Adaptive Grid-Voltage Sensor Less Control Scheme___05075661

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 3, SEPTEMBER 2009 683

Adaptive Grid-Voltage Sensorless Control Schemefor Inverter-Based Distributed Generation

Yasser Abdel-Rady Ibrahim Mohamed, Member, IEEE, Ehab F. El-Saadany, Senior Member, IEEE,and Magdy M. A. Salama, Fellow, IEEE

Abstract—This paper presents an adaptive grid-voltage sensor-less control scheme for inverter-based distributed generation unitsbased on an adaptive grid-interfacing model. An adaptive grid-interfacing model is proposed to estimate, in real time, the inter-facing parameters seen by the inverter and the grid-voltage vectorsimultaneously. A reliable solution to the present nonlinear estima-tion problem is presented by combining a grid-voltage estimatorwith an interfacing parameter estimator in a parallel structure.Both estimators adjust the grid-interfacing model in a mannerthat minimizes the current error between the grid model and theactual current dynamics, which acts as a reference model. Theestimated quantities are utilized within the inner high-bandwidthcurrent control loop and the outer power controller to realize anadaptive grid-voltage sensorless interfacing scheme. Theoreticalanalysis and simulation results are provided to demonstrate thevalidity and usefulness of the proposed interfacing scheme.

Index Terms—Adaptive identification, digital currentcontrol, distributed generation (DG), grid-voltage sensorlesscontrol, pulsewidth-modulated (PWM) inverters.

I. INTRODUCTION

DRIVEN BY economical, technical, and environmentalreasons, the energy sector is moving into an era where

large portions of increases in electrical energy demand will bemet through widespread installation of distributed resources orwhat’s known as distributed generation (DG) [1]. Unlike largegenerators, which are almost exclusively 50–60 Hz synchronousmachines, DG units include variable frequency (variable speed)sources (such as wind energy sources), high-frequency (highspeed) sources (such as microturbines), and direct energy con-version sources producing dc voltages (such as fuel cells andphotovoltaic sources). The majority of the distributed resourcesare interfaced to the utility grid via dc–ac inverter systems[2], [3]. However, the control performance of the interfacingsystem depends on the interfacing impedance seen by the in-verter and the grid voltage at the point of common coupling(PCC).

To reduce system’s cost and to increase its reliability, it ishighly desirable to realize a grid-interfacing scheme with theminimum number of sensing elements. Along with the reliabilityand cost enhancements, significant performance enhancements

Manuscript received May 19, 2007; revised December 22, 2007. First pub-lished June 16, 2009; current version published August 21, 2009. Paper no.TEC-00170-2007.

The authors are with the Department of Electrical and Computer Engi-neering, University of Waterloo, Waterloo ON N2L 3G1, Canada (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2008.2001448

can be obtained by eliminating the grid-voltage sensors in aninverter-based DG interface. Among these are: 1) the eliminationof the residual negative sequence and voltage feedforward com-pensation errors (the injected currents are so sensitive to minutevariations in the reference voltage vector, which highly dependson the feedforward compensation control) and 2) the positivecontribution to the robustness of the power sharing mechanismin paralleled inverter systems, where the power-sharing mecha-nism is generally based on open-loop controllers.

Recently, grid-voltage sensorless techniques have been inves-tigated especially in three-phase pulsewidth-modulated (PWM)voltage-sourced converter systems [4]–[9], where the basicnumber of sensors needed is five (two ac currents and voltages,and the dc-link voltage). By avoiding the use of grid-voltage orgrid-current measurements, the number of sensors is reduced. Itis commonly desirable to use grid-voltage sensorless schemes,where an inherent overcurrent protection is provided. Differentsensorless interfacing schemes have been reported. The posi-tion of the grid-voltage vector is estimated in [4] by modify-ing the proportional-integral (PI) current regulator and using amodel-based observer. The principle of direct power control isapplied to realize voltage sensorless control of a PWM rectifiersystem [5]. The dc-link voltage information is considered theonly measured variable in a PWM active rectifier in [6], and astate–space observer is proposed to estimate unknown quanti-ties. However, the control algorithm is very complex, and thestability is not verifiably guaranteed under parametric uncertain-ties. An input current model-based observer is proposed in [7]for input current estimation in PWM converters. A direct con-trol of the converter instantaneous current, based on the directpower control, and the estimation of the line voltage waveformis proposed in [8].

However, the aforementioned voltage-sensorless controlschemes assume precise knowledge of the interfacingimpedance parameters at the PCC. On the distribution level,distribution system parameters are time-varying and directly im-pact the performance of the control and estimation algorithms.For example, the current delivered by a grid-connected inverter-based DG unit passes though a filter inductor and possibly acoupling transformer. Interfacing parameters, such as the equiv-alent inductance and resistance of the coupling transformer,filter inductors, and connection cables, vary with temperature,transformer saturation, cable overload, and other environmentalconditions. On the other hand, depending on the grid configura-tion, a large set of grid impedance values (as DG is commonlyinstalled in weak grids such as remote areas with radial distribu-tion feeders) challenge the stability and control of the interfacing

