Review and improvements to the measurements of the VSC impedance transfer matrix

Review and improvements to the measurements of the VSC impedance transfer matrix Journal: IEEE Transactions on Power Delivery Manuscript ID TPWRD-00568-2023.R1 Manuscript Type: Transactions Date Submitted by the Author: 10-Nov-2023 Complete List of Authors: Pedra, Joaquin; Universitat Politecnica de Catalunya, Eng. Electrica Sainz, Luis; ETSEIB , UPC, Department of Electrical Engineering Monjo, Lluís; Universitat Politecnica de Catalunya, Department of Electrical Engineering Technical Topic Area : Harmonics and power quality < Transmission and Distribution Key Words: Impedance measurement, Converters, Stability IEEE PES Transactions on Power Delivery 1 Abstract- Impedance/admittance transfer matrix characterization of voltage source converters is an important issue in frequency domain stability studies. Several approaches in the dq- and pn-frame are proposed to obtain this matrix with either experimental or numerical simulation measurements. Although these are well-established approaches, various issues require further study. The paper presents a detailed review and discussion on the dq- and pn-frame approaches for converter impedance/admittance transfer matrix measurements and contributes to clarifying the influence of the mirror frequency effect and the starting point of the FFT window on these measurements. The contributions are validated by PSCAD/EMTDC simulation with a VSC model containing PLL and outer controls. Index Terms— Impedance measurements, converter impedance, voltage source converters. I. INTRODUCTION ver the past decade, the development of power electronics has led to the rising integration of voltage source converters (VSCs) and modular multilevel converters (MMCs) into traditional power systems, e.g., microgrids, distributed generation, renewables and AC/DC links of HVDC grids [1]. Although this integration greatly improves power transmission control and quality, the interaction between converters and traditional grids also brings several technological challenges, such as stability issues of power systems [2], [3]. Stability assessment of power systems with multiple connected converters is a difficult task and is currently drawing increasing interest in electric utilities, transmission system operators, and power electronics manufacturers and researchers. State-space and frequency domain methods are usual techniques to assess stability issues in power systems. Frequency domain methods relying on impedance-based characterization of power systems can be applied using either analytical models or measurements and with less computational effort than the state-space method. Another significant advantage is that, unlike in state-space analysis, changes in one part of the circuit only affect this part. On the other hand, the state-space method is based on the study of the eigenvalues of the system. This allows a more thorough study, This work is part of the project PID2021-123633OB-C33 (J-02895) supported by the MCIN (Proyectos de Generación de Conocimiento). J. Pedra, L. Sainz and Ll. Monjo are with the Dep. of Elect. Eng., UPC, in Av. Diagonal 647, 08028 Barcelona, Spain (mails: joaquin.pedr[email protected]du, [email protected]). and Víctor Balaguer s/n, 08800 Vilanova i La Geltrú, Spain (mail: [email protected]). since the determination of instability is limited to checking the presence of positive real-part eigenvalues, which is easy to automate. The main drawback is that this method requires a detailed description of the circuit and its parameters. In [4], the advantages and disadvantages of both types of methods are compared. At present, frequency domain methods are becoming a powerful tool for stability studies because they overcome state-space drawbacks by frequency domain impedance-based characterization of power systems with multiple connected converters [4] − [7]. Thus, appropriate frequency domain impedance/admittance characterization of power system components, in particular grid-connected converters, is a key task in stability studies [5] − [7]. This frequency domain characterization can be achieved by setting s = j ω in the Laplace- (s-) domain transfer impedance/admittance of white-box analytical converter transfer function models [6]. It is important to keep in mind that, for intellectual property reasons, manufacturers only supply protected models (i.e., black boxes). These cannot be introduced in a state-space model. However, manufacturers often provide the frequency domain impedance/admittance characterization of converters. This characterization can also be obtained by experimental measurements in real converterbased systems or EMT simulation measurements made with PSCAD and MATLAB/Simulink tools in black-box converter models. Many approaches derived from DC impedance measurement methods [9] − [12] for extracting frequency domain impedance/admittance in power systems either with experimental or numerical simulation measurements are proposed in the literature [13] − [34]. In general, these approaches inject an AC small-signal shunt current or series voltage at several frequencies, and subsequently, currents and voltages are measured at the terminals of the studied device in order to calculate its impedance/admittance over a frequency range. Although the above procedure has been widely investigated, several issues, such as the experimental measurement framework [19], the FFT window effect on the terms of the impedance/admittance transfer matrix and the influence of the