Virtual-Flux-Based Voltage-Sensor-Less Power Control for ... - NTNU

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 9, SEPTEMBER 2012

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Virtual-Flux-Based Voltage-Sensor-Less Power Control for Unbalanced Grid Conditions Jon Are Suul, Member, IEEE, Alvaro Luna, Member, IEEE, Pedro Rodr´ıguez, Senior Member, IEEE, and Tore Undeland, Fellow, IEEE

Abstract—This paper presents a virtual flux-based method for voltage-sensor-less power control of voltage source converters under unbalanced grid voltage conditions. The voltage-sensor-less grid synchronization is achieved by a method for virtual flux estimation with inherent sequence separation in the stationary reference frame. The estimated positive and negative sequence (PNS) virtual flux components are used as basis for calculating current references corresponding to the following objectives for control of active and reactive powers under unbalanced conditions: 1) balanced positive sequence currents, 2) elimination of doublefrequency active power oscillations, and 3) elimination of doublefrequency reactive power oscillations. For simple implementation and flexible operation, the derived current references are synthesized into one generalized equation where the control objectives can be selected by real coefficients. Since the converter has a limited current capability, a simple, generalized, method for current limitation is also presented with the purpose of maintaining the intended power flow characteristics during unbalanced grid faults. The proposed strategies for virtual flux-based voltage-sensor-less operation have been investigated by simulations and laboratory experiments, verifying the expected performance of active and reactive power control with different objectives. Index Terms—Control of active and reactive powers, three-phase voltage source converters, unbalanced grid voltage, virtual flux, voltage-sensor-less control.

I. INTRODUCTION VER the past decades, there has been a continuous growth in the use of three-phase voltage source converters (VSCs) for grid-connected power conversion systems [1], [2]. The established applications of VSCs cover a wide power range and include, among others, variable speed wind turbines and other

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Manuscript received September 26, 2011; revised December 28, 2011; accepted February 26, 2012. Date of current version May 15, 2012. This work was supported in part by the Project ENE2011-29041-C02-01 funded by the Spanish Ministry of Science and Innovation. Recommended for publication by Associate Editor V. Staudt. J. A. Suul was with the Department of Electric Power Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway. He is now with SINTEF Energy Research, 7465 Trondheim, Norway (e-mail: [email protected]). A. Luna is with the Department of Electrical Engineering, Technical University of Catalonia, 08034 Barcelona, Spain (e-mail: [email protected]). P. Rodr´ıguez is with the Department of Electrical Engineering, Technical University of Catalonia, 08034 Barcelona, Spain, and also with the Electrical Engineering Division, Abengoa Research, E-41014 Seville, Spain (e-mail: [email protected]). T. Undeland is with the Department of Electric Power Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2012.2190301

distributed generation systems, regenerative loads, energy storage systems, high-voltage dc-transmission, and various types of compensation devices. In an increasing number of these applications, the VSCs are required to be capable of operating during grid voltage disturbances [3]–[5]. The performance of control systems designed for balanced three-phase conditions will, however, degenerate if the grid voltage becomes unbalanced [6]. One of the main challenges for unbalanced operation of threephase VSCs is to achieve both fast and accurate grid synchronization. The simple and well-known synchronous reference frame (SRF) phase-locked loop (PLL) can, however, not achieve satisfactory operation during unbalanced conditions without significantly reducing the bandwidth, as demonstrated by the results presented in [7]. As reviewed in [8], several solutions to overcome this problem have been suggested, for instance, by adding digital filters in the traditional SRF PLL structure, or by applying various techniques for separating PNS components in synchronous or stationary reference frames. The performance of the inner control loops, i.e., the current control loop in a traditional cascaded control structure, must also be ensured under unbalanced conditions. This is usually achieved by either implementing separate current controllers in the PNS SRFs or by applying current controllers designed for operating in the stationary reference frame [9]–[11]. With satisfactory performance of the current controllers, the operational characteristics of a converter under unbalanced conditions will mainly be influenced by the objectives and techniques used for calculating the current references. In early publications related to control of VSCs during unbalanced conditions, the focus was mainly on avoiding second harmonic oscillations in the active power flow of the converter, and by that reducing or eliminating oscillations in the dc-link voltage [9], [10], [12], [13]. In other cases, the priority has been to assure balanced sinusoidal currents from the converter, independently of the grid voltage unbalance. With the increased use of VSCs in renewable energy systems and the emergence of grid codes requiring capability for delivering reactive power to the grid during voltage disturbances, other control objectives have become relevant [3], [14]. Several recent publications have, therefore, presented generalized discussions on how to derive current references corresponding to different objectives for control of active and reactive powers during unbalanced conditions, as well as comparative studies investigating converter operation with various control objectives [5], [14]–[20]. The available studies of generalized and flexible power control strategies for VSCs during unbalanced conditions are based on utilizing measured voltages to fulfill the specified control

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objectives, and similar studies have not been conducted for VSCs operating in voltage-sensor-less mode. This paper will, therefore, present a voltage-sensor-less approach and will derive the equations needed for flexible control of active and reactive powers under unbalanced conditions based on the concept of virtual flux (VF). The analysis will be based on the VF estimation method proposed in [21] and will result in VF-based active and reactive power control strategies, analogous to the voltage-based strategies presented in [15]–[19].

II. VOLTAGE-SENSOR-LESS GRID SYNCHRONIZATION UNDER UNBALANCED GRID VOLTAGE CONDITIONS In parallel to the development of control strategies for operation under unbalanced voltage conditions, there has been increasing focus on voltage-sensor-less control of VSCs, motivated by cost reduction, increased modularity in hardware and software and, in some cases, by increased reliability [22]–[30]. Among the methods for voltage-sensor-less operation, the concept of “flux-based” or “VF-based” control, applied and discussed in various ways in [26]–[29], has become well known due to its simplicity and easy interpretation in analogy to the flux of electrical machines.

A. Review of Voltage-Sensor-Less Grid Synchronization and VF-Based Control Under Unbalanced Conditions Considering the relatively large number of publications discussing voltage-sensor-less operation of VSCs, it is remarkable that few studies have until now explicitly considered voltagesensor-less grid synchronization and control under unbalanced grid voltage conditions. The majority of the available publications related to unbalanced operation are based on mathematical observers, as discussed in [31]–[33], without considering the inherent simplicity of the VF concept. The first study of VF-based voltage-sensor-less control during unbalanced conditions was based on direct power control (DPC) with space vector modulation in combination with a slow PLL for tracking the positive sequence of the estimated VF [34]. Later, a double decoupled synchronous reference frame PLL has been used to track the PNS components of the estimated VF [35], [36]. A method based on notch filters and low-pass filters in the positive sequence SRF, to cancel the influence from negative sequence voltage components on the estimation of the positive sequence VF, has also been recently presented [37]. The first study of VF-based voltage-sensor-less control that considered explicit estimation of both PNS VF components in the stationary reference frame was presented in [38]. A simpler method for PNS VF estimation, with improved transient response and explicitly frequency-adaptive operation, was later presented and analyzed in [21]. However, these studies did not present any generalized analysis of how to derive current references corresponding to specific objectives for voltage-sensorless control of active and reactive powers under unbalanced conditions.

B. Basic Principles of VF Estimation for Voltage-Sensor-Less Grid Synchronization The concept of “VF” is based on the voltage integral in (1) and can be applied to processing of voltage measurements, [39]–[41], or as an estimation method for voltage-sensor-less control of VSCs [27], [28], [38], [42] (1) Ψ = V dt + Ψ0 . Considering the simple system in Fig. 1, the voltage Vf at the grid side of the filter inductor is given by (2), expressed in the stationary αβ reference frame Vf ,α β = Vc,α β − R1 · Ic,α β − L1 ·

dIc,α β . dt

(2)

Estimating the converter output voltage from the modulation index mr ef , used as reference signal for the pulse width modulation (PWM) algorithm, multiplied by the dc-link voltage VDC , and applying the voltage integral from (1) to (2), the estimated VFs at the filter terminals can be expressed by (3) [28], [38]. Using base values as defined by (4), the per unit expression for the grid side VF is given by (5) 1 mref ,α β · VDC − R1 · Ic,α β dt Ψf ,α β = 2 − L1 · Ic,α β Vb = Vˆphase ,

(3) Vb,DC = 2 · Vb ,

Ψb =

ψf ,α β = ωb

Vb ωb

(4)

(mref ,α β · vDC − r1 · ic,α β ) dt

− l1 · ic,α β .