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684 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 3, SEPTEMBER 2009

scheme [10]–[14]. In particular, there is a strong trend toward theuse of current control for grid-connected PWM inverters in DGsystems [2], [15], due to the need to control both the harmoniccontent and the power factor of the current. In this approach,it is commonly desired to design the inner current control loopwith high-bandwidth characteristics to ensure accurate currenttracking, shorten the transient period as much as possible, andforce the voltage-source inverter (VSI) to equivalently act asa current source amplifier within the current loop bandwidth.In addition, the current controller should not allow system un-certainties and disturbances, such as preexisting grid-voltagedistortion and parameter variations, to drive harmonic currentsthrough the inverter. However, if the current control loop isdesigned with high-bandwidth characteristics (e.g., deadbeatcontrol performance [16]–[20]), the sensitivity of the dominantpoles of the closed-loop current controller becomes very high touncertainties in the total interfacing inductance and resistance.In addition, in a current-controlled DG interface, the sensitiv-ity of the current controller to system uncertainties remarkablyincreases when the grid voltage is estimated [19], [20], wherethe grid-voltage estimator dynamics will be dependent on sys-tem parameters. These facts challenge the stability and con-trol effectiveness of a grid-voltage sensorless current-controlledsystem.

Motivated by the aforementioned limitations, this paperpresents an adaptive grid-voltage sensorless interfacing schemefor inverter-based DG units based on an adaptive grid-interfacing model. A novel adaptive grid-interfacing modelis proposed to provide real-time estimates of the interfacing(impedance) parameters seen by the inverter and the grid-voltage vector simultaneously. The adaptive model utilizes agrid-voltage estimator in parallel with an interfacing parame-ters estimator. Both estimators adjust the grid-interfacing modelin a manner that minimizes the current error between the gridmodel and the actual current dynamics, which acts as a referencemodel. Due to the nonlinear nature of the estimation problemand the periodic time-varying nature of the grid voltage, the grid-voltage estimator utilizes a neural network (NN) based adapta-tion algorithm, which works as a real-time optimization agent.The self-learning feature of the NN adaptation algorithm allowsa feasible and easy adaptation design at different operating con-ditions [21]–[24]. In the proposed scheme, the online weightadaptation rules are synthesized in the sense of Lyapunov sta-bility theory [25]. This approach guarantees the stability of thelearning algorithm in a systematic manner. The estimated gridvoltage can be regarded as a quasi-input signal; hence, undis-turbed model is yielded. Relying on the undisturbed model, asimple parameter estimator is used to estimate unknown inter-facing parameters by minimizing the parameter estimation errorby an iterative gradient algorithm offered by the projection al-gorithm (PA) [26]. The estimated quantities are utilized withinthe inner high-bandwidth current control loop and the outerpower controller to realize an adaptive grid-voltage sensorlessinterfacing scheme. Theoretical analysis and simulation resultsare provided to demonstrate the validity and usefulness of theproposed interfacing scheme.

Fig. 1. Grid-connected three-phase VSI with an inner current control loop andLC filter.

II. MODELING OF THREE-PHASE CURRENT-CONTROLLED

GRID-CONNECTED VSI

A common topology of a grid-connected three-phase current-controlled VSI with an LC filter is depicted in Fig. 1, where Rand L in Fig. 1 represent the equivalent resistance and induc-tance of the filter inductor, the coupling transformer (if any),and the equivalent grid resistance and inductance seen by theinverter, respectively, C is the filter capacitance, and vs is thegrid voltage. To impose an arbitrary current in the inductive R–Limpedance, a current controller is usually adopted to shape thevoltage applied on the inductor so that minimum current er-ror is achieved. A PWM scheme would ensure that the invertervoltage is free from low-order harmonic distortion. However, thehigh-frequency current distortion due to the switching frequencymust be attenuated to cope with the power quality standards forconnection of an inverter to the grid [27]. Also, the current con-troller should not allow system uncertainties and disturbances,such as preexisting grid-voltage distortion and parameter varia-tions, to drive harmonic currents through the inverter. The firstrequirement can be achieved with the second-order LC filter,while the second requirement calls for a robust current regula-tion scheme. A robust current regulation scheme, which satisfiesthese requirements, is presented in Section IV.

In the stationary reference frame αβ, the current dynam-ics can be reasonably represented by the following state–spaceequations:

x = Acx + Bc (v − vs) , y = Cx (1)

with

x = [ iα iβ ]T , v = [ vα vβ ]T , vs = [ vsα vsβ ]T

and

Ac =[−R

L 0

0 −RL

], Bc =

[ 1L 00 1

L

], C =

[1 00 1

]

ABDEL-RADY IBRAHIM MOHAMED et al.: ADAPTIVE GRID-VOLTAGE SENSORLESS CONTROL SCHEME FOR INVERTER-BASED DG 685

where Ac , Bc , and C are the system matrices of continuoustime system (1); vα , vβ , iα , and iβ are the α- and β-axis in-verter’s voltages and currents; vsα and vsβ are the α- and β-axiscomponents of the grid-voltage vector.

Since the harmonic components included in the inverter out-put voltage are not correlated with the sampled reference cur-rents, the PWM VSI can be assumed as a zero-order hold circuitwith a transfer function H(s)

H(s) =1 − e−sT

s(2)

where T is the discrete-time control sampling period and s isthe Laplace operator.