mirror frequency effect on the mathematical processing of measured current and voltage signals [5], [26], still require further study. Pioneer impedance/admittance measurement studies of three-phase power systems are developed in the abc-frame [13] but need multiple experimental tests due to the couplings between phases. Later studies are classified into three main Review and improvements to the measurements of the VSC impedance transfer matrix J. Pedra, L. Sainz, and Ll. Monjo O Page 1 of 16 IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2 approaches: measurements in the synchronous- (dq-) [14] − [20], sequence- (pn-) [21] − [34] and stationary- ( αβ -) [35] frame. Other methods based on time domain step responses [36] are also proposed in the literature but are out of the scope of this paper. The relationship between the impedance model in the αβreal or αβ-complex frames and the dq-real or dq-complex frames is not studied either because they are not equivalent for unsymmetric systems. By not equivalent, it is understood that a VSC model cannot be transformed from one reference frame to another. In symmetric systems, the impedance matrix satisfies Zdd(s) = Zqq(s) and Zdq(s) = −Zqd(s). As discussed in [8], in “Section IV. Modeling of unsymmetric systems, Subsection A. Linearity and temporal invariance”, when transformed to the αβ-frame, a dq-unsymmetric system described by a linear and time-invariant dq model, y=G(s).u, becomes a linear but time variant system. The same problem appears when transforming an αβ-frame unsymmetric system to the dq-frame. From a physical point of view, it can be observed that they are not equivalent because an αβ-frame unsymmetric system produces positive and negative sequence currents of the same frequency when a pure positive sequence voltage is applied (e.g., in case of unbalanced faults in ac grids [8]). On the other hand, as will be seen in equations (6) and (7), unsymmetric systems cause voltages and currents of different frequencies (see Section III.C). This work focuses on the relationship between the dq-real frame and the dq-complex frame. The latter gives rise to the pn-frame. This paper studies the measurement methods of the converter impedance/admittance transfer matrix in the dq- and pn-frame. The fundamentals of the study are revisited in Section II and the main contributions are presented in the next Sections as follows: • a complete and detailed review and discussion on the measurement methods highlighting their advantages and drawbacks and describing the main concerns about their implementation are presented in Section III; • the FFT window influence on the off-diagonal terms of the pn-frame impedance/admittance transfer matrix is studied in Section IV.A, and the impact of the phase difference between the fundamental frequency component and the perturbation on the measurements is analytically discussed; • the influence of the mirror frequency effect on the measurements of the dq-frame impedance transfer matrix is discussed in Section IV.B; • the influence of the mirror frequency effect on the measurements of the pn-frame admittance transfer matrix is studied in Section IV.C, and an improved procedure addressing the admittance measurements at DC frequencies lower than the fundamental frequency is proposed; • a validation and a detailed comparison of the use of the pnframe versus the dq-frame are presented in Section VI and Appendix I, respectively. The above contributions are validated from a complete VSC model with the PLL, outer controls and DC grid. Frequency analysis verification is performed by PSCAD/EMTDC simulation with FFT frequency domain steady-state analysis. II. FUNDAMENTALS A. Coordinate transformations It is important to emphasize that only normalized or power invariant transformations were used in this work. The normalized Park (dq-real) transformation [8] of a three-phase set of voltages Uabc = [Ua Ub Uc]T is Udq = P(θ)Uabc, where cos( ) cos( 2 3) cos( 2 3) 2 () , sin( ) sin( 2 3) sin( 2 3) 3 P θ θπ θπ θθ θπ θπ −+  = − −− −+  (1) and Udq = [Ud Uq]T is the dq-real space voltage vector Ud and Uq. This voltage is related to the dq-real current vector Idq = [Id Iq]T through the impedance matrix as Udq = Zdq(s)Idq. The variable θ is the transformation angle obtained from the PLL or θ = ω1t if the PLL is not considered, where ω1 = 2πf1 and f1 is the fundamental frequency. The inverse Park transformation becomes Uabc = P(θ)TUdq. The αβ-real space voltage vector Uαβ = [Uα Uβ]T can be obtained from the three-phase set of voltages Uabc by applying the Park (dq-real) transformation with θ = 0; hence Uαβ = P(0)Uabc. The relation between the two transformations P(θ) = G(θ) P(0) is ( ) ( ) ( ) ( ) ( ) cos sen . sen cos G θ θ θ=  −θ θ  (2) The normalized Ku (dq-complex) transformation [10], [11] of a three-phase set of voltages Umdq = [Udq Udq*]T is Umdq = K(θ)Uabc, where 2 2 1 () , 3 jj j j jj eaeae K e a e ae θθ θ θ θθ θ −− −  ⋅⋅ = ⋅⋅   (3) and a = ej·2π/3, Uabc = (K(θ)T)* Umdq and Udq = (Ud + j·Uq)/√2 is the dq-complex space voltage vector. This vector and its conjugate are also the forward and backward dq-complex space vectors (i.e., Uf = Udq and Ub = U*dq) in the literature on electrical machines. Disregarding the 0-sequence, a set of abc voltages (x = u) and currents (x = i) {xa(t), xb(t), xc(t)} = {Xa·cos( ω ·t + φ a), Xb·cos( ω ·t + φ b), Xc·cos( ω ·t + φ c)}, where ω = 2π·f is the angular frequency and f is the frequency, can be decomposed in a sum of p- and n-sequence components [6]: ( ) cos( ) cos( ) ( ) cos( 2 3) cos( 2 3) ( ) cos( 2 3) cos( 2 3), a p pp n nn b p pp n nn c p pp n nn xt X t X t xt X t X t xt X t X t ωφ ωφ ωφ π ωφ π ωφ π ωφ π = ++ + = +− + ++ = ++ + +− (4) where ω p = ω n = ω . The p- and n-sequence complex phasors {Xp = Xp∠ φ p, Xn = Xn∠ φ n} of the pn-sequence sinusoidal components are obtained by applying the symmetrical component transformation (or Fortescue transformation) to the Page 2 of 16IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 3 abc complex phasors {Xa = Xa∠ φ a, Xb = Xb∠ φ b, Xc = Xc∠ φ c} of the abc sinusoidal components [5], i.e., 2 2 23 ( ,) 1 1. 