(5)

VF estimation based on ideal integration according to (5) will be sensitive to drift and saturation of the estimated values. The integral part of the VF estimation is, therefore, usually implemented by digital filters designed to emulate integration by having phase and amplitude characteristics resulting in 90◦ phase shift and unity gain for fundamental frequency signals [28], [38], [42]. Such filter-based methods can, however, be sensitive to grid frequency variations. In this paper, a method for explicitly frequency-adaptive, filter-based, VF estimation designed to achieve inherent sequence separation under unbalanced conditions, as proposed and analyzed in [21], will be applied. C. VF Estimation With Inherent Sequence Separation The applied method for VF estimation is based on utilization of a second-order generalized integrator (SOGI) configured as a quadrature signal generator (QSG) according to [43], as a generic, frequency-adaptive, building block. 1) Basic Properties of SOGI-QSG for VF Estimation: The structure of the SOGI-QSG is shown to the lower left of Fig. 2, where it can be seen how the fundamental angular frequency ω of the grid is used as an explicit input to keep the structure frequency adaptive. The figure also indicates how the SOGI-QSG

SUUL et al.: VIRTUAL-FLUX-BASED VOLTAGE-SENSOR-LESS POWER CONTROL FOR UNBALANCED GRID CONDITIONS

Fig. 1.

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Basic system under consideration.

Fig. 2. Structure of frequency-adaptive DSOGI-based Virtual Flux (DSOGI-VF) estimation with inherent Sequence Separation, designed on basis of the SOGI-QSGs from [43] and the FLL from [45].

has two output signals, where the signal v , usually referred to as the direct output signal, is a bandpass-filtered version of the input signal v, according to the transfer function given in (6). The second output signal from the SOGI-QSG, qv , is usually labeled as the quadrature output signal since it is given by the integral of v according to the transfer function in (7) and will therefore be a 90◦ phase-shifted version of v for fundamental frequency sinusoidal signals. These properties were utilized for stationary frame sequence separation of voltage measurements when the SOGI-QSG was proposed in [43] but are also corresponding to the phase and amplitude characteristics required for VF estimation as discussed in [21] kε · ω · s v (s) = 2 v (s) s + kε · ω · s + ω 2 ω v (s) kε · ω 2 qv (s) = · = 2 . v (s) s v (s) s + kε · ω · s + ω 2

(6)

Considering (7) and the structure of the SOGI-QSG, it can be understood that the quadrature output signal qv corresponds to the VF integral of the bandpass-filtered signal v , scaled by the per unit grid angular frequency, as indicated in (8). According to [21], this signal can be defined as the “frequency-scaled VF,” labeled χ as given by (8), and will be a convenient basis for VF-based grid synchronization and control 1 1 · ωb · · v = ωpu · ψ = χ. (8) qv = ω · · v = ωpu s s ψ

By applying the definition from (8), estimation of the frequency-scaled VF at the filter terminals in Fig. 1 can be expressed in the Laplace domain by kε · ω 2 χf ,α β (s) = 2 s + kε · ω · s + ω 2 · [mref ,α β (s) · vDC − r1 · ic (s)]

(7)

− ωpu · l1 · ic,α β (s) .

(9)

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2) Structure for VF Estimation With Inherent Sequence Separation: By utilizing both of the output signals from two SOGIQSGs, it is possible to achieve VF estimation with inherent sequence separation in the stationary αβ reference, as demonstrated in [21]. The output signals from the α-axis and β-axis SOGI-QSGs are then used as basis for positive and negative sequence calculation (PNSC) according to the theory of symmetrical components in the time domain [21], [43], [44]. The resulting structure is shown in Fig. 2, and the expressions for the PNS VF components based on the outputs of the two SOGIQSGs are given by

−

+

χα χα 1 χα + vβ 1 χα − vβ = , = . (10) 2 χβ − vα 2 χβ + vα χ+ χ− β β This method is labeled as DSOGI-VF estimation, since it is based on two parallel SOGI-QSGs, and the main advantage compared to other available methods for VF estimation under unbalanced conditions is that delays originating from cascaded VF estimation and sequence separation are avoided [21]. As shown by the block diagram in Fig. 2, sequence separation of the current measurements is necessary for estimating the individual PNS VF components at the point of common coupling in Fig. 1, and this requires two additional SOGI-QSGs in the structure. The sequence separation of the currents is, however, operated in parallel to the estimation of the PNS VF components, based on the same configuration of SOGI-QSGs and will, therefore, not slow down the dynamic response of the VF estimation [21]. In Fig. 2, a frequency-locked loop (FLL), with the structure and tuning proposed by Rodriguez [45], is used to track the grid frequency. The frequency estimation of this FLL is based on the error signals of the SOGI-QSGs. Gain normalization based on the amplitude of the estimated positive sequence VF and the estimated angular frequency is included in the structure to obtain a transient response that is independent of the grid voltage conditions. However, any other method for frequency tracking could be used, like for instance, a traditional SRF-PLL operating on the estimated positive sequence VF component. 3) Transient Response and Steady-State Accuracy of the DSOGI-VF Estimation: From the block diagram in Fig. 2, it can be found that the dynamic response of the VF estimation will be dominated by the second-order transfer function of the SOGIQSG. According to traditional √ approximations for second-order systems, a gain constant kε = 2, corresponding to a critically damped response of (6) and (7), should result in a rise time of approximately a quarter of a fundamental frequency period (5 ms for a 50-Hz system) and a settling time of abut 1 period (20 ms in a 50-Hz system). Simulations and experiments presented in [21] have verified the validity of this tuning and the corresponding transient response. It should be noted that this transient response achieved by DSOGI-VF estimation for voltage-sensor-less grid synchronization is the same as for grid synchronization by sequence separation of voltage measurements according to [43], and similar to the response of other well-known methods for sequence separation like the delayed signal cancellation [46]. From the structure in Fig. 2, it can also be seen that the steady-state accuracy of the DSOGI-VF estimation will depend on the accuracy of the available parameters for the inductance l1

and the equivalent resistance r1 of the filter inductor. Inaccurate values of these parameters will cause deviations in the phase and amplitude of the estimated PNS VF signals but will not introduce any stability problems to the DSOGI-VF estimation method itself. The consequence of inaccurate VF estimation will, therefore, mainly be to introduce steady-state deviations in the active and/or reactive power flow control at the point of synchronization to the grid. This is, in principle, the same effect as for VF-based grid synchronization under balanced conditions, as analyzed in [42]. 4) Initialization and Start-Up of DSOGI-VF Estimation: Initialization and start-up of the converter, the control system, and the estimation method needed for grid synchronization is a general problem in case of voltage-sensor-less operation of VSCs. A common approach is to estimate the phase angle and frequency of the grid voltage by presampling techniques and then predict values that can be used to initialize the grid synchronization method and the control loops at the time when switching of the converter is enabled [24], [38]. The same approach can be used to initialize the values of the integrators of the SOGI-QSGs used for VF estimation in Fig. 2. Detailed investigations of start-up procedures are, however, considered outside the scope of these investigations, and a simple approach is, therefore, applied by considering a control system with inner loop current controllers in the stationary reference frame. If the current controllers are fast enough to prevent overcurrents at the time of enabling the converter switching, and if the converter dc-link is precharged to a voltage above the peak value of the line voltages, the converter can then be started with zero as current reference, without requiring any information from the grid synchronization method. With the converter in operation with zero current, the DSOGI-VF estimation will converge quickly so that the rest of the control system can be enabled.

III. VF-BASED CURRENT REFERENCE CALCULATION FOR ACTIVE AND REACTIVE POWER CONTROL UNDER UNBALANCED GRID VOLTAGE CONDITIONS The following sections will present the foundation for power calculations based on PNS VF components and a systematic derivation of current references corresponding to different objectives of active and reactive power control.

A. Active and Reactive Powers Expressed by VF The starting point for the following derivations will be the formulation of instantaneous active and reactive powers, well known from the work of Akagi et al. [47]. Only three-phase three-wire systems will be considered, and vector notation based on the representation introduced in [48] and [49], according to the conventions applied in [15]–[19], will be used for the derivations. The active and reactive power components are then expressed by (11) and (12), respectively, where “•” denotes the inner-product while “×” represents the cross-product of two vectors. The vector operation “⊥” indicates lagging of 90◦ with


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respect to the original vector p = vα · iα + vβ · iβ = v · i

(11)

q = vβ · iα − vα · iβ = − |v × i| = v⊥ · i.