For digital implementation of the control algorithm, the cur-rent dynamics in (1) can be represented in discrete-time withthe conversion H(s) in (2) as follows:

iαβ (k + 1) = Aiαβ (k) + B vαβ (k) − vsαβ (k) (3)

where A, B are the sampled equivalents of Ac , Bc . If thecontinuous system is sampled with interval T , which is at leastten times shorter than the load time constant, then the matricesof the discrete-time system A, B can be obtained by Euler’sapproximation as follows:

A = eAc T ≈ I + AcT =[ 1 − T R

L 0

0 1 − T RL

](4)

B =∫ T

0eAc τ dτ · Bc ≈ BcT =

[ TL 0

0 TL

]. (5)

Considering the physical constraints, the preceding model issubjected to the following limits. The load current is limitedto the maximum continuous current of the inverter or to themaximum available current of the inverter in a limited short-time operation. Also, the load voltage is limited to the maximumavailable output voltage of the inverter depending on the dc-linkvoltage.

III. ADAPTIVE GRID-INTERFACING MODEL

A. Estimation Algorithm

Generally, any implementation strategy of a model-basedgrid-interfacing scheme is by nature parameter dependent. Inparticular, accurate knowledge of the interfacing parametersR,L, and the grid voltage is required in order to implementa high-performance voltage-sensorless interfacing scheme.

Fig. 2 shows the proposed estimation algorithm. The algo-rithm utilizes an adjustable current dynamics reference modelin the stationary reference frame, which runs in parallel withthe actual inverter current dynamics; the later acts as a referencemodel. The grid-voltage estimator utilizes an NN-based adap-tation algorithm, which employs a three-layer feedforward NNto work as a real-time optimization agent for the present esti-mation problem. The estimated grid voltage can be regarded asa quasi-input signal, which can be included in the model excita-tion voltage; hence, undistributed model is yielded. Relying onthe undistributed model, a simple parameter estimator is usedto estimate unknown interfacing parameters by minimizing the

Fig. 2. Proposed estimation algorithm.

parameter estimation error by an iterative gradient algorithmoffered by the PA.

To estimate the grid-voltage vector, suppose that theimpedance parameters R and L are exactly known; and let usconstruct an adjustable model with the following input/outputrelation:

˙x = Acx + Bc(v∗ − vs) (6)

where superscripts (ˆ) and (∗) denote estimated and referencequantities, respectively. Assuming that the nonlinearities associ-ated with the inverter operation—particularly the blanking timeand the voltage limitation effects—are properly compensated,the actual voltage components can be replaced with the refer-ence ones, denoted by v∗ in (6). This assumption is justified byconsidering that the inverter’s switching period is much smallerthan the circuit time constant. As a result, the direct measure-ments, which are affected by the modulation and acquisitionnoise, are avoided.

The convergence of the adjustable model in (6) can beachieved with an appropriate adaptation algorithm using theestimation error. The dynamics of the estimation error vectore ≡ x − x = [ eα eβ ]T can be obtained as

e = Bc (vs − vs) . (7)

A three-layer NN, as shown in Fig. 3, which comprises aninput layer (the i layer), a hidden layer (the j layer), and anoutput layer (the k layer), is adopted to implement the proposedNN-based adaptation algorithm. The inputs of the NN adapta-tion algorithm are e(k) and ∆e(k), whereas the output is theestimated grid-voltage vector vs . The connective weights of theNN are adjusted online to adjust the model in (6) so that thatthe estimation error in (7) is minimized. The signal propagationand the fundamental function of each layer are summarized asfollows:


Fig. 3. NN structure.

Input layer:

neti = xi , Oi =[Oα

i Oβi

]T = fi(neti), fi(ϕ) = ϕ

i = 1, 2 (8)

where x1 = e(k) and x2 = (1 − z−1)e(k) are the network in-puts.Hidden layer:

netj =[∑

i WαjiO

αi

∑i Wβ

jiOβi

]TOj =

[Oα

j Oβj

]T = fj (netj ), fj (ϕ) =1

1 + e−ϕ(9)

j = 1, . . . , 4

where W j i =[Wα

ji Wβji

]Tare the connective weights be-

tween the input and the hidden layers and fj is the activationfunction.Output layer:

netk =[∑

j WαkjO

αj

∑i Wβ

kjOβj

]TOk =

[Oα

k Oβk

]T = fk (netk ), fk (ϕ) = ϕ

k = 1, 2 (10)

where W kj =[Wα

kj Wβkj

]Tare the connective weights be-

tween the hidden and the output layers.The selection of the NN learning algorithm dictates the net-

work performance. Recently, several NN learning approacheshave been proposed based on Lyapunov stability theory [28].This approach guarantees the stability of the learning algorithmin a systematic manner. In the present online learning algo-rithm, the weight adaptation rules are synthesized in the senseof Lyapunov stability theory.

A discrete-time Lyapunov candidate function is selected as

Ve(e(k), k) =12eT (k)e(k). (11)

The Lyapunov’s convergence criterion must be satisfied suchthat

Ve(k)∆Ve(k) < 0 (12)

where ∆Ve(k)is the change in the Lyapunov function.

The stability condition in (12) is satisfied when ∆Ve(k) < 0as ∆Ve(k) is defined as an arbitrary positive, as shown in (11).