31 j aa aa ae π  =    =      =   ai ii p b n c XX UI XX XX (5) The abc voltages (x = u) and currents (x = i) {xa(t), xb(t), xc(t)} of angular frequency ω can also be transformed by the dq-complex transformation (or Ku transformation) into the dq-complex space vectors as a sum of two p- and n-sequence complex space vectors xdq, p and xdq, n of angular frequencies ω − ω 1 and ω + ω 1 [8]: 1 ,, 11 2 () () * 1(() () ()) 23 33 ( ), 22 pn dq jt ab c jt jt pn x jx x t x ta x ta e ee ω ωω ωω ωω ω ωωω − −−+ + = = ++ =+== dq p dq n dq xx pn x XX   (6) where the constant √3/2 comes from the power−invariant scaling constant (3/2)1/2 in [8], ω 1 = 2πf1 is the fundamental angular frequency of the sinusoidal components with f1 being their fundamental frequency, and X*n is the complex conjugate of the n-sequence complex phasor Xn. In the case at hand, ω p = ω + ω 1 and ω n = ω − ω 1 in (4). Hence, substituting these values in (6) results in * 11 33 22 ( ; ), dq dq jt jt p dq n dq ee ωω ω ω ωω ω ω − = + =+=− dq p n xX X (7) where the angular pulsation of the disturbance corresponds to that of the DC side, ω = ω dq. Also of interest is the expression of d and q components as a function of the positive and negative sequence magnitudes, respectively. ( ) ( ) () ( ) ( ) ( ) 11 3cos cos 2 3sin sin 2 ( ; ), d dq p dq n q dq p dq n p dq n dq x tt x tt ωφ ωφ ωφ ωφ ω ω ωω ω ω = ++ + = +− + =+=− pn pn XX XX (8) The dq-complex space vector xdq in (6) is related to the dqreal space vector xdq = [xd, xq]T as [5], [8], [32], [33] ** 1 11 11 , 1 22 dd qq xx j xx j jj        = =        −−          dq dq dq dq xx xx (9) where x*dq is the complex conjugate of the dq-complex space vector xdq. It must be noted that (9) also applies in the sdomain, with xdq → Xdq(s), x*dq → X*dq(s), xd → Xd(s) and xq → Xq(s). B. Impedance/admittance transfer matrix The dq-real frame Ohm’s law is expressed in the s-domain as () () () () () () (). () () () () m dq dq dq Us Zs Is d dd dq d q qd qq q Us Z s Z s Is Us Z s Z s Is      =                          (10) According to [5], [8], [32], [33], (10) can be transformed into dq-complex frame by applying (9): ( ) ( ) () () () ** ** () () () (), () () () () s ss ss ss ss ss +− −+      =            m mm dq dq dq Z UI dq dq dq dq dq dq dq dq ZZ UI UI ZZ      (11) where U*dq(s) and I*dq(s) are the complex conjugate of the dqcomplex voltage and current space vectors Udq(s) and Idq(s), and (Z+dq(s))* and (Z − dq(s))* are the complex conjugate of the equivalent dq-complex transfer functions. Note that the variable s must be considered real when the complex functions Z+dq(s) and Z − dq(s) in the s-domain are conjugate in (11), i.e., a complex conjugate function of Z(s) = A(s) + jB(s) becomes Z*(s) = A(s) – jB(s) [33]. According to this, the frequency response of a conjugate transfer function can be computed as ( ) ( ) ( ) ( ) * * .Zsj Zs j Asj jBsj ω ω ωω ===− ==−=   (12) The relationships between the impedance matrices in the dqreal frame, Zmdq(s), and the dq-complex frame, Zmdq(s), are ( ) ( ) ** () () () () 1 11 1. () () 1 2 () () dd dq qd qq ss ZsZs j ZsZs j jj ss +− −+       =     −−      dq dq dq dq ZZ ZZ (13) The dq-complex impedances Z+dq(s) and Z − dq(s) are derived from (13) as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () 22 () . 22 dd qq qd dq dd qq qd dq ZsZs ZsZs sj ZsZs ZsZs sj + − +− = + −+ = + dq dq Z Z (14) For symmetric systems, Z − dq(s) = 0. C. Frequency coupling and pn-admittance transfer matrix According to (6), a p-sequence voltage perturbation of angular frequency ω p = 2πfp on the VSC AC side becomes a dq-frame voltage perturbation of angular frequency ω dq = 2πfdq = ω p − ω 1 which, in turn, is reflected in two p- and n-sequence perturbations of angular frequencies ω p and ω n = 2πfn = ω p − 2 ω 1, respectively, in the VSC AC currents. The same is true of an n-sequence voltage perturbation of angular frequency ω n on the VSC AC side which is reflected in two p- and n-sequence perturbations of angular frequencies ω p = ω n + 2 ω 1 and ω n, respectively, in the VSC AC currents. The above coupling between the p- and n-sequence frequencies is called the mirror frequency effect [5]. The best frequency to study this coupling phenomenon between p- and n-sequence frequencies is the dq-frame angular frequency ω dq. Hence, a dq-frame perturbation at angular frequency ω dq Page 3 of 16 IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4 becomes two p- and n-sequence perturbations (4) of angular frequencies ω p = ω dq + ω 1 and ω n = ω dq − ω 1. Their corresponding dq-complex space vectors (6) can be expressed as the sum of two p- and n-sequence complex space vectors of angular frequencies ω dq and − ω dq [8] as , , ,, ** ( ) ( ), dq dq dq dq jt jt jt jt K e e Ke e ω ω ωω −− =+=+ dq p dq n dq p dq n u u ii dq p n dq p n uU U iI I        (15) where K = √3/2, Up = Up∠ φ Up and Ip = Ip∠ φ Ip are the psequence complex voltage and current phasors of the original angular frequency ω dq + ω 1 and U*n, I*n