(12)

Starting from the definition of VF from (1), the voltage vector needed for the power equations of (11) and (12) can be found by differentiation according to (13). As supported by the results and discussions presented in [27] and [35], sufficient accuracy of power calculations is usually achieved by neglecting the derivative of the VF vector amplitude, as given by the approximation shown in (13). Utilizing the definition of the frequency-scaled VF according to (8), and introducing the results from (13) into (11) and (12), the active and reactive powers can be expressed by (14) and (15) d |ψ| d d |ψ| · ej ω t = + jω · ψ v= ψ= dt dt dt ⎡ ⎤ d |ψ|

⎢ dt − ω · ψβ ⎥ α −ω · ψβ −χβ ⎢ ⎥ =⎢ = = −χ⊥ ⎥≈ ω · ψα χα ⎣ d |ψ| ⎦ + ω · ψ α dt β

(13) p ≈ χα · iβ − χβ · iα = −χ⊥ · i

(14)

q ≈ χα · iα + χβ · iβ = χ · i.

(15)

Considering sinusoidal but unbalanced three-phase voltages and currents at the fundamental frequency, the active and reactive powers can be expressed by the PNS components of the voltages and currents as given by (16) and (17), respectively [15], [16]. The two last terms in these equations are oscillating power components at twice the fundamental frequency, as indicated by the annotation in the equations, originating from the interaction between PNS currents and voltages p = v · i = v + + v − · i+ + i− − − + − − + = v+· i+ + v · i + v · i + v · i

P+

P−

P

p+ −

p −+

(16)

p˜2 ω

+ + − · i + i− + v⊥ q = − |v × i| = v⊥ · i = v⊥ + · i+ + v − · i− + v + · i− + v − · i+ . = v⊥ ⊥ ⊥ ⊥

Q+

Q

Q−

q+ −

q −+

(17)

q˜2 ω

For expressing these power equations by the estimated PNS VF components, it has to be considered that the VF is lagging the voltage by 90◦ in the direction of rotation, while the vector operation “⊥” will be applied in the fixed, stationary, αβ reference frame. A vector diagram illustrating the relationships between the PNS voltage vectors, the corresponding VF vectors, and their orthogonal components is given in Fig. 3, and the relationship between the voltage vectors and the PNS VF components is given by (18). The resulting expressions for calculating the active and reactive powers, when neglecting the derivative terms,

Fig. 3. Conventions and orientations used for power calculations based on voltage and VF.

are then given by (19) and (20) − v = v+ + v− ≈ −χ+ ⊥ + χ⊥ + − v ⊥ = v⊥ + v⊥ ≈ χ+ + χ − + + − p = v · i ≈ −χ⊥ + χ− ⊥ · i +i

(18)

+ + − − − − + = −χ+ ⊥ · i + χ⊥ · i −χ⊥ · i + χ⊥ · i

P+

P−

P

p+ −

p −+

(19)

p˜2 ω

q = v⊥ · i = χ+ − χ− · i+ + i− = χ+ · i + − χ − · i − + χ + · i − − χ − · i + .

Q+

Q

Q−

q+ −

q −+

(20)

q˜2 ω

The active and reactive powers of (19) and (20) can be calculated from the PNS VF and current components, available directly from the DSOGI-VF estimation in Fig. 2. Moreover, it was verified in [21] that the DSOGI-VF estimation has the same transient response and frequency characteristics as DSOGI-based sequence separation of measured voltages. Active and reactive power calculations according to (21) and (22), as well as control strategies derived by starting from these equations, will therefore yield similar dynamic response as voltage-based calculations and derivations according to the discussions in [15]–[19], as long as the VF and current components are estimated by the DSOGI-VF structure from Fig. 2. B. Active and Reactive Power Control Objectives Under Unbalanced Conditions For derivation of current references corresponding to different power control objectives, the active and reactive powers from (19) and (20) should be expressed on basis of the active and reactive current components. Considering that reactive current components will cause zero average active power flow, while active current components will produce zero average reactive power, the resulting active and reactive power equations are

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given by (21) and (22), respectively + − − + − p = v · i ≈ −χ+ ⊥ + χ⊥ · i p + i p + i q + i q

These three objectives can be applied separately when deriving current references for controlling the active and reactive power flow, although it can be seen from (21) and (22) that the active and reactive power flows under unbalanced conditions + + + + − − − − + − − + = −χ⊥ · ip +χ⊥ · ip −χ⊥ · ip +χ⊥ · ip −χ⊥ · iq +χ⊥ · iq are not inherently decoupled. +− −+ +− −+ P− From (21) and (22), it can also be seen that in total six distincP+ p˜p p˜p p˜q p˜q tive components of active and reactive powers are defined, while P p˜2 ω , p p˜2 ω , q there are only four controllable current components, given by p˜2 ω the PNS components of the active and reactive currents. Thus, (21) only four of the different active and reactive power terms can be controlled at the same time. Assuming that the average active − + − q = v⊥ · i = χ + − χ − · i + p + ip + iq + iq and reactive powers should always be controlled, there will be + + − − + − − + + − − + = χ · iq −χ · iq + χ · iq −χ · iq + χ · ip −χ · ip . only two degrees of freedom, which can be used to select the objectives for control of the active and reactive power flow. In Q− Q+ q˜+ − q˜q−+ q˜p+ − q˜p−+ case of nonzero references for both average active and average q reactive powers, any of the three different control objectives deQ q˜2 ω , q q˜2 ω , p fined previously can, therefore, only be achieved as long as the q˜2 ω same objective is applied for both the active and reactive power (22) controls.

Thus, the active and reactive powers can be expressed by one constant term corresponding to the average active/reactive power flow and two oscillating terms at twice the grid frequency, as well known from voltage-based power calculations [14]. The annotations in the equations also indicate how the active current components corresponding to a specific average active power flow during unbalanced conditions can result in a doublefrequency oscillating term p˜2ω ,p in the active power flow as well as a double-frequency oscillating term q˜2ω ,p in the reactive power flow. Similarly, the reactive current components corresponding to a specific average reactive power flow during unbalanced conditions can result in a double-frequency oscillating term q˜2ω ,q in the reactive power as well as a double-frequency oscillating term p˜2ω ,q in the active power flow. From (21), it can also be understood that controlling average active power P with balanced, positive sequence, three-phase currents will require that the negative sequence active current components are controlled to zero. By properly controlling the will cancel PNS active current components so that the term p˜+− p , it can also be possible to eliminate the active the term p˜−+ p power oscillations q˜2ω ,p caused by the average active power flow P during unbalanced conditions. Another option can be to control the negative sequence active current components with the purpose of eliminating the reactive power oscillations q˜2ω ,p caused by the flow of average active power. However, when requiring sinusoidal currents, the double-frequency oscillations in both the active and reactive powers caused by the average power flow P cannot be eliminated simultaneously, as discussed in [14]–[17], and [20]. The same line of considerations can also be followed for the control of the reactive current components corresponding to a specific flow of average reactive power Q. Thus, the following three objectives for control of active or reactive power flow under unbalanced conditions can be defined: O1) operation with balanced positive sequence currents; O2) elimination of double-frequency active power oscillations; O3) elimination of double-frequency reactive power oscillations.