For the error dynamics in (7), and with the chosen Lyapunovfunction, the sensitivity of the controlled system is not requiredin the online learning algorithm. On the other hand, complexidentification techniques are needed to identify the Jacobian ofthe controlled plant in the traditional back-propagation learningalgorithm [22], [23]. The learning algorithm aims at evaluatingthe derivatives of the Lyapunov energy function with respect tothe network parameters so that ∆Ve(k) < 0 is satisfied. Accord-ingly, the output layer weights W kj are updated as follows:

∆W kj (k) = −ηkj∂Ve

∂vs

∂vs

∂Ok

∂Ok

∂netk

∂netk∂W kj

= −ηkjB

[eα (k)Oα

j

eβ (k)Oβj

](13)

where ηkj is the learning rate of the connected weight vectorW kj .

Similarly, the hidden layer weights are updated as follows:

∆W j i(k) = −ηji∂Ve

∂netk

∂netk

∂Oj

∂Oj

∂netj

∂netj

∂W j i

= −ηjiB

eα (k)Wα

kjf′(∑

i WαjiO

αi

)Oα

i

eβ (k)Wβkj f

′(∑

i WβjiO

βi

)Oβ

i

(14)

where ηji is the learning rate of the connected weight vectorW j i .

The update rules in (13) and (14) provide an iterative gradientalgorithm designed to minimize the energy function in (11).Since the gradient vector is calculated in the direction oppositeto the energy flow, the convergence of the NN is feasible.

In the aforementioned analysis, it has been assumed that theinterfacing parameters R and L are exactly known. However, inpractical applications of the control system, the actual parame-ters a = 1 − TR/L and b = T/L are assumed to be unknown,and they should be adjusted in real time by a parameter estimatorthat can provide estimated values a and b.

With the NN grid-voltage estimator, the estimated grid volt-age can be regarded as a quasi-input signal, which can be used forfeedforward control. Subsequently, the following undistributedmodel can be derived from (3) as follows:

iαβ (k + 1) = Aiαβ (k) + Bu∗αβ (15)

where u∗αβ = v∗

αβ − vsαβ is the equivalent excitation voltage.Due to the decoupling symmetry of the system matrices Ac

and Bc in the stationary reference frame, there are only twoparameters to be estimated: a and b; therefore, either the α- orthe β-current dynamics can be used in the estimation phase. Bythis method, the computational demand is reduced. Using theα-axis current dynamics,

iα (k) = aiα (k − 1) + bu∗α (k − 1) = RT (k − 1)θ(k) (16)


where R(k − 1) = [ iα (k − 1) u∗α (k − 1) ]T is the input/

output measurement vector and θ(k) = [ a(k) b(k) ]T is a pa-rameter vector. The dynamics in (16) can be used in a re-cursive estimation process to provide an estimate θ(k) =[ a(k) b(k) ]T of unknown plant parameters. The estimation er-ror iα (k) − RT (k − 1)θ(k) will be produced mainly by pa-rameter variation. Therefore, this error can be used to adaptivelyadjust the estimated parameters in a manner that minimizes theerror. At this condition, the estimated parameters will convergeto their real values. To achieve this objective, an iterative gradi-ent algorithm based on the PA is used. The PA suits the presentestimation problem, where rapid parameter estimation with lowcomputational complexity is required.

The parameter vector θ(k) is recursively updated using thePA [23] as follows:

θ(k + 1) = θ(k) +rR(k − 1)iα (k) − RT (k − 1)θ(k)

ε + RT (k − 1)R(k − 1)(17)

where r ∈ [0, 2] is a reduction factor and ε is a small value toavoid division by zero if RT (k − 1)R(k − 1) = 0.

The estimate θ is used to update the adjustable model param-eters in (6); therefore, an unbiased grid-voltage estimate vs canbe obtained. The estimated grid voltage is fedforward to (15)resulting in equivalent excitation signal to cancel the voltagedisturbance. As a result, the unbiased parameter estimate θ canbe reliably obtained. Using the preceding recursive process, thegrid voltage and the interfacing parameters will quickly con-verge into their real values. The estimated quantities can bereliably used to realize an adaptive grid-voltage sensorless con-trol scheme.

B. Convergence Analysis

The estimation scheme is based on parallel estimators strat-egy to linearize the present nonlinear estimation problem. Themain potential of the parallel estimation strategy is the inher-ent decoupling of the disturbance parameter estimation prob-lems; therefore, the augmented nonlinear error dynamics areavoided and both estimators can be designed separately [25],[29]. Accordingly, two Lyapunov function candidates for theerror vector [ eα eβ ]T and the parameter estimator error vectorθ(k) ≡ θ(k) − θ(k) are utilized. The total Lyapunov functionis selected as

VT (eα (k), eβ (k), θ(k), k)

= V1 (eα (k), eβ (k), k) + V2(θ(k), k)

=12[eα (k)2 + eβ (k)2]+ θ(k)T θ(k). (18)

The Lyapunov’s convergence criterion must be satisfied suchthat

VT (k)∆VT (k) < 0 (19)

where ∆VT (k) = ∆V1(k) + ∆V2(k)is the change in the totalLyapunov function.

The stability condition in (19) is satisfied when ∆VT (k) < 0as VT (k) is defined as an arbitrary positive as shown in(18).