are the complex conjugate of the n-sequence complex voltage and current phasors Un and In of the original angular frequency ω dq – ω 1. Note that the phasors of the original angular frequencies ω dq + ω 1 and ω dq – ω 1 are transformed into the phasors of the angular frequency ω dq when the dq-complex transformation is applied and (15) is obtained. The modified sequence-domain impedance matrix in the frequency domain and its relationship with the dq-real domain impedance matrix (10) are presented in [5] and discussed in [30]. The frequency dependence of the modified sequencedomain impedance matrix is not well solved and, in [5], it is stated that Zpp(ωp), Znn(ωn), Zpn(ωn → ωp) and Znp(ωp → ωn) (see Table I in [5]). The pn-sequence domain impedance matrix is written as ( ) ( ) ( ) ( ) 11 11 () () , () () pp pn pp np nn nn ZsZs Vsj Isj Zs Zs Vsj Isj ωω ωω + +  =    −−    (16) and the admittance matrix as ( ) ( ) ( ) ( ) 11 11 () () . () () pp pn pp np nn nn YsYs Isj Vsj Ys Ys Isj Vsj ωω ωω + +  =    −−    (17) Note that, although the p- and n-sequence are mixed in the dq-real (7) and dq-complex (8) frames, they are clearly separated in (16) and (17). The derivation of (16) from (11) is presented below. By replacing the dq-complex space vectors (15) in the sdomain Ohm’s law (11), the following relationships are obtained: ( ) ( ) , , ,, * ,, * ** , , ,, ** ,, () () () () () () () () () () () () () () () () , s s ss s ss s s s s ss s ss + − − +  += ++   +   += ++   +  dq p dq n dq dq p dq n dq dq p dq n dq p dq n dq dq p dq n dq dq p dq n U U ZI I ZI I U U ZI I ZI I (18) where Udq, p(s), Udq, n(s), Idq, p(s) and Idq, n(s) are the s-domain complex signals obtained from the complex space vectors udq, p, udq, n, idq, p and idq, n in (15), i.e., * ,, () () ( , ). dq dq ss sj sj ωω = = = +− pn dq p dq n XX X X X UI (19) Separate equations corresponding to the complex space vectors of angular frequencies ω dq and − ω dq in (15) (i.e., the sdomain complex signals with the denominators s + j ω dq and s − j ω dq in (19)) must be true separately because (18) must be valid for all values of the Laplace variable s. By considering the complex space vectors of angular frequency ω dq (i.e., the signals with the denominators s + j ω dq in (19)), the following complex transfer matrix function is derived from (18) [8], [32]: ( ) ( ) *, ,, *, () () () * () ** ** , *, () () () () () () () s ss dq dq dq s dq dq dq ss sj sj sj ss sj sj sj s s s ωω ω ω ωω +− −+ +  = +   ++ −    = +     − +−    ⇒=    dq n dq p dq p dq n I UI pp n dq dq U p nn dq dq dq dq dq p dq n UI I ZZ I UI ZZ ZZ U U           ( ) ( ) , ** *, () (), () () () ss s ss − −+         dq p dq n dq dq I I ZZ (20) which can be expressed in the frequency domain by setting s → jωdq as [5], [8], [32]. ( ) ( ) ** () () , () () dq dq dq dq ωω ωω +− −+      =         dq dq pp nn dq dq ZZ UI UI ZZ (21) where (Up and Ip) and (Un and In) are the p- and n-sequence complex phasors of angular frequencies ω dq + ω 1 and ω dq – ω 1, respectively. Note that (Z+dq(ωdq))* = (Z+dq(s))*s = jωdq and (Z − dq(ωdq))* = (Z − dq(s))*s = jωdq, i.e., the conjugate is applied first to the s-domain impedances, and subsequently these impedances are expressed in the frequency domain by setting s → jωdq.. The relationships in (21) are also obtained for the complex space vectors of angular frequencies − ω dq in (18). As a summary, and using a pn-frame notation, we have ( ) ( ) 1 1 ** () () ,( ) () () ,( ) () () () () () () () (). dq dq p dq dq dq n dq dq dq dq dq dq dq dq dq ω ω ωω ω ω ω ωω ω ωωωω ωω ωω +− −+ = +     =    = −     = = = = pp pn p p pp np nn n n nn pp dq pn dq np dq nn dq Z Z UI UI Z Z UI UI ZZZZ ZZ ZZ (22) The same relationships apply for pn-sequence admittance transfer matrices, i.e., 1 1 () () ,( ) . () () ,( ) dq dq p dq dq dq n dq ω ω ωω ω ω ω ωω ω = +     =    = −     pp pn p p pp np nn n n nn Y Y UI IU Y Y UI IU (23) Note that the mirror frequency effect is characterized by the off-diagonal (or coupling) terms of the pnimpedance/admittance transfer matrix (22), (23), which are equivalent to (16) and (17) used in the literature. Page 4 of 16IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 5 III. VSC IMPEDANCE/ADMITTANCE MEASUREMENT METHODS The impedance/admittance measurements of converters can be mainly classified into the dq- and pn-frame approaches. A small number of studies in the αβ -frame approach is also available [35], [36]. The following Subsections present the dq- and pn-frame approaches in detail. A. The dq-frame approach [14] − [19] The dq-frame approach is the first one to be used in the literature for these measurements because most stability studies are performed in this frame. In general, this approach injects two single-frequency linearly independent three-phase current perturbations over a frequency range and obtains the dq-real frame impedance transfer matrix at each frequency with the linear system derived from the dq-real frame Ohm’s law. References related to this dq-frame approach contribute with (i) different procedures to obtain the independent dq-real frame current perturbations [14], [15], [16], and (ii) alternative methods for injecting [17], measuring [18], [19] and analytically handling [19] the perturbations and measurements. The dq-frame approach has the advantage that the derivations of the analytical impedance models are less complicated than in other frames [26] but has several drawbacks related to the measurements, Park transformations, FFT analysis and physical interpretation [26], [30]. All works in the dq-frame focus on the injection of a shunt current perturbation to measure impedance transfer matrix of converters. 