unit is connected to the gridfrom island running mode. The proposed power controller isillustrated in Fig. 5 where the integration is implemented indiscrete-time.

C. The Stability Issue

In the proposed power regulation approach, P and Qcontrols are decoupled under steady state due to the integrationof the errors. However, in the transient, the P Q couplingcannot be eliminated. Moreover, both P and Q are nonlinearfunctions of Vout and φ, which increases the complexity toanalyze the system behavior. Due to the coupling issue andthe nonlinearity, the stability of the power control must beinvestigated.

Due to the strong regulation of the DG output voltage Vout

and its phase angle φ, the dynamics of the voltage controlloop can be simplified into a first-order system with a transferfunction representation

φ(s) =aφ

s + aφφref (s), (15)

Vout(s) =aV

s + aVVref (s), (16)

where aφ and aV are the inverses of the time constants of φand Vout dynamics. Since the scope of this discussion is aboutpower control stability of a DG unit connected to a large powersystem, whose time constant is in a range of seconds and muchgreater than that of the voltage tracking response measured ina range of 0.01 sec. or less, the DG power response has theroom to be much slower than the voltage tracking and stillfast compared to the power system. Therefore, it is reasonableto ignore the dynamics of the voltage tracking when powercontrol is concerned, i.e. aφ → ∞, aV → ∞, and φ → φref ,Vout → Vref .

At the moment that the DG is switched from island modeto grid-connected mode, V0 and φ0 should match E and φE ,where φE denotes the phase angle associated to E. Therefore,(13) and (14) can be rewritten into

δ =∫

[KP (Pref − P ) + ωn]dt, (17)

∆V =∫

KQ(Qref − Q)dt, (18)

where δ = φref − φE and ∆V = Vref − E. Assuminglarge capacity power system with small power angle δ, it is

0.08 0.1 0.12 0.14 0.16 0.18 0.2−200

0

200

vou

tAB

C (

V)

0.08 0.1 0.12 0.14 0.16 0.18 0.2−20

0

20

i outA

BC (

A)

0.08 0.1 0.12 0.14 0.16 0.18 0.2100

120

140

Time (s)

Vou

tAB

C (

V)

Fig. 6. Transient response of Vout in instantaneous and RMS at step loadincrease from 0 to 100% and decrease from 100% to 0.

reasonable to have sin δ ≈ δ and cos δ ≈ 1. Equations (17)and (18) can be rewritten in differential format

δ = KP (Pref − EVout

Xδ), (19)

∆V = KQ(Qref − Vout

X∆V ). (20)

Since the dynamics of DG voltage tracking is ignored, thestability of the power loop can be evaluated using Lyapunov’sdirect method where there is no external excitation, i.e.,Pref = 0 and Qref = 0.

A Lyapunov function can be defined as

ξ(∆V, δ) =12∆V 2 +

12δ2, (21)

where ξ > 0 holds unless ∆V = 0 and δ = 0. The derivativeof the above function is

ξ(∆V, δ) = ∆V · ∆V + δ · δ= −KQ

Vout

X∆V 2 − KP

EVout

Xδ2. (22)

From (22), it can be observed that ξ(∆V, δ) < 0 holds when∆V = 0 or δ = 0, given positive values of KP , KQ, E,Vout, and X . Therefore, the proposed power control loopis asymptotically stable at vicinity of the equilibrium point∆V = 0 and δ = 0.

III. SIMULATION RESULTS

Simulations have been run on a 5kVA DG unit with atopology shown in Fig. 1 connected to a 120V line-to-neutralpower system with an equivalent reactance 0.1Ω under anumber of difference scenarios as shown below.

A. Island Mode

Under island mode, a 100% step load increase is applied tothe DG output terminal. After steady state is reached, the loadsteps back down to zero. The voltage response under thesetransients are shown in Fig. 6.

1 1.5 2 2.5 3 3.5 4 4.5 5−5

0

5

P &

Q(k

W|k

Var

) P Q

5 5.5 6 6.5 7 7.5 8 8.5 9−5

0

5

Time (s)

P &

Q(k

W|k

Var

) P Q

Fig. 7. P , Q regulation under nonlinear local load.

Simulation data shows that the voltage tracking error understeady state is nearly zero and the THD is 0.4%. It can beobserved from Fig. 6 that the load disturbance has little impacton Vout waveform and the RMS transients last for only 0.02second with 3% or so peak variations.

B. Grid-connected Mode

Under grid-connected mode, P and Q output to the utilitygrid need to be controlled for system stabilizing, compen-sation, or handling local load disturbances. Fig. 7 illustratesthe P and Q regulation under various step references undernonlinear local load. It can be observed from Fig. 7 that thesettling time is about 0.5-0.8 sec. for P response and 0.3-0.5sec. for Q response under reference step changes with size of60%-100% unit rating, which is significantly faster than thework in [11] with much greater step sizes in per unit value.Even though some minor coupling between P and Q is seenin the transients, the steady state decoupling is achieved withthe control.

When nonlinear local load exists, the line current iLine issupposed not be affected with the proposed control. Fig. 8exhibits the current waveforms at three different locations ofthe system including the line current iLine, the unit outputcurrent iout, and the inverter current iinv . The waveforms showthat all current harmonics are taken by the DG unit and thesystem line current is clean. This is because the voltage controlloop eliminates the voltage harmonics at the DG output whichavoids harmonic current flow to or from the utility grid.

C. The Switching Transients

The transients as the DG unit is switched between islandmode and grid connected mode are presented in Fig. 9 andFig. 10. Both figures show seamless switching as far as thevoltage is concerned. The power transient shown in Fig. 9 iscaused by the circuit topology change with a nonzero systemequivalent impedance Z.

2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3−40

−20

0

20

40 i ou

tAB

C (

A)

2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3−20

0

20

i Lin

eAB

C (

A)

2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3−40

−20

0

20

40

Time (s)

i invA

BC (

A)

Fig. 8. Current waveforms of DG unit output current iout, system linecurrent iLine, and inverter current iinv under nonlinear local load.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

P &

Q(k

W|k

Var

)

0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13−200

0

200

vou

tAB

C (

V)

0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13−10

0

10

Time (s)

i Lin

eAB

C (

A)

P Q

Fig. 9. Transients of P , Q, vout, and iout when the DG unit is switchedfrom island mode to grid-connected mode with zero Pref and Qref .

1.99 1.995 2 2.005 2.01 2.015 2.02 2.025 2.03

0

2

4

P &

Q(k

W|k

Var

)

P Q

1.99 1.995 2 2.005 2.01 2.015 2.02 2.025 2.03−200

0

200

vou

tAB

C (

V)

1.99 1.995 2 2.005 2.01 2.015 2.02 2.025 2.03−20

0

20

Time (s)

i Lin

eAB

C (

A)

Fig. 10. Transients of P , Q, vout, and iout when the DG unit is switchedfrom grid-connected mode to island mode with nonzero P and Q initialoperating condition.

IV. CONCLUSION

This paper has presented a power flow control approach fora single distributed generation unit connected to utility gridwith a nonlinear local load. The proposed control techniquecombines an independent integral P and Q controller, a robustservomechanism voltage controller, and a discrete-time slidingmode current controller. Simulation results under various sce-narios have demonstrated the effectiveness of the proposedtechnique in power regulation and harmonic elimination ofline current. The stability of the power control loop is provedusing Lyapunov direct method.

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