First, the change in the Lyapunov function ∆V1(k) is givenby

∆V1(k) = V1(eα (k + 1), eβ (k + 1)) − V1(eα (k), eβ (k)) < 0.(20)

The change in the error ∆eα (k) and ∆eβ (k)due to the adap-tation of the weight vector W kj can be given by [30]

[∆eα (k)

∆eβ (k)

]=[

eα (k + 1) − eα (k)

eβ (k + 1) − eβ (k)

]=

∂eα (k)∂W α

k j∆Wα

kj

∂eβ (k)∂W β

k j

∆Wβkj

.

(21)Since

[∂eα (k)∂W α

k j

∂ eβ (k)∂W β

k j

]T

= bηkj

[Oα

j

Oβj

]

then the following incremental error dynamics can be obtained:

[∆eα (k)

∆eβ (k)

]= −b2ηkj

[eα (k)

(Oα

j

)2

eβ (k)(Oβ

j

)2

]. (22)

Accordingly, ∆V1(k) can be represented as

∆V1(k) = eα (k)∆eα (k) + eβ (k)∆eβ (k)

+12(∆eα (k)2 + ∆eβ (k)2)

= −b2ηkj eα (k)2 (Oαj

)2

(1 −

b2ηkj

(Oα

j

)2

2

)

− b2ηkj eβ (k)2(Oβj

)2

1 −

b2ηkj

(Oβ

j

)2

2

.

(23)

To satisfy the convergence condition ∆V1(k) < 0, the learn-ing rate ηkj should satisfy

0 < ηkj <2

maxk

[b2∥∥Oα

j (k)∥∥2

, b2∥∥Oβ

j (k)∥∥2] . (24)

Since 0 < Oαj < 1 and 0 < Oβ

j < 1, j = 1, . . . , Rkj , whereRkj is the number of weights between the output and hiddenlayers, then by the definition of the usual Euclidean norm in n ,‖Oα

j ‖ ≤√

Rkj and ‖Oβj ‖ ≤

√Rkj .

The change in the error ∆eα (k)and ∆eβ (k) due to the adap-tation of the weight vector W j i can be given by

[∆eα (k)

∆eβ (k)

]=[

eα (k + 1) − eα (k)

eβ (k + 1) − eβ (k)

]=

∂eα (k)∂W α

j i∆Wα

ji

∂eβ (k)∂W β

j i

∆Wβji

.

(25)


Further,

∂eα (k)∂W α

j i

∂ eβ (k)∂W β

j i

= b

Wα

kjf′(∑

i WαjiO

αi

)Oα

i

Wβkj f

′(∑

i WβjiO

βi

)Oβ

i

therefore, the following incremental error dynamics can beobtained:[

∆eα (k)∆eβ (k)

]

= −b2ηji

eα (k)

(Wα

kj

)2(f ′(∑

i WαjiO

αi

))2(Oα

i )2

eβ (k)(Wβ

kj

)2(f ′(∑

i WβjiO

βi

))2 (Oβ

i

)2

.

(26)

Accordingly, ∆V1(k)can be represented as

∆V1(k) = eα (k)∆eα (k) + eβ (k)∆eβ (k)

+12(∆eα (k)2 + ∆eβ (k)2)

= −b2ηjieα (k)2 (Mαji

)2

(1 −

b2ηji

(Mα

ji

)2

2

)

− b2ηjieβ (k)2(Mβji

)2

(1 −

b2ηji

(Mβ

ji

)2

2

)

(27)

where(Mα

ji

)2

(Mβ

ji

)2

=

(Wα

kj

)2(f ′(∑

i WαjiO

αi

))2(Oα

i )2

(Wβ

kj

)2(f ′(∑

i WβjiO

βi

))2 (Oβ

i

)2

.

To satisfy the convergence condition ∆V1(k) < 0, the learn-ing rate ηji should satisfy

0 < ηji <2

maxk

[∥∥Mαji(k)

∥∥2,∥∥Mβ

ji(k)∥∥2] . (28)

Provided that fj (ϕ) ∈ [0, 1] and f ′j (ϕ) = fj (ϕ) − (fj (ϕ))2 ,

then max[f ′

j (ϕ)]

= 1/4. Therefore, the following inequalitiescan be deduced:

∥∥Mαji(k)

∥∥2 ≤(Wα

kj−max

)2(Oα

i−max

)2

16∥∥∥Mβji(k)

∥∥∥2≤

(Wβ

kj−max

)2(Oβ

i−max

)2

16. (29)

Since the weights update rules are synthesized in the directionopposite to the energy flow, the weights between the hidden andoutput layers are bounded. Since the parameters of the NN arebounded, the convergence is guaranteed.

Second, the change in the Lyapunov function for the param-eter estimator ∆V2(k) is given by

∆V2(k) = V2

(θ(k + 1)

)− V2

(θ(k)

)< 0. (30)

By using the parameter estimation error dynamics with theupdate law (17), ∆V2(k) can be evaluated as

∆V2(k) =

∥∥∥∥∥θ(k)− rR(k− 1)R(k − 1)θ(k)ε + RT (k − 1)R(k − 1)

∥∥∥∥∥2

− θ(k)T θ(k)

=r[θ(k)T R(k − 1)

]2ε + RT (k − 1)R(k − 1)

×[−2 +

rRT (k − 1)R(k − 1)ε + RT (k − 1)R(k − 1)

]. (31)

Now, if ε > 0 and 0 < r < 2 are assumed, the bracketed termin (31) is negative, and consequently, the stability condition in(30) is satisfied and the following convergence properties aresatisfied:

‖θ(k) − θo‖ ≤ ‖θ(k − 1) − θo‖ ≤ ‖θ(0) − θo‖ , k ≥ 1

(32)

limk→∞

R(k − 1)θ(k)T√ε + RT (k − 1)R(k − 1)

= 0 (33)

where θo is the parameter vector obtained at perfect convergenceof the estimator.