C. Current Reference Calculation for Active Power Control Current references for control of average active power according to the three defined control objectives will be derived separately and labeled with subscripts O1–O3 according to the numbering above. 1) Active Power Control With Balanced Positive Sequence Currents: The active current reference i∗ p, O 1 resulting in average active power flow equal to the reference value P∗ , while controlling balanced three-phase currents, can be found from (21) by requiring the negative sequence active current component to be zero. The resulting current reference expressed by the positive sequence VF is given by i∗p,O 1 =

P∗ + −χ⊥ . |χ+ |2

(23)

This equation for current reference calculation can be considered as the VF-based equivalent to the voltage-based strategy for active power control labeled balanced positive sequence control (BPSC) in [15] and [16]. 2) Active Power Control With Elimination of Corresponding Double-Frequency Active Power Oscillations: The doublefrequency active power oscillations caused by a specified flow of average active power can be eliminated by imposing a reference value of zero for the oscillating power component, p˜∗2ω ,p as given by (24). The negative sequence current reference components can then be expressed by (25). Substituting (25) into the expression for the average active power flow P in (21), the positive sequence active current reference can be expressed by the reference value P∗ as given by (26) ∗− − ∗+ p˜∗2ω ,p = −χ+ ⊥ · i p + χ⊥ · i p = 0

i∗− p = i∗+ p =

− χ+ ⊥ · χ⊥

|χ+ |2

|χ+ |2

i∗+ p

P∗ . · −χ+ ⊥ 2 − |χ− |

(24) (25) (26)


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The total active current reference i∗p,O 2 for elimination of the double-frequency active power oscillations caused by the average active power reference P∗ can be found by introducing (26) into (25) and calculating the sum of the PNS current references as given by

oscillations, this control strategy can be considered as the VFbased equivalent to the voltage-based active power control strategy labeled as average-active-reactive control (AARC) in [15] and [16].

∗− i∗p,O2 = i∗+ p + ip

D. Current Reference Calculation for Reactive Power Control

=

|χ+ |2

+ P∗ − 2 −χ⊥ − χ⊥ , − − |χ |

+ χ χ− . (27)

Since the negative sequence active current reference is calculated to compensate for the active power oscillations generated by the positive sequence active current component, (27) can be considered as the VF-based equivalent to the voltage-based active power control strategy labeled PNSC in [15] and [16]. It should also be noted that this equation will only be valid as long as the positive sequence VF component is significantly larger than the negative sequence component. This is as expected, since equal amplitudes of PNS VF components will correspond to “single-phase” operating conditions, where nonzero active power flow cannot be achieved without double-frequency active power oscillations. 3) Active Power Control With Elimination of Corresponding Double-Frequency Reactive Power Oscillations: The doublefrequency reactive power oscillations caused by a specified flow of average active power can be eliminated by imposing a reference value of zero for the oscillating reactive power component ∗ q˜2ω ,p as given by (28). The negative sequence current reference can then be expressed by the positive sequence current reference as given by (29) ∗ + ∗− − ∗+ q˜2ω ,p = χ · ip − χ · ip = 0

i∗− p

χ+ · χ− + = i . |χ+ |2

(28) (29)

By substituting the expression from (29) into the expression for the average active power flow P in (21), and by using the vector property of (30), the positive sequence current reference required for the average active power flow to be equal to the reference value P∗ can be derived to be given by (31) − + χ− · χ+ ⊥ = −χ⊥ · χ

i∗+ p =

(30)

∗

|χ+ |2

P · −χ+ ⊥. + |χ− |2

(31)

The total current reference required for following the active power reference P∗ while fulfilling the objective of eliminating double-frequency reactive power oscillations is given by P∗ + ∗− − −χ (32) + i = + χ i∗p,O 3 = i∗+ p p ⊥ . ⊥ |χ+ |2 + |χ− |2 ≈v

It can be noticed that the active current reference vector in this case will follow a steady-state vector trajectory with amplitude proportional to the trajectory of the grid voltage. If the reactive current component is zero, this characteristic will correspond to zero instantaneous reactive power flow. By controlling the average active power flow while eliminating the reactive power

The same approach as presented for the active power control can be followed for the derivation of reactive current reference equations, by applying the same three control objectives. 1) Reactive Power Control With Balanced Positive Sequence Currents: By imposing the condition that the negative sequence reactive current component should be zero, the reactive current reference resulting in average reactive power flow equal to the reference Q∗ can be found from the expression of the average reactive power flow in (22). The resulting reactive current reference i∗q ,O 1 is given by (33) and can be considered as the VF-based equivalent to the voltage-based reactive power control strategy labeled BPSC in [15] and [17] i∗q ,O1 =

Q∗ + χ . |χ+ |2

(33)

2) Reactive Power Control With Elimination of Corresponding Double-Frequency Active Power Oscillations: Reactive power control with the objective of eliminating the corresponding double-frequency active power oscillations p˜∗2ω ,q requires that the condition specified by (34) is fulfilled. By following the same line of derivation as described for the case of active power control, the resulting expression for reactive current reference calculation can be derived to be given by (35) ∗− − ∗+ p˜∗2ω ,q = −χ+ ⊥ · iq + χ⊥ · iq = 0 ∗− i∗q ,O2 = i∗+ q + iq =

∗

|χ+ |2

Q + |χ− |2

(34)

χ+ + (−χ− ) .

(35)

≈v ⊥

In this case, the average reactive power is controlled while the current reference is calculated to follow a vector trajectory that is proportional to the trajectory of the orthogonal voltage v ⊥ . If the active power reference is zero, this condition will correspond to zero active power flow. This control strategy can, therefore, be considered as the VF-based equivalent to the voltage-based reactive power control strategy labeled as AARC in [15] and [17]. 3) Reactive Power Control With Elimination of Corresponding Double-Frequency Reactive Power Oscillations: Reactive power control with the objective of eliminating the correspond∗ ing double-frequency reactive power oscillations q˜2ω ,q requires the condition specified by (36) to be fulfilled. By following the same steps as described for case of active power control, the reactive current reference can be derived to be given by (37) ∗ + ∗− − ∗+ q˜2ω ,q = χ · iq − χ · iq = 0

(36)

∗− i∗q ,O3 = i∗+ q + iq

=

Q∗ (χ+ − (−χ− )), |χ+ |2 − |χ− |2

|χ+ | |χ− |. (37)

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As the negative sequence reactive current in this case is used to compensate for the reactive power oscillation caused by the positive sequence component, this strategy can be considered as a VF-based equivalent to the voltage-based reactive power control strategy labeled PNSC in [15] and [17]. As for active power control with elimination of double-frequency active power oscillations, this equation can only be applied as long as the positive sequence VF component is significantly larger than the negative sequence component. E. Generalized Expressions for Current Reference Calculation Considering the expressions for the active current reference given in (23), (27), and (32), and the expressions for the reactive current reference given in (33), (35), and (37), it can be found that it is possible to develop generalized expressions that include the three presented objectives and any gradual transition between them. By introducing a gain constant that is limited in the range between −1 and 1 for weighting the contribution from the negative sequence VF component on the final current reference, the corresponding expressions for the active and reactive current references are given in (38) and (39), respectively. These expressions can be established intuitively from the described control strategies, but can also be derived in a similar way as described for voltage-based control of active and reactive power flow in [18] and [19] ∗− i∗p = i∗+ p + ip =

P∗ , · −χ+ + kp χ− ⊥ ⊥ 2 + kp |χ− | + χ χ− for kp = −1 (38)

|χ+ |2

− 1 ≤ kp ≤ 1,

+ Q∗ − , 2 2 · χ − kq · χ + − |χ | + kq |χ | − 1 ≤ kq ≤ 1, χ+ χ− for kq = −1. (39)

∗− i∗q = i∗+ q + iq =

From these expressions, it is clearly seen that selecting a gain constant kp equal to −1 will result in (27), corresponding to active power control under objective O2 with elimination of corresponding double-frequency active power oscillations. Selecting kp equal to 0 corresponds to O1 with the current reference equation simplifying to (23) for achieving positive sequence balanced three-phase currents, while selecting kp equal to 1 corresponds to O3 and will result in (32). Similarly, reactive current reference calculation with the gain constant kq equal to −1 corresponds to O3 and will result in (37), kq equal to 0 corresponds to O1 and will simplify the current reference calculation to (33), while kq equal to 1 corresponds to O2 and will result in (35). The total current reference with nonzero reference values for both average active and reactive powers is given by + P∗ − i∗ = i∗p + i∗q = 2 2 · −χ⊥ + kp χ⊥ + − |χ | + kp |χ | ∗

· χ+ − kq · χ− ,

Q + kq |χ− |2 − 1 ≤ kq ≤ 1, χ+ χ−

+

|χ+ |2

−1 ≤ kp ≤ 1,

for kp , kq = −1. (40)