1) State of the art Fig. 1(a) shows the flowchart of this approach, which is typically divided into the following steps: (i) two linearly independent three-phase sinusoidal current perturbations (subindexes 1 and 2) are injected at frequency ω k over a frequency range; (ii) the steady-state three-phase voltages of frequency ω k caused by the current injections are measured at the converter terminals; (iii) the three-phase currents and voltages are transformed into the dq-real frame currents and voltages of frequency ω dq, k (6); (iv) the FFT is applied to extract the magnitude and phase values of the dq-real frame current (Idi, Iqi) and voltage (Udi, Uqi) phasors (i = 1, 2) at each frequency ω dq, k; (v) the dq-real frame impedance transfer matrix of the converter in the frequency domain ((10) with s → jωdq) is obtained at each frequency ω dq, k with the linear system derived from the dq-real frame Ohm’s law, i.e., 1 ,, ,, () () . () () dd dq k dq dq k qd dq k qq dq k ZZ ZZ ωω ωω −     =          d1 d2 d1 d2 q1 q2 q1 q2 UUII UUII (24) The impedance (22) / admittance (23) transfer matrices in the pn-frame are obtained from (24) and (13), respectively. Note that the resolution of (24) requires two linearly independent measurements. References related to this dqframe approach present different improvements over the previous steps. In [14], [15], two different angles are used for each single-frequency current injection to obtain the independent dq-frame perturbations. Another alternative method only injects a single-phase current between the phases instead of the three-phase currents [17]. The work in [17] is extended in [18] by implementing a measurement equipment that determines dq-frame impedances using a single-phase wide-bandwidth injection algorithm. In [19], a three-phase impedance analyzer and the corresponding algorithm are constructed for the impedance measurement. Moreover, multiple sweeps at single frequencies are proposed to determine the dq-real frame impedance transfer matrix (24) from larger voltage and current matrices in case of linearly dependent measurements. Recently, [16] correctly obtains the two independent dq-frame current perturbations by injecting p- and n-frame perturbations of frequencies ω dq + ω 1 and ω dq − ω 1, respectively. Finally, in [20] a very detailed description of the DQ scan is made, which proposes to disturb the system with a pure d voltage and a pure q voltage. In this case, the terms of the admittance matrix can be easily determined, i.e., (d) (q) (d) (q) (d) (q) (d) (q) () () () ; () () () () () () ; ( ) () () ii dd dd i dq i qi i d qi qi qd i qq dq qi i d ff ff f f ff ff f ω = = = = II YY U U II YY U U , (25) where fi is the d or q voltage perturbation frequency, Yes [·] −1 Inject current perturbation ω k : i abc1 , i abc2 Measure voltage abc ( ω k ) v abc1 , v abc2 i abc1 , i abc2 dq ( ω dq,k ) u dq1 , u dq2 i dq1 , i dq2 ω k : u abc1 , u abc2 FFT: Extract terms ω dq,k v dq1 , v dq2 i dq1 , i dq2 U d1 , U q1 , U d2 ,U q2 I d1 , I q1 , I d2 ,I q2 Solve (24) to obtain the dq-real frame impedances at ω dq,k ω dq,k > ω dq,kmax ω dq,k = ω dq,k + ∆ ω dq,k No a) (i) (ii) (iii) (iv) (v) Inject voltage perturbation Solve (26) to obtain Y pp ( ω dq, k ) Y nn ( ω dq, k ) ω dq,k > ω dq,kmax ω dq,k = ω dq,k + ∆ ω dq,k b) (i) Yes Measure voltage and current u (p) = v (p) i (p) a a a FFT: u (p) a ω p, k = ω dq, k + ω 1 : v (p) ( ) abc ω n, k = ω dq, k − ω 1 : v (n) ( ) abc u (n) = v (n) i (n) a a a u (p) i (p) abc abc u (n) i (n) abc abc (ii) i (p) u (n) i (n) a a a U p I p U n I n (p) (p) (n) (n) ( ω p, k ) ( ω p, k ) ( ω n, k ) ( ω n, k ) U p I p U n I n (p) (p) (n) (n) ( ω p, k ) ( ω p, k ) ( ω n, k ) ( ω n, k ) I n (p) ( ω n, k ) I p (n) ( ω p, k ) U p I p U n I n (p) (p) (n) (n) ( ω p, k ) ( ω p, k ) ( ω n, k ) ( ω n, k ) I n (p) ( ω n, k ) I p (n) ( ω p, k ) U n (p) ( ω n, k ) U p (n) ( ω p, k ) (iii) Obtain the pn-frame admittances at ω dq,k No coupling Coupling Z g ≠ 0 Zg = 0 No Solve (27) Solve (28) (iv) G NCY GNCN GYCY u (p) abc i (p) u (n) i (n) abc abc abc ,, ,, () () () () dd dq k dq dq k qd dq k qq dq k ZZ ZZ ωω ωω     ,, ,, () () () () dq k dq k dq k dq k ωω ωω     pp np pn nn YY YY (13) ,, ,, ()() ()() dd dq k dq dq k qd dq k qq dq k YY YY ωω ωω     Plot impedances/admittances vs frequency Fig. 1. Flowchart of measurement methods: a) the dq-frame approach ; b) the pn -frame approach. Page 5 of 16 IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 6 superscripts d and q indicate the voltage perturbation and subscripts d and q the response current. 2) Comments and discussion The dq-frame approach has the following advantages: (i) the frequency of the dq-frame variables is unique; (ii) most stability studies are in the dq-frame; (iii) the derivations of the analytical impedance models are less complicated in the dqframe than in other frames. It, however, has drawbacks too [26], [30]: (i) experimental measurements require special equipment; (ii) the transformations of the three-phase perturbations to the dq-frame by the Park transformation and the PLL may introduce errors in the impedance measurements; (iii) no physical interpretations can be clearly made in the dqframe; (iv) two linearly independent dq-current perturbations are required in (24); (v) although the unique frequency of the dq-frame variables allows the impedance measurements to be addressed correctly, the injection of dq-current perturbations at angular frequencies below