Using the aforementioned conditions, it can be shown that∆VT (k) = ∆V1(k) + ∆V2(k) < 0, and it follows that the aug-mented error is monotonically nonincreasing. Therefore, theconvergence is guaranteed and the estimates can be reliablyused in the control system design.

IV. PROPOSED CONTROL SCHEME

Fig. 4 shows the proposed adaptive grid-voltage sensorlesscontrol scheme for a current-controlled PWM-VSI. The schemeconsists of the proposed adaptive grid-interfacing model, whichprovides real-time estimates of the interfacing parameters andthe grid-voltage vector at the PCC; a current control loop, whichis realized in the rotational reference frame to null the phaseerrors, and an average power controller to generate the referencecurrent vector.

A. Current Control Loop

The inner current loop is necessary to obtain high powerquality in grid-connected inverters. Currently, there is a strongtrend toward fully digital control of power converters based ondeadbeat current control techniques [16]–[20]. As compared toother current control techniques, such as the hysteresis controlscheme, ramp comparison, and stationary and synchronous PIcontrol schemes [16], deadbeat controllers offer the potentialfor achieving the fastest transient response, precise current con-trol, zero steady-state error, the lowest distortion, and the fullcompatibility with digital-control platforms [18]. However, asa high-bandwidth model-based controller, the deadbeat currentcontrol scheme shows a high sensitivity to plant uncertainties. Inaddition, the sensitivity to system uncertainties increases whenthe grid voltage is estimated. To alleviate this limitation, the


Fig. 4. Proposed control scheme for the DG interface.

outputs of the estimation unit are used to redesign the deadbeatcontroller in real time.

In the synchronous reference frame that rotates syn-chronously with the grid angular speed ω1 , and by using thetime-delay compensation method developed in [28], the currentdynamics can be controlled to yield a deadbeat current controlresponse, in the presence of system delays with the followingcontrol law:

v∗dq (k + 1) =

1b

i∗dq (k + 2) − Adq

(Adq i

∗dq (k)

+ bv∗

dq (k) − vsdq (k))

+ vsdq (k + 1)

(34)

where

Adq =[

a −Tω1Tω1 a

].

Using the estimated quantities, the control voltage can beadaptively calculated as follows:

v∗dq (k + 1) =

1

b

i∗dq (k + 2) − Adq

(Adq i

∗dq (k)

+ bv∗

dq (k) − vsdq (k) )

+ vsdq (k + 1)

(35)

where

Adq =[

a −Tω1Tω1 a

].

To achieve higher dc-link voltage utilization and lower distor-tion in the output current, the space vector modulation (SVM)technique is employed to synthesize the control voltage in (35).

B. Power Control

To ensure high power quality, the outer power control loopshould offer a relatively slowly changing current reference tra-

Fig. 5. Proposed voltage-sensorless power controller.

jectory. Since the required power transient response is muchslower than the current dynamics, the reference current can befiltered to ensure high-quality inductor current.

Based on the time-scale separation between the power andcurrent dynamics, the output power variation depends only onthe variation of the grid voltage within the control cycle. Us-ing the active and reactive power references p∗ and q∗ andgrid-voltage components vsq and vsd , the reference currents arecalculated as follows:[

i∗di∗q

]=

1‖v2

s ‖

[vsd −vsq

vsq vsd

] [p∗

q∗

]. (36)

To compensate for the filter-capacitor current component, theinductor current references are calculated by adding a simplefeedforward compensation term as follows:[

i∗di∗q

]=

1‖v2

s ‖

[vsd −vsq

vsq vsd

] [p∗

q∗

]+

1Zc

[vsd

vsq

](37)

where Zc is the capacitor impedance.The voltage-sensorless power controller can be realized using

the estimated voltage components as[i∗di∗q

]=

1‖v2

s ‖

[vsd −vsq

vsq vsd

] [p∗

q∗

]+

1Zc

[vsd

vsq

]. (38)


Fig. 6. Developed Simulink model of the proposed DG interface with a microturbine source.

To provide a sufficient attenuation gain for the harmonic con-tent in the reference current vector, a low-pass filter (LPF) isadopted and digitally implemented as follows:[

i∗df (k)

i∗qf (k)

]=

2σ − T

2σ + T

[i∗df (k − 1)i∗qf (k − 1)

]

+T

2σ + T

[i∗d(k) − i∗d(k − 1)i∗q (k) − i∗q (k − 1)

](39)

where i∗df , i∗qf are the filtered d- and q-axis reference currentcomponents, respectively, and 1/σ is the filter cutoff frequency.It should be noted that the filter cutoff frequency should be lowenough to provide sufficient attenuation of current referenceharmonics caused by voltage harmonics. At the same time, itshould be high enough to provide a reasonable dynamic responseof the power control loop. Fig. 5 depicts the structure of theproposed voltage-sensorless power controller.