The objectives of the current reference calculation can be selected independently for the active and reactive components by specifying the gain factors kp , and kq to 0, −1, or 1, or any value between −1 and 1. From the previous discussions, it should, however, be clear that simultaneous control of both average active and average reactive powers with balanced three-phase currents will only be achieved if both kp and kq are 0. Similarly, elimination of double-frequency active power oscillations will only be possible if kp = −1 while kq = 1, and elimination of double-frequency reactive power oscillations can only be achieved if kp = 1 while kq = −1. F. Generalized Expressions for Active and Reactive Power Flow Characteristics Assuming that the currents of the converter are equal to the reference values, it is possible to analyze the active and reactive power flow characteristics resulting from different power control objectives in a generalized way. This can be achieved by introducing the generalized current reference equations from (38) and (39) back into the active and reactive power equations from (21) and (22). The expressions for the total active and reactive powers derived by following this approach are given by (41) and (42), respectively, where φ+− is the phase angle between the PNS VF components p = P ∗ + p˜2ω ,p + p˜2ω ,q p˜2 ω

= P∗ + +

P ∗ · (1 + kp ) · |χ− | · |χ+ | · cos 2ω · t + φ+− 2 2 |χ+ | + kp · |χ− |

Q∗ · (1 − kq ) |χ− | · |χ+ | · sin 2ω · t + φ+− 2 2 |χ+ | + kq · |χ− |

(41)

q = Q∗ + q˜2ω ,q + q˜2ω ,p q˜2 ω

= Q∗ + −

Q · (1 + kq ) · |χ− | · |χ+ | · cos 2ω · t + φ+− 2 2 + − |χ | + kq · |χ | ∗

P ∗ · (1 − kp ) · |χ− | · |χ+ | · sin 2ω · t + φ+− . (42) 2 2 + − |χ | + kp · |χ |

The implications of these equations are summarized in Table I, where the first two rows show the expressions for the amplitude of double-frequency active and reactive power oscillations when either Q∗ or P∗ is zero. In case of nonzero values of both P∗ and Q∗ , the oscillating active and reactive power components are given by the square sums shown in the last row of the table. G. Current Reference Calculation for Operation Under Current Limitation Converter operation with current reference calculation according to the generalized equations presented in Section III-E will result in average active and/or reactive power flow according to the reference values and power flow characteristics according to the control objectives specified by kp and kq . In case of severe


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TABLE I OVERVIEW OF ACTIVE AND REACTIVE POWER FLOW CHARACTERISTICS FOR DIFFERENT OPERATING CONDITIONS

voltage drops, this approach can result in current references exceeding the current capability of the converter. A simple method for limiting the current references while maintaining the control objectives specified by kp and kq , can, however, be developed by accepting that the average active and/or reactive power flow have to be reduced when the active and/or reactive current references reach a specified limitation. Starting from the generalized active current reference in (38), the maximum vector amplitude is given by (43). From this expression, the maximum average active power transfer Plim within the current limitation ip ,lim can be expressed by (44) as a function of the control objective specified by kp and the amplitudes of the PNS VF components, χ+ and χ− . By substituting Plim from (44) into (38), the active current reference when operating under current limitation can be found as given by (45) ∗ i p =

|χ+ |2

|P ∗ | · χ+ + |kp | · χ− , 2 + kp · |χ− |

− 1 ≤ kp ≤ 1

(43)

− 2 ip,lim · Sign(P ) + 2 χ , · χ = + + k · p |χ | + |kp | · |χ− | ∗

Plim

− 1 ≤ kp ≤ 1 i∗p,lim =

(44)

ip,lim · Sign (P ) + · −χ⊥ + kp χ− ⊥ , |χ+ | + |kp | · |χ− | ∗

− 1 ≤ kp ≤ 1.

(45)

From (44) and (45), it can be seen that these expressions will be generally valid for all operating conditions without the limitation described for (27) and (38). Considering a fault condition with χ+ = χ− and a kp = −1, it can however be seen from (44) that the maximum average active power flow in such a situation will be equal to zero. Thus, the control objective of eliminating double-frequency active power oscillations can be maintained, on cost of the average active power flow being reduced to zero under “single-phase” operating conditions. The same approach can also be followed for imposing a current limitation on the reactive current reference calculation. The maximum average reactive power transfer Qlim can then be ex-

pressed by (46), and the resulting current reference equation for operation under reactive current limitation is given by (47) Qlim =

iq ,lim · Sign(Q∗ ) · (|χ+ |2 + kq |χ− |2 ), −1 ≤ kq ≤ 1 |χ+ | + |kq | · |χ− | (46)

i∗q ,lim =

iq ,lim · Sign (Q∗ ) + · χ − kq · χ− , −1 ≤ kq ≤ 1. + − |χ | + |kq | · |χ | (47)

Limitation of the active and reactive current components according to these equations can be implemented by selecting (45) and/or (47) as the source of the current reference inputs to the current controllers when the current vector amplitudes resulting from (38) and/or (39) exceeds the active and/or reactive current limitations specified by ip ,lim and iq , lim . IV. SIMULATION STUDY OF ACTIVE POWER CONTROL WITH CURRENT LIMITATION UNDER SEVERE UNBALANCED CONDITIONS To verify the validity of the developed strategies for current reference calculation and current limitation, and to illustrate the converter operation under severe unbalanced conditions, a simple configuration similar to the structure shown in Fig. 1 has been simulated by the PSCAD/EMTDC software. For simplicity, the converter was simulated by an average model neglecting the PWM operation of the converter and connected to an ideal, controllable, voltage source Vg at the grid side of the filter inductor while an ideal dc voltage source was connected to the dc-link capacitor. A. Case Description and Operating Conditions For the presented simulations, the converter rating is specified to be 2.23 MVA at 690-VRM S line voltage, although all variables will be presented in per unit values. The PNS VF components corresponding to the grid voltage Vg are estimated by the DSOGI-VF structure from Fig. 2, and the estimated signals are used as basis for calculating the current references according to the equations in Sections III.E and III.G. The resulting current references in the stationary αβ reference frame are used

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Fig. 4.


Simulation results illustrating active power control with different control objectives during single-phase faults as well as the influence of current limitation.

as input signals to a set of proportional-resonant (PR) current controllers. As a starting point for the simulations, the converter is operated under balanced grid voltage conditions with 0.5 p.u. as the active power reference P∗ , and with a limitation of the active current component given by ip ,lim = 1.0 p.u. The average reactive power reference Q∗ is in this case set to zero. The converter operation corresponding to different power control objectives is then investigated in case of a single-phase grid fault with zero remaining voltage in phase b.

B. Simulation Results The main results from the simulations are presented in Fig. 4. The three-phase grid voltages are shown in Fig. 4(a), and it can be seen that the voltages are balanced with 1.0 p.u. amplitude until a fault resulting in zero remaining voltage in phase b occurs at t = 0. The estimated PNS VF components χ+ and χ− are shown in Fig. 4(b) and (c), respectively. When the fault occurs, the amplitude χ+ is reduced from 1.0 to 0.5 p.u., while χ− is increased from 0 to 0.5 p.u., with the expected dynamic response.


Fig. 5.

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Overview of laboratory setup.

To illustrate the full range of possible control objectives and operating conditions, the value of kp is changed during the simulation, as shown in Fig. 4(d). At the beginning of the simulation, kp is equal to 1, corresponding to objective O3 of eliminating double-frequency reactive power oscillations caused by the average active power flow. The resulting currents are plotted in Fig. 4(e), where it can be seen that zero current is injected in the faulted phase as long as kp = 1. The active and reactive powers calculated by (11) and (12) are shown in Fig. 4(f). From these curves, it can be seen that a double-frequency active power oscillation with amplitude of 0.5 p.u. occurs when the single-phase fault occurs. The reactive power flow is, however, kept at zero, after a small transient, as long as kp = 1. The average active and reactive powers, corresponding to P and Q in (19) and (20), calculated from the PNS VF and current components, are also plotted in Fig. 4(g), and it can be seen how the average active power flow remains at 0.5 p.u. after a small transient. From the time t = 0.04 s, the control parameter kp is ramped down toward zero. From Fig. 4(e), it can be seen how this results in increasing current in phase b, until three-phase balanced currents occur when kp reaches zero. As the current in phase b is increasing, double-frequency reactive power oscillations starts to occur, as shown by Fig. 4(f), but the average value of the reactive power flow is still kept at zero. For operation with balanced three-phase currents, both the active and reactive power flow has double-frequency oscillations with amplitude equal to P∗ , corresponding to the expressions given in the first row of Table I. When the control parameter kp is reduced from 0 toward −1, the vector amplitude of the active current reference resulting from (38) exceeds the specified limit of 1.0 p.u. Thus, current reference calculation with current limitation according to (45) is activated. From Fig. 4(e), it can be seen how the maximum current is limited to 1.0 p.u., while Fig. 4(e) and (f) shows how the average active power flow is reduced as kp is approaching −1. For operation with kp = −1, the control objective of eliminating double-frequency active power oscillations is then achieved by reducing the average active power flow to zero. The converter is, however, providing maximum short-circuit current in the faulted phase. The reactive power flow resulting from the current injection in the faulted phase still maintains the average value of 0 but has a double-frequency oscillating component with amplitude of 0.5 p.u. The amplitudes of the oscillations can also be verified by substituting the expression for the max-