the fundamental frequency (i.e., at ωdq < ω 1) must be carefully analyzed because the frequency of the required n-sequence current perturbations becomes negative, i.e., ω n = ω dq − ω 1 < 0, when ωdq < ω 1. The last drawback is not completely addressed in the literature (no references to it could be found). The present paper contributes to solving this issue and also sheds light on the linearly independent dq-current perturbation injection from the pn-sequence currents in Section IV.B. B. The pn-frame approach [21] − [34] The following studies about measurements are developed in the pn-frame [21] – [34]. This approach avoids the Park transformation and the PLL and can be related to physical phenomena in three-phase power systems. In general, it injects p- and n-sequence voltage perturbations over a frequency range and obtains the pn-frame admittance transfer matrix of converters at each frequency from the measured values of the p- and n-sequence complex current and voltage phasors. References related to this approach contain different approximations and contributions. The first study neglects the pn-sequence coupling and only measures the diagonal terms of the pn-frame admittance transfer matrix [21] − [23]. However, this coupling cannot be neglected in stability studies because this could lead to inaccuracies [24] − [26]. Accordingly, other studies consider the pn-sequence coupling but assume strong grids [25], [30]. The most complete studies measure the pnframe admittance transfer matrix without disregarding the coupling phenomenon and the grid impedance [26] − [29]. Recent works [16], [31] obtain the pn-frame admittance transfer matrix by applying the modified-sequence frame transformation to the dq-frame admittance transfer matrix [5], [32], [33]. While the pn-frame approach offers the above advantages over the dq-frame approach, it has also several drawbacks, such as the number of frequencies (p-sequence, nsequence and DC frequencies) involved in the perturbation measurements [27], the influence of the FFT window on the time dependence of the off-diagonal terms of the pn-frame admittance transfer matrix [30], and the impact of the mirror frequency effect on admittance transfer matrix characterization [27]. The above discussion illustrates that, although the procedures for impedance/admittance measurements are extensively known, several issues are not yet fully solved. One of the most important is the impact of the mirror frequency effect on the handling of measured voltages and currents when the pn-frame approach is applied [5], [26]. This phenomenon is characterized by the off-diagonal terms of the pn-frame impedance/admittance transfer matrix and must be carefully addressed in the matrix measurement below the fundamental frequency [5], [24]. In [34], a method is described, which is the closest to the correct solution. It works correctly when the phase angles between the fundamental frequency source uabc, fund and the perturbation uabc, harm are zero. Section V analyzes the problem in detail. In [34], the phases are not calculated correctly for fdq<f1, since the complex conjugates must be applied in the coupling terms to the measurements (see (36)) and not to the admittances, as in (5) in [34]. This problem is more obvious when comparing (12) in [34] with (38) in this paper. Most works in the pn-frame focus on the injection of a series voltage perturbation to measure the admittance transfer matrix of converters. 1) State of the art Fig. 1(b) shows the flowchart of this approach. It presents three different procedures: GNCN, which neglects the grid impedance and pn-sequence couplings; GNCY, which neglects the grid impedance and considers the pn-sequence couplings; and GYCY, which considers both the grid impedance and pnsequence couplings. Broadly speaking, they are divided into the following steps (see Section II.C for fundamentals): (i) p- and n-sequence three-phase voltage perturbations of angular frequencies ω p, k = ω dq, k + ω 1 and ω n, k = ω dq, k − ω 1 are injected over a frequency range (p- and n-sequence measurement tests identified with super indexes (p) and (n), respectively); (ii) the steady-state three-phase voltages and currents at the converter terminals caused by the voltage perturbations are measured. Note that there is no interaction between the grid and the converter in the GNCN and GNCY procedures, and therefore, measurements of the voltages at the converter terminals are not required because they are equal to the voltage perturbations; (iii) FFT frequency domain analysis is applied to extract the magnitude and phase values of the pn-sequence current and voltage phasors at each frequency ω p, k (p-sequence signal) and ω n, k (n-sequence signal); (iv) the pn-frame admittance transfer matrix of converters is obtained at each frequency ω dq, k from the above current and voltage values. Regarding step (iv), the GNCN procedure neglects the grid impedance and the pn-sequence couplings and only measures the diagonal terms of the pn-frame admittance transfer matrix from two single-phase voltages corresponding to the p- and nsequence perturbations (see the pn-frame admittance transfer matrix in (23) [21] − [23], [30], i.e., Page 6 of 16IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 7 (p) 1 (p) (n) 1 (n) () ( ) () ( ) () () ( ) ( ) ( ), () p dq p p dq p n dq n n dq n ω ω ω ωωω ω ω ω ω ωω ω ω = = = + = = = − p pp p p n nn n n I YY U I YY U (26) where Yp( ω p) and Yn( ω n) denote the standard pn-sequence admittances at the frequencies of the pn-sequence