V. SIMULATION RESULTS

To evaluate the performance of the proposed control scheme,a 10-kV·A three-phase grid-connected microturbine DG inter-face incorporated with the proposed control scheme, as reportedin Fig. 4, has been used. The system parameters are as follows:grid phase voltage = 120 V at 60 Hz, grid resistance = 0.05 Ω,grid inductance = 0.2 mH, dc-link voltage = 500 V, nominalinterfacing inductance Lo = 2.5 mH, nominal interfacing re-sistance Ro = 1.0 Ω, nominal filter capacitance Co = 45 µF.The LC-filter parameters are chosen to attenuate the switchingfrequency components in the output current much below thenominal fundamental current component. An attenuation factor50–70 dB can be selected as a design constraint to cope withstandards in [27]. The developed Simulink model of the overallDG interface is shown in Fig. 6. The real-time code of the pro-posed control scheme is generated by the real-time workshop(RTW), under Matlab/Simulink environment. The TMS320C30

Fig. 7. Robustness of the conventional and proposed current controllers withmeasured and estimated grid voltage.

digital signal processor (DSP) has been chosen as an embeddedplatform for real-time digital simulation experiments. The esti-mation and control algorithm is coded as a Matlab S-functionwritten in C++. The C++ S-function facilitates straightfor-ward real-time coding via the RTW. The execution time of thecurrent control interrupt routine is about 130 µs. Subsequently,a control period T = 150 µs is selected. With this setting, a safeCPU load coefficient 86% and a switching frequency 6.7 kHzhave been obtained. As these figures reveal, the processing de-mand of the proposed control scheme is relatively modest for aDSP system, making it possible to achieve quite high switchingfrequencies. It should be noted that the control period is mainlydependent on the instruction time of the hardware. Further, codeoptimization is another factor in determining the execution timeof the control interrupt service routine.

The proposed interfacing scheme is initially tuned using thenominal system parameters. For the adaptive estimation unit,r = 0.65, ε = 0.001, and an initial parameter vector: θ(0) =[ao, bo ] = [0.94, 0.06] are chosen. The initial parameter vectoris set to the nominal value.


Fig. 8. Performance of proposed and conventional current control with 20% mismatch in L. (a) and (b) Phase-a steady-state current response and the correspondingspectrum of the conventional controller. (c) and (d) Phase-a steady-state current response and the corresponding spectrum of the proposed controller.

Fig. 9. Dynamic performance of the proposed adaptive estimation unit. (a) Phase-a current command and the corresponding output current. (b) Actual andestimated grid voltages. (c) Estimated interfacing resistance. (d) Estimated interfacing inductance.

To verify the feasibility of the proposed controller, differ-ent operating conditions have been considered. Some selectedresults are presented as follows.

A. Stability and Robustness to Parameter Variation

The stability and robustness of the inner current control ismainly affected by parameter variation. The sensitivity to pa-rameter variation increases when the grid voltage is estimated.Fig. 7 shows the change in the radius of the pole with maximumamplitude (in the z-plane) versus the uncertainty in the interfac-ing inductance, for both the conventional deadbeat current con-troller [17] and proposed controllers. Two cases are considered,

Fig. 10. Current control performance with the nominal inductance and 50%reduction in R.


Fig. 11. Dynamic performance of the proposed interfacing scheme. (a) Step response in the active power. (b) Reactive power response due to zero command. (c)Generated reference d-axis current and output phase-a current. (d) Output phase-a current and grid voltage.

first when the grid voltage is measured and second when thegrid voltage is estimated. It can be seen that the conventionaldeadbeat current controller is driven into instability with about10% mismatch in L when the grid voltage is measured. Whenthe grid voltage is estimated, the conventional deadbeat con-troller becomes unstable at only 5% mismatch in L. On theother hand, the proposed controller is stable at any value of theload inductance, as shown in Fig. 7. This is due to the self-tuningfeature, which enables the “redesign” of the deadbeat controllerin real time at different load parameters. When both parametersR and L and the grid voltage are reliably estimated and used forthe self-tuning control, the stability of the deadbeat controllerbecomes independent of system parameters.

Fig. 8(a) shows the phase-a steady-state current responseobtained with the conventional deadbeat controller with 20%mismatch in L. In this scenario, the d-axis current command isset to 20 A at t ≥ 0.0167 s, and the q-axis current commandis set to zero. Sustained oscillations in the current response areobvious in Fig. 8(a). These oscillations are indeed the result ofthe instability of the control system. The control loop limiterconstrains the magnitude of these oscillations. Fig. 8(b) showsthe corresponding current spectrum. The total harmonic distor-tion (THD) is 66.12% up to 8.16 kHz (i.e., up to the 136thharmonic). This result does not meet the IEEE Standard 1547requirement of THD [27], which is below 5%. Fig. 8(c) showsthe phase-a steady-state current response obtained with the pro-posed current controller with 20% mismatch in L. It is clear thatthe algorithm is stable and the output current tracks its refer-ence command precisely. The corresponding current spectrumis given in Fig. 8(d). In this case, the THD up to 8.16 kHz is0.95%.