TABLE II DETAILS OF LABORATORY SETUP

imum power flow under current limitation from (44) into the expression given in the first row of Table I. V. EXPERIMENTAL VERIFICATION OF VF-BASED POWER CONTROL UNDER UNBALANCED CONDITIONS As the idealized simulation results presented in the previous section indicate validity of the developed strategies for VF-based current reference calculation under unbalanced grid conditions, the same approach has been tested in a small-scale laboratory setup. The experimental results are presented in the following sections to verify the feasibility of VF-based voltage-sensorless grid synchronization and power control under unbalanced conditions. A. Description of the Laboratory Setup The laboratory setup used for the presented experiments is outlined in Fig. 5. Power to the converter dc-link was supplied by a diode rectifier with an adjustable transformer, and an LC filter was used on the ac side as the interface between the converter and a grid emulator with the ability to generate balanced and unbalanced voltage sags. The main parameters of the system are listed in Table II. The structure and the main parts of the control system are outlined in Fig. 6. All the necessary functionality was implemented in Simulink/MATLAB and operated on the dSPACE DS1103 platform used to control the converter. Measured values and

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Fig. 6.


Overview of control system implemented in laboratory setup.

internal signals of the control system were logged through the dSPACE interface for postprocessing and plotting by MATLAB. The experiments were carried out in a laboratory environment with a significant background distortion in the grid voltage. As the simple PR current controllers used in these experiments achieve a high gain only at the grid frequency, they are sensitive to grid voltage distortions. The PR controllers are in this case also operated with relatively low proportional and resonant gains to avoid causing oscillations in the LC filter, since active damping of filter oscillations have not been considered. Feed-forward terms based on measured voltages for relieving the current controllers have neither been applied, since the converter is operated in voltage-sensor-less mode. Additionally, the converter is operated with relatively low margin between the dc-link voltage and the grid voltage. Therefore, the current waveforms in the presented results will show significant levels of both low-frequency distortions, as well as higher frequency disturbances caused by the background distortion in the grid voltage. Since these experiments are intended as a general proof-ofconcept, no further effort has, however, been made in this case to improve the current control of the converter during distorted voltage conditions. Thus, the resulting distortions in the voltages and currents are influencing the active and reactive power components calculated from voltages and current measurements and will also challenge the performance of the applied method for VF estimation and current reference calculation. Successful grid synchronization with accurate estimation of fundamental frequency PNS VF components, and corresponding calculation of sinusoidal current references, can therefore be interpreted as a verification of robust performance of the proposed approach for voltage-sensor-less control under unbalanced conditions.

B. Experimental Results With Different Strategies for Current Reference Calculation In the following sections, results illustrating the performance of VF-based voltage sensor-less power control for unbalanced conditions are presented and discussed for five selected cases with different active and reactive power references and con-

trol objectives. All results were logged when the same voltage sag, corresponding to a single-phase voltage drop of about 50% under no-load conditions, was imposed by the voltage sag generator. The active and reactive current limitations where set to high values so that the current reference calculation was based directly on (40) as indicated in Fig. 6. 1) Active Power Control With Balanced Three-Phase Currents (kp = 0): Results from operation with an active power reference P∗ of 1.0 p.u. while kp is set to zero are shown in Fig. 7. As can be seen from Fig. 7(a) and (b), the threephase currents remain balanced when the unbalanced drop in the grid voltage occurs. The current amplitude is, however, increased to maintain the average active power injected to the grid. Fig. 7(c) and (d) shows that the estimated PNS VF components have a fast, but well damped, response to the unbalanced voltage drop, as expected from the discussion in Section II.C.3. It can also be noticed that the influence of the harmonic distortion of the currents and voltages are significantly attenuated in the VF signals, since the sequence separation and VF estimation have a filtering effect for frequency components above the fundamental grid frequency, as discussed in [21]. The current references shown in Fig. 7(e) are, therefore, sinusoidal signals, indicating that the distortions observed in the measured currents are caused by the current controllers and not by the VF-based grid synchronization and reference current calculation. Fig. 7(f) shows the active and reactive powers calculated from measurements by using (11) and (12), and Fig. 7(g) shows the powers calculated from the PNS VF and current components according to (19) and (20). Inspecting these plots, it can be verified that the average power is maintained when the unbalanced voltage dip occurs, while second harmonic oscillations appear in both active and reactive powers as expected. From the plotted power signals, it can be noticed that there is an influence from the distortions in the grid voltages and currents both before and during the voltage dip. These distortions also influence the active and reactive powers so that the second harmonic oscillations during the unbalanced conditions are not perfectly sinusoidal. As expected, the distortions have a larger influence on the active and reactive powers plotted in Fig. 7(f)


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Fig. 8. Results from experiment with P∗ = 1.0 p.u., Q∗ = 0 p.u. and current reference calculation for eliminating double-frequency active power oscillations (kp = −1).

Fig. 7. Results from experiment with P∗ = 1.0 p.u., Q∗ = 0 and active current reference calculation for achieving balanced three-phase currents (kp = 0).

than on the curves in Fig. 7(g), but the powers calculated from the measurements and from the estimated PNS VF components are corresponding reasonably well. Since the results in Fig. 7 has illustrated the performance of the VF estimation and current reference calculations, only measured voltages and currents together with active and reactive powers calculated from (19) and (20) will be shown in the following sections to illustrate how the different specified power control objectives are achieved. 2) Active Power Control With Elimination of DoubleFrequency Active Power Oscillations (kp = −1): Fig. 8 shows the results with an active power reference P∗ of 1.0 p.u. when kp is set to −1 with the purpose of eliminating double-frequency active power oscillations. From the plots of the voltages and currents, it can be seen that the converter is injecting the largest current in the phase with the lowest voltage. Comparing to Fig. 7, it can also be seen that the voltages at the filter capacitors during the unbalanced conditions are influenced by the unbalanced current injection due to the significant grid impedance of the

Fig. 9. Results from experiment with P∗ = 1.0 p.u., Q∗ = 1.0 p.u. and current reference calculation for eliminating double-frequency reactive power oscillations (kp = 1).

voltage sag generator. Observing the resulting active and reactive powers during the unbalanced conditions, it can be noticed that the objective of eliminating double-frequency active power oscillations is achieved, while the reactive power oscillations are almost doubled in amplitude compared to Fig. 7. 3) Active Power Control With Elimination of DoubleFrequency Reactive Power Oscillations (kp = 1): For the results in Fig. 9, the active power reference P∗ is still specified to 1.0 p.u., but kp is set to 1 when calculating the current references, with the intention of eliminating double-frequency oscillations in the reactive power flow. Thus, the lowest current is injected in the phase with the lowest voltage, corresponding to an “impedance-like” distribution of the currents between the three phases.

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Fig. 10. Results from experiment with P∗ = 1.0 p.u., Q∗ = 1.0 p.u. and active and reactive current reference calculation for elimination of double-frequency active power oscillations (kp = −1, kq = 1).