perturbations while Ypp( ω dq) and Ynn( ω dq) denote the elements of the pn-sequence admittance matrix at dq-frame frequency (23) [30]. However, it is well known that the pn-sequence coupling cannot be neglected in stability studies because this could lead to inaccuracies [24] − [26]. Accordingly, [25] − [30] propose the admittance transfer matrix measurement taking into account this coupling (GNCY and GYCY procedures). The GNCY procedure is simpler because it considers the grid impedance as zero in the measurement tests (i.e., the grid is strong enough to avoid considering interactions with the converter). This way, the impedance transfer matrix can be measured from two single-phase voltages corresponding to the p- and nsequence perturbations, as follows [25], [30]: (p) (n) (p) (n) (p) (n) (p) (n) () () () () () () () () () () . () () pp dq dq pn nn dq dq pn ωω ωω ωω ωω ωω ωω = = = = pp pp pn pn nn np nn pn II YY UU II YY UU (27) Note that the pn-sequence relationship in (26) can be expressed at the angular frequencies ω p and ω n or at the angular frequency ω dq because they relate phasors of the same frequency, while the coupling phenomenon between p- and nsequence frequencies in (27) must be expressed only at the dqframe angular frequency ω dq, according to (23). The GYCY procedure is more complex because it considers the grid impedance and the pn-sequence coupling. In this procedure, the three-phase voltages at the converter terminals are different from the perturbations, and these voltages and the three-phase currents are characterized by the pn-sequence complex voltage and current phasors. Thus, the pn-frame admittance transfer matrix is measured by injecting two single-frequency linearly independent p- and n-sequence voltage perturbations over a frequency range and by solving the linear system derived from the pn-frame voltage and current phasor relationship (23) [26] − [29], i.e., 1 (p) (n) (p) (n) (p) (n) (p) (n) () () () () () () () (). () () () () dq dq dq dq pp p p nn n n ωω ωω ωω ω ω ωω ω ω −  =               pp pn np nn pp p p nn n n YY YY IIUU IIUU (28) Both procedures, GNCY and GYCY, are compared in [27] − [29]. It is observed that (27) can be derived from (28) and that (26) can be derived from (27). Recent studies [16], [31] obtain the dq-real frame admittance transfer matrix by injecting pn-frame perturbations, and subsequently determine the pn-frame impedance transfer matrix by applying the modified-sequence frame transformation (11) to the dq-real frame admittance transfer matrix (see Fig. 1) [5], [32], [33]. Some works present the characterization of a single pn-frame admittance transfer function by means of the single-frequency linearly independent pn-sequence voltage/current perturbations to avoid solving the pn-frame linear voltage and current system (28) [27], [30]. 2) Comments and discussion While the pn-frame approach offers several advantages over the dq-frame approach (e.g., it avoids the Park transformation and the PLL and is related to physical phenomena in threephase power systems), it also has the following drawbacks [27], [30]: (i) experimental measurements require twofrequency measurement equipment in order to acquire the voltages and currents at the frequency of the perturbations and at the mirror frequency effect caused by the pn-sequence coupling; (ii) resolution of the linear system of the pn-frame voltage and current in the GYCY procedure requires two independent pn-frame current perturbations; (iii) the FFT window must be aligned to the same point on wave of the fundamental three-phase voltages to avoid the time dependence of the off-diagonal terms of the pn-frame admittance transfer matrix [30]; (iv) the mirror frequency effect must be carefully addressed when the pn-frame impedance transfer matrix is characterized from measurements. As mentioned in Section II.C, the pn-sequence impedance (22) or admittance (23) transfer matrix derived from the dqcomplex transfer matrix function (21) in the frequency domain must be applied in order to explicitly identify the mirror frequency effect from measurements. These matrices, (22) and (23), relate the pn-sequence complex phasors of the angular frequency ω dq, which come from the pn-sequence complex phasors of the different angular frequencies ω dq + ω 1 and ω dq – ω 1 (mirror frequency effect). On the other hand, the s-domain complex transfer matrix function (21) based on the complex space vectors (15) is still in the dq-frame and the mirror frequency effect is overlooked in it [32]. This means that frequency domain conventional theory expressed in terms of complex voltage and current phasors in the pn-sequence frame must be used to reveal this effect in the measurements [26] − [29]. However, various works related to the measurements of the pn-sequence impedance/admittance transfer matrix consider the pn-sequence voltage and current relationships in the s-domain [21], [25], [30], [31], which can lead to misinterpretations: (i) the angular frequency of the svariable in the pn-sequence impedance transfer matrix is not defined [21]; (ii) the terms of the pn-sequence admittance transfer matrix are frequency shifted at two different frequencies [25], [31]; (iii) the p- and n-sequence voltage and current signals are frequency shifted as s + j ω 1 and s − j ω 1, respectively, to characterize their frequency ( ω p = ω dq + ω 1 and ω n = ω dq – ω 1) [30], but these frequency translations must be Page 7 of 16 IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 8 applied to the transfer matrices instead of the