B. Dynamic Performance of Proposed Estimation Unit

To evaluate the performance of the proposed adaptive esti-mation unit, uncertainties in the interfacing parameters as 60%

mismatch in L and 50% mismatch in R are considered. Fig. 9depicts the dynamics of the proposed adaptive estimation unit.In this scenario, the d-axis current command is set to 20 A att ≥ 0.0167 s, and the q-axis current command is set to zero.Fig. 9(a) shows phase-a current response with the proposed cur-rent controller. It can be seen that the actual current tracks itsreference trajectory precisely with zero steady-state error, zeroovershoot, and with a rise time around 280 µs. Fig. 9(b) showsthe estimated grid phase voltage. The estimate converges to itsreal value within 2.5 ms. Fig. 9(c) shows the estimated interfac-ing resistance. The estimate smoothly converges in less than 2ms to the expected value 1.5 Ω. Fig. 9(d) shows the estimatedinterfacing inductance. The estimate smoothly converges in lessthan 2 ms to 4.15 mH that is the total inductance seen by theinverter.

The control performance is examined with the nominal inter-facing inductance and 50% reduction in R, which yields 100%increase in the power circuit time constant. The current controlperformance is shown in Fig. 10. Stable control performancewith high power quality is preserved due to the self-tuningcontrol.

C. Power Control

To evaluate the performance of the overall control system, astep change in the demanded active power from 0 to 5.2 kWis given at t ≥ 0.2 s, while the reference reactive power is setto zero to maintain a unity power factor. Fig. 11(a) shows thereference and actual active power. It can be seen that the activepower output of the inverter can be correctly estimated and theactual power follows its reference correctly. Fig. 11(b) shows thereactive output power. The reactive power is well regulated toits reference value. Fig. 11(c) shows the reference d-axis currentcomponent and the output phase-a current. The actual currenttracks its reference trajectory precisely with zero steady-stateerror and zero overshoot. Because the injected power is only


active, the output current appears in phase with the estimatedgrid voltage, as shown in Fig. 11(d).

The reported results indicate that the proposed scheme resultsin an adaptive grid-voltage sensorless operation and robust cur-rent and power tracking responses even under the occurrence oflarge uncertainties in the interfacing parameters.

VI. CONCLUSION

In this paper, an adaptive grid-voltage sensorless controlscheme for inverter-based DG has been presented. An adap-tive grid-interfacing model has been designed to estimate, inreal time, the interfacing parameters seen by the inverter andthe grid-voltage vector simultaneously. A reliable solution tothe present nonlinear estimation problem is presented by com-bining a grid-voltage estimator with an interfacing parameterestimator in a parallel structure. The estimated quantities havebeen utilized within the inner high-bandwidth current controlloop and the outer power controller to realize an adaptive grid-voltage sensorless interfacing scheme. Theoretical analysis andsimulation results have been provided to demonstrate the va-lidity and usefulness of the proposed interfacing scheme. Theproposed adaptive grid-interfacing model is independent of thetype of the current controller; therefore, it can be used to en-hance the robustness of the existing controllers. In addition, theproposed grid-voltage sensorless interfacing scheme is inher-ently self-commissioning/self-tuning and guarantees high con-trol performance, without the constraint conditions and detailedprior knowledge of the interfacing parameters at the PCC.

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Yasser Abdel-Rady Ibrahim Mohamed (M’06)was born in Cairo, Egypt, on November 25, 1977. Hereceived the B.Sc. (with honors) and M.Sc. degreesin electrical engineering from Ain Shams University,Cairo, in 2000 and 2004, respectively, and the Ph.D.degree in electrical engineering from the Universityof Waterloo, Waterloo, ON, Canada, in 2008.

He is currently a Research Associate in the De-partment of Electrical and Computer Engineering,University of Waterloo. His current research interestsinclude distributed energy resources interfacing and

control, high-performance motor drive systems for aerospace actuators, and ro-bust and adaptive control theories and applications.

Dr. Mohamed is an Associate Editor of the IEEE TRANSACTIONS ON INDUS-TRIAL ELECTRONICS.


Ehab F. El-Saadany (M’01–SM’05) was born inCairo, Egypt, in 1964. He received the B.Sc. andM.Sc. degrees in electrical engineering from AinShams University, Cairo, in 1986 and 1990, respec-tively, and the Ph.D. degree in electrical engineer-ing from the University of Waterloo, Waterloo, ON,Canada, in 1998.

He is currently an Associate Professor in the De-partment of Electrical and Computer Engineering,University of Waterloo. His current research inter-ests include distribution system control and opera-

tion, power quality, power electronics, digital signal processing (DSP) applica-tions to power systems, and mechatronics.

Magdy M. A. Salama (S’75–M’77–SM’98–F’02)received the B.Sc. and M.Sc. degrees from Cairo Uni-versity, Cairo, Egypt, and the Ph.D. degree from theUniversity of Waterloo, Waterloo, ON, Canada, in1971, 1973, and 1977, respectively, all in electricalengineering.

He is currently a Professor and the UniversityResearch Chair in the Department of Electrical andComputer Engineering, University of Waterloo. Hiscurrent research interests include the areas of the op-eration and control of distribution systems, power

quality analysis, artificial intelligence, electromagnetics, and insulation sys-tems. He has consulted widely with government agencies and industrial plants.

Prof. Salama is a Registered Professional Engineer in the Province ofOntario.

Adaptive Grid-Voltage Sensor Less Control Scheme___05075661

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