Studying the plot of the reactive power flow, it can be seen that the second harmonic oscillations during the unbalanced conditions are almost completely eliminated. In this case, there is a significant influence from the distortion in the voltages and currents, but the active and reactive powers calculated from the sequence separated currents and the estimated VF are still corresponding well to the specified control objective. 4) Active and Reactive Power Control With Elimination of Double-Frequency Active Power Oscillations (kp = −1, kq = 1): Fig. 10 shows results from a case when the active and reactive power references are specified as P∗ = 1.0 p.u. and Q∗ = 1.0 for injection of equal amounts of active and reactive powers into the grid, while the current references are calculated to eliminate double-frequency active power oscillations. Studying the measured voltages and currents, it can be seen that this results in maximum current injection in the phase with the lowest voltage, and since the reactive power reference has a nonzero value, the currents are phase-shifted with respect to the voltages. The plots of the active and reactive powers show that the objective of eliminating second harmonic oscillations in the active power is well achieved, although the distortions in the currents and voltages are causing some disturbances. However, these distortions are not undermining the fundamental frequency behavior and the overall control objectives under investigation. 5) Active and Reactive Power Control With Elimination of Double-Frequency Reactive Power Oscillations (kp = −1, kq = −1): A last example of control objectives during unbalanced conditions is shown in Fig. 11. In this case, the active and reactive power references are still specified as P∗ = 1.0 p.u. and Q∗ = 1.0, while the current references are calculated to eliminate second harmonic oscillations in the reactive power flow. Thus, the results show how the lowest current is injected into the phase with the lowest voltage, while the currents are phase-shifted with respect to the voltages due to the nonzero reactive power reference. The lower plot shows how the second

Fig. 11. Results from experiments with P∗ = 1.0 p.u., Q∗ = 1.0 p.u., and active and reactive current reference calculation for elimination of double-frequency reactive power oscillations (kp = 1, kq = −1).

harmonic oscillations are eliminated from the reactive power and instead appear in the active power. Although the significant distortions in the voltages and currents occurring in this case are reflected to some extent in the active and reactive power flow, the overall control objectives are still achieved. C. Summary of Experimental Results Some of the information that can be extracted from the presented experimental results is summarized in Table III. From this table, it can also be noticed that the amplitudes of the active and reactive power oscillations calculated from the expressions in Table I are corresponding reasonably well with the active and reactive powers plotted in the presented figures. The estimated PNS VF amplitudes listed in Table III also indicate how the inductance of the grid emulator is influencing the voltage or the VF at the grid side of the filter inductor differently depending on the active and reactive power references and control objectives. As expected, reactive power injection to the grid is increasing the positive sequence VF component both before and during the voltage sag, as can be seen from Cases 4 and 5. In Case 4, it can, however, be noticed that the negative sequence reactive currents resulting from the objective of eliminating double-frequency active power oscillations is reducing the amplitude of the negative sequence VF component. Thus, the double-frequency reactive power oscillations are actually reduced in amplitude compared to Case 2. For Case 5, the influence of the negative sequence reactive currents is the opposite, causing an increase in the amplitude of the negative sequence VF component and, therefore, also an increase in the amplitude of the double-frequency active power oscillations. D. Further Implications of the Presented Results Although the presented results have shown only a few selected cases, the VF-based current references from Section IIIE, as well as (41) and (42), are valid for any combinations of


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TABLE III SUMMARY OF EXPERIMENTAL RESULTS

active and reactive power references and control objectives. By introducing the active and reactive current limitation strategies defined by (45) and (47), the power control objectives specified by kp and kq can also be maintained under generalized unbalanced conditions, including single-phase faults. By the results from simulations and experiments, it has also been verified that the transient response of the presented approach for grid synchronization and current reference calculation is fast and well damped when utilized for active and reactive power control under unbalanced conditions. This is mainly due to the DSOGI-VF estimation that achieves VF estimation and sequence separation with a response time in the range of 5 ms, which is similar to the performance of conventional sequence separation methods based on voltage measurements. Thus, the presented approach for voltage-sensor-less operation under unbalanced conditions does not imply additional delays as usually associated with sensor-less control strategies. The derived expressions for VF-based current reference calculation can be easily adapted for and investigated together with other types of current controllers than the simple PR-controllers applied for the presented simulations and experiments. Thus, the poor performance of the current controllers observed in the experimental results is not a limitation with respect to the applicability of the developed approach for VF-based active and reactive power control. It can also be noted that the active and reactive power flow characteristics described by (41) and (42) can be used as a starting point for specifying active and reactive power references in control structures based on the concept of DPC discussed in [28] and [34]. Thus, the same range of flexible power control objectives under unbalanced conditions as discussed for control structures with inner loop current controllers could be achieved with the DPC concept. VF estimation by using the DSOGI-VF structure from [21] as a basis for implementing power control strategies derived from (21) and (22) can, therefore, be considered a general approach that can be combined with different types of control strategies and applied for voltage-sensor-less operation of VSCs in a wide range of applications.

VI. CONCLUSION This paper has proposed a general approach for VF-based voltage-sensor-less control of VSCs under unbalanced grid voltage conditions. A method for VF estimation with inherent sequence separation in the stationary reference frame is used to estimate PNS VF components at the point of connection to the grid. On this basis, expressions for current reference calculation corresponding to three different objectives of active and reactive power control have been derived and analyzed. The derived current reference equations have been synthesized into generalized equations where the objectives of achieving balanced three-phase currents, elimination of double-frequency active power oscillations, or elimination of double-frequency reactive power oscillations can be selected separately for the active and reactive current reference components by specifying the value of two scalar coefficients. Under severe grid fault conditions, current references calculated directly from active and/or reactive power references can exceed the current capability of the converter. A simple strategy for limiting the active and/or reactive current references within the converter’s current capability, while at the same time maintaining the specified power control objectives, has, therefore, been developed. By applying this strategy for current limitation, the presented approach for VF-based voltage-sensor-less control will be valid for operation under generalized unbalanced conditions, including single-phase fault conditions, for any combinations of active and reactive power references and control objectives. To verify the derived expressions for current reference calculation, VF-based voltage-sensor-less operation of a VSC under unbalanced conditions has been investigated by simulations and laboratory experiments. The presented results indicate general validity of the proposed approach for derivation of VF-based power control strategies, which can potentially be adapted to different control system configurations and utilized for voltagesensor-less operation under unbalanced conditions in a wide range of applications.

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ACKNOWLEDGMENT The authors would like to thank Prof. M. Molinas of the Norwegian University of Science and Technology (NTNU) for initiating and facilitating the cooperation between NTNU and the Technical University of Catalonia (UPC). The authors would also like to thank the Ph.D. students and researchers of the Research Center on Renewable Electrical Energy Systems at UPC for providing equipment and facilitating the experimental verification of the presented concept during a one-month research stay of J. A. Suul in Terrassa, Spain, in November 2010. REFERENCES [1] S. Chakraborty, B. Kramer, and B. Kroposki, “A review of power electronics interfaces for distributed energy systems toward achieving lowcost modular design,” Renewable Sustainable Energy Rev., vol. 13, no. 9, pp. 2323–2335, Dec. 2009. [2] J. R. Rodr´ıguez, J. W. Dixon, J. R. Espinoza, J. Pontt, and P. Lezana, “PWM regenerative rectifiers: State of the art,” IEEE Trans. Ind. Electron., vol. 52, no. 1, pp. 5–22, Feb. 2005. [3] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus, “Overview of control and grid synchronization for distributed power generation systems,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1398–1409, Oct. 2006. [4] L. Xu, B. R. Andersen, and P. Cartwright, “VSC transmission operating under unbalanced AC conditions: Analysis and control design,” IEEE Trans. Power Del., vol. 20, no. 1, pp. 427–434, Jan. 2005. [5] Y. Suh, Y. Go, and D. Rho, “A comparative study on control algorithms for active front-end rectifier of large motor drives under unbalanced input,” IEEE Trans. Ind. Appl., vol. 47, no. 3, pp. 1419–1431, May/Jun. 2011. [6] A. Sannino, M. H. J. Bollen, and J. Svensson, “Voltage tolerance testing of three-phase voltage source converters,” IEEE Trans. Power Del., vol. 20, no. 2, pp. 1633–1639, Apr. 2005. [7] V. Kaura and V. Blasko, “Operation of a phase locked loop system under distorted utility conditions,” IEEE Trans. Ind. Appl., vol. 33, no. 1, pp. 58– 63, Jan./Feb. 1997. [8] M. Boyra and J.-L. Thomas, “A review on synchronization methods for grid-connected three-phase VSC under unbalanced and distorted conditions,” in Proc. 14th Eur. Conf. Power Electron. Appl., Birmingham, U.K, Aug. 30–Sep. 1, 2011, pp. 1–10. [9] H.-S. Song and K. Nam, “Dual current control scheme for PWM converter under unbalanced input voltage conditions,” IEEE Trans. Ind. Electron., vol. 46, no. 5, pp. 953–959, Oct. 1999. [10] Y. Suh and T. A. Lipo, “Control scheme in hybrid synchronous stationary frame for PWM AC/DC converter under generalized unbalanced operating conditions,” IEEE Trans. Ind. Appl., vol. 42, no. 3, pp. 825–835, May/Jun. 2006. [11] A. Timbus, M. Liserre, R. Teodorescu, P. Rodriguez, and F. Blaabjerg, “Evaluation of current controllers for distributed power generation systems,” IEEE Trans. Power Electron., vol. 24, no. 3, pp. 654–664, Mar. 2009. [12] P. N. Enjeti, P. D. Ziogas, and M. Ehsani, “Unbalanced PWM converter analysis and corrective measures,” in Proc. Conf. Record 1989 IEEE Ind. Appl. Soc. Annu. Meet., San Diego, CA, Oct. 1–5, 1989, vol. 1, pp. 861– 870. [13] P. Rioual, H. Pouliquen, and J.-P. Louis, “Regulation of a PWM rectifier in the unbalanced network state using a generalized model,” IEEE Trans. Power Electron., vol. 11, no. 3, pp. 495–502, May 1996. [14] R. Teodorescu, M. Liserre, and P. Rodr´ıguez, Grid Converters for Photovoltaic and Wind Power Systems. New York: Wiley, 2011. [15] P. Rodr´ıguez, A. V. Timbus, R. Teodorescu, M. Liserre, and F. Blaabjerg, “Independent PQ control for distributed power generation systems under grid faults,” in Proc. 32nd Annu. Conf. IEEE Ind. Electron. Soc., Paris, France, Nov. 6–10, 2006, pp. 5185–5190. [16] P. Rodriguez, A. V. Timbus, R. Teodorescu, M. Liserre, and F. Blaabjerg, “Flexible active power control of distributed power generation systems during grid faults,” IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2583– 2592, Oct. 2007. [17] P. Rodr´ıguez, A. Timbus, R. Teodorescu, M. Liserre, and F. Blaabjerg, “Reactive power control for improving wind turbine system behavior under grid faults,” IEEE Trans. Power Electron., vol. 24, no. 7, pp. 1798– 1801, Jul. 2009.