signals [8]. Regarding the works that use complex phasors in the frequency domain, [29] correctly addresses the involved angular frequencies but [28] and [27] define a pn-sequence admittance transfer matrix of two different angular frequencies, and [26] does not define the angular frequency of the matrix. However, it is worth emphasizing that the influence of the mirror frequency effect on the characterization of the pn-sequence admittance transfer matrix at DC frequencies below the fundamental angular frequency is not addressed in the above approaches and requires further study. Only [5] and [26] analyze this issue: [5] briefly comments on the angular frequency conflict and points out the relationship between the p- and n-sequence impedances at these frequencies; and [26] correctly addresses the pn-sequence voltage and current relationships at these frequencies but neither provides a systematic procedure for determining the pn-sequence admittance transfer matrix nor defines the angular frequency for which this matrix is calculated. IV. IMPROVEMENTS TO THE IMPEDANCE/ADMITTANCE MEASUREMENTS Considering the detailed discussion on the dq- and pn-frame approaches for the impedance/admittance transfer matrix measurements in Section III, the present Section contributes to bridging several gaps about these measurements. A. FFT window effect As discussed in [30], the off-diagonal terms of the admittance transfer matrix become time dependent when the FFT window slides over time a time step tv because the difference between the phases of the p- and n-sequence voltages and currents changes with the window. This is due to the different frequency of the p- and n-sequence signals, i.e., 1 1 1 11 1 11 1 1 ' 1 1 1 1 () ( ,FFT () ( ) cos( ) ( ) cos( ) ( ') cos ' ( ') cos ' v v v nn vv np n p v pp t tt t p p pp n n nn v p p pp p v n n nn n j j nn jp p ut U t it I t ut U t it I t Ie I e U Ue φ ω φ φω φφ ωφφω ω ωω φ φω ω ωφ ωφ φ ωω φ ω φ ωω φ ω =−=− +−+ − + = + ⇒  = +   = ++     = ++    ⇒= = np Y ), (29) where φ v1 is the starting point of the FFT window with respect to the fundamental component of the phase a voltage. This issue can be solved by keeping this starting point equal to zero during the admittance measurement, as follows [30]: 11 11 11 11 ( )() () ,FFT ( )() . vn nn v np p v pp v jj j nn jj p p Ie e I e U Ue e φω φω φ ωω φφ ω φ φω φ ωω +− − +− ⋅ = = = ⋅ np np YY (30) In order to avoid compensating the phase of the FFT components to eliminate the time dependence of the offdiagonal term of the admittance transfer matrix, the FFT window can slide over time with a time step equal to the sampling time, i.e., tv = Ts = Tw/Ns, where Ns ∈ N is the number of samplings and Tw is the window time length. Note that the frequencies of the frequency response terms correspond to multiples of the window time length, i.e., fp = kp·fw = kp/Tw (and fn = kn·fw = kn/Tw) with kp and kn ∈ N. Hence, () (2 2) ,FFT () ( 2 2) () . ww n nv np n p ss p pv p n np np ss TT jt j ff NN nn jt p p k k jj NN nn pp Ie I e U Ue II ee UU φω φφ π π φω φφ π π φφ +−+ − + −+ − − = = = = = np np Y Y (31) In (31), the FFT window effect is avoided when the number of samplings Ns and ratios fp /fw and fn /fw is an integer number. The terms (kp /Ns)2π and (kn /Ns)2π are multiples of 2π and the off-diagonal terms of the admittance transfer matrix become time independent. The use of a constant DFT window size is fundamental to keep ratios fp /fw and fn /fw as integer values. This was validated by PSCAD simulation (results not shown for space reasons). Regarding the number of samplings, simulation packages such as PSCAD typically fix them to an integer value per cycle of base frequency, and therefore the FFT window effect is not an issue. It is worth noting that, according to [6], the off-diagonal term of the admittance transfer matrix Ynp, FFT (29) expressed at the DC frequency by ω p = ω dq + ω 1 and ω n = ω dq – ω 1 is related to the off-diagonal term of the admittance transfer matrix Ynp (30) by the rotation e−j2 φ v1, i.e., 11 11 1 11 () () 2 ,FFT . vv dq v dq v np v j jj n p Iee e U φφ ωφωφ φφ φ ωω −− − −− = = np np YY (32) The same is true of the off-diagonal admittance Ynp and the rotation e j2 φ v1. B. Influence of the mirror frequency effect on the measurements of the dq-frame admittance transfer matrix The dq-frame perturbations in (8) can be expressed as the well-known voltage perturbations: cos( ) cos( ) sin( ) sin( ), d p dq Up n dq Un q p dq Up n dq Un uU t U t uU t U t ωφ ωφ ωφ ωφ = ++ + = +− + (33) where Up and φ Up / Un and φ Un are the magnitude and phase angle of the p- / n-sequence complex voltage phasors of original angular frequencies ω p = ω dq + ω 1 / ω n = ω dq – ω 1, respectively. Hence, the following issues (not clearly described in the literature) must be highlighted from (33): • there are several attempts in the literature to obtain two linearly independent dq-current perturbations, such as using two different angles in the perturbations [14], [15], injecting a single-phase current between the phases [17] and performing multiple sweeps at single frequencies to Page 8 of 16IEEE PES Transactions on Power Delivery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 15 it is more complicated to observe the frequency shift. The currents, ia, ib, ic, have a mix of positive sequence of ω p = ω dq + ω 1 and negative sequence of ω n = ω dq – ω 1. The product of cosines causes angular pulsations to appear for the positive sequence of ω 1 = ω dq and ω 2 = ω dq + 2 ω 1. 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