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[39] M. Weinhold, “A new control for optimal operation of a three-phase voltage dc link PWM converter,” in Official Proc. 19th Int. Intell. Motion Conf., Nürnberg, Germany, Jun. 25–27, 1991, pp. 371–383. [40] M. C. Chandorkar, D. M. Divan, and R. Adapa, “Control of parallel connected inverters in standalone ac supply systems,” IEEE Trans. Ind. Appl., vol. 29, no. 1, pp. 136–143, Jan./Feb. 1993. [41] J. L. Duarte, A. van Zwam, C. Wijnands, and A. Vandenput, “Reference frames fit for controlling PWM rectifiers,” IEEE Trans. Ind. Electron., vol. 46, no. 3, pp. 628–630, Jun. 1999. [42] R. Pöllänen, “Converter-flux-based current control of voltage source PWM rectifiers: Analysis and implementation,” Ph.D. dissertation, Lappeenranta Univ. Technol., Lappeenranta, Finland, Dec. 2003. [43] P. Rodriguez, R. Teodorescu, I. Candela, A. V. Timbus, M. Liserre, and F. Blaabjerg, “New positive-sequence voltage detector for grid synchronization of power converters under faulty grid conditions,” in Proc. 37th Annu. IEEE Power Electron. Spec. Conf., Jun. 18–22, 2006, pp. 1–7. [44] W. V. Lyon, Transient Analysis of Alternating-Current Machinery. Cambridge, MA: MIT Press, 1954. [45] P. Rodriguez, A. Luna, I. Etxeberria, R. Teodorescu, and F. Blaabjerg, “A stationary reference frame grid synchronization system for three-phase grid-connected power converters under adverse grid conditions,” IEEE Trans. Power Electron., vol. 27, no. 1, pp. 99–112, Jan. 2012. [46] J. Svensson, M. Bongiorno, and A. Sannino, “Practical implementation of delayed signal cancellation method for phase-sequence separation,” IEEE Trans. Power Del., vol. 22, no. 1, pp. 18–26, Jan. 2007. [47] H. Akagi, E. Watanabe, and M. Aredes, Instantaneous Power Theory and Applications to Power Conditioning. New York: Wiley, 2007. [48] F. Z. Peng and J.-S. Lai, “Generalized instantaneous reactive power theory for three-phase power system,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 293–297, Feb. 1996. [49] M. Depenbrock, V. Staudt, and H. Wrede, “A theoretical investigation of original and modified instantaneous power theory applied to four-wire systems,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 1160–1167, Jul./Aug. 2003.

Jon Are Suul (M’11) received the M.Sc. and the Ph.D. degrees from the Department of Electric Power Engineering at the Norwegian University of Science and Technology, Trondheim, Norway, in 2006 and 2012, respectively. In 2006, he was employed by SINTEF Energy, Trondheim, where he was engaged in the simulation of power electronic systems. In 2007, he was granted a leave of absence and returned to NTNU for studying towards the Ph.D. degree. Since 2012, he has resumed his position in SINTEF Energy. In 2008, he was a guest Ph.D. student for two months with the Energy Technology Research Institute of the National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan. He was also a visiting Ph.D. student for two months with the Research Center on Renewable Electrical Energy Systems, within the Department of Electrical Engineering, Technical University of Catalonia (UPC), Terrassa, Spain, during 2010. His research interests are mainly related to control of power electronic converters in power systems and for renewable-energy applications. Alvaro Luna (S’07–M’10) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2001, 2005, and 2009, respectively. He joined the faculty of UPC, in 2005, where he is currently an Assistant Professor. His research interests include wind turbine control, photovoltaic systems, integration of distributed generation, and power conditioning. Dr. Luna is a member of the IEEE Power Electronics Society, the IEEE Industrial Electronics Society, and the IEEE Industry Applications Society.

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Pedro Rodr´ıguez (S’99–M’04–SM’10) received the M.Sc. and Ph.D. degrees in electrical engineering from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1994 and 2004, respectively. He was a Postdoctoral Researcher at the Center for Power Electronics Systems, Virginia Tech, Blacksburg, in 2005, and in the Department of Energy Technology, Aalborg University (AAU) in 2006. He joined the faculty of UPC as an Assistant Professor in 1990, where he became the Director of the Research Center on Renewable Electrical Energy Systems in the Department of Electrical Engineering. He was also a Visiting Professor at the AAU from 2007 to 2011, acting as a cosupervisor of the Vestas Power Program. He still lectures Ph.D. courses at the AAU every year. Since 2011, he has been the Head of Electrical Engineering Division, Abengoa Research, Seville, Spain, although he is still with the UPC as a Part-Time Professor. He has coauthored one book and more than 100 papers in technical journals and conference proceedings. He is the holder of seven licensed patents. His research interests include integration of distributed generation systems, smart grids, and design and control of power converters. Dr. Rodr´ıguez is a member of the administrative committee of the IEEE Industrial Electronics Society (IES), the general chair of IEEE-IES Gold and Student Activities, the Vice-Chair of the Sustainability and Renewable Energy Committee of the IEEE Industry Application Society, and a member of the IEEE-IES Technical Committee on Renewable Energy Systems. He is an Associate Editor of the IEEE TRANSACTION ON POWER ELECTRONICS.

Tore Undeland (M’86–SM’92–F’00) was born in Bergen, Norway, in 1945. He received the M.Sc. and Ph.D. degrees from the Norwegian University of Science and Technology (NTNU), Trondheim, Norway, in 1970 and 1977, respectively. Since 1970, he has been with NTNU, where he has been a Full Professor since 1984. He has also served as an Adjunct Professor with Chalmers University of Technology, Gothenburg, Sweden, since 2000, and as a Scientific Advisor to SINTEF Energy, Trondheim. His research interests include power electronics and wind energy systems. He is a coauthor of the well-known textbook Power Electronics: Converters, Applications, and Design (New York: Wiley, 1989, 1995, and 2003). Dr. Undeland has served as the President of the European Power Electronics and Drives Association and is a member of the Norwegian Academy of Technological Sciences.

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