A Passivity-Based Small-Signal DEF Analysis for Low-Frequency Oscillation Source Characterization of VSC-HVdc

In bulk power systems, detecting the sources of low-frequency oscillations can be challenging. The energy-based approaches for oscillation source localization are known to have better accuracy compared to others. The dissipating energy flow (DEF) method is one such approach with notable success in localizing real-world oscillation cases. While the mathematical justifications for the method's characterization of a device as an oscillation source (or sink) is well-exposed for synchronous generators and standalone inverters, no insights have been developed for power-electronics-interfaced dc transmission systems. To fill this gap, this article presents a theoretical analysis of the DEF in a voltage source converter-based high voltage direct current (VSC-HVdc) system. Passivity-based analysis is performed to explain why a VSC-HVdc system, operating at unity power factor, with the commonly used control strategy involving constant real power control, dc-link voltage control, and ac voltage–reactive power droop control is a source of oscillation energy. Supporting case studies are performed on the IEEE 4-machine and IEEE 16-machine 68-bus test systems.

A Passivity-Based Small-Signal DEF Analysis for Low-Frequency Oscillation Source Characterization of VSC-HVdc Kaustav Chatterjee , Member, IEEE, Sayan Samanta, Student Member, IEEE, and Nilanjan Ray Chaudhuri , Senior Member, IEEE Abstract-In bulk power systems, detecting the sources of lowfrequency oscillations can be challenging.The energy-based approaches for oscillation source localization are known to have better accuracy compared to others.The dissipating energy flow (DEF) method is one such approach with notable success in localizing real-world oscillation cases.While the mathematical justifications for the method's characterization of a device as an oscillation source (or sink) is well-exposed for synchronous generators and standalone inverters, no insights have been developed for powerelectronics-interfaced dc transmission systems.To fill this gap, this article presents a theoretical analysis of the DEF in a voltage source converter-based high voltage direct current (VSC-HVdc) system.Passivity-based analysis is performed to explain why a VSC-HVdc system, operating at unity power factor, with the commonly used control strategy involving constant real power control, dc-link voltage control, and ac voltage-reactive power droop control is a source of oscillation energy.Supporting case studies are performed on the IEEE 4-machine and IEEE 16-machine 68-bus test systems.Index Terms-Dissipating energy flow, low-frequency oscillations, oscillation source, passivity, VSC-HVdc.

I. INTRODUCTION
I N THE monitoring of bulk power systems it is common to observe low-frequency electromechanical oscillations (typically, 0.2−2 Hz) in the system variables.The oscillatory behaviour can be attributed to the transient energy associated with the system's poorly-damped natural modes and/or the external perturbations [1], [2].These oscillations, if sustained, can impose serious threat to system stability and security [3].Once detected, as a first-step in mitigating these, it is important to localize their sources in the grid [4].For forced oscillations (FOs), defining their source is relatively easier.The component 'generating' the oscillation by means of external perturbations, unambiguously, is the source.However, for sustained oscillations from poorly-damped natural modes, the source characterization is more complicated [5].There could be more than one element, which depending on their control designs, may negatively impact the damping of a mode by shifting its eigenvalue to the right.Such elements with negative damping contributions could be characterized as sources [4].
In this article, we focus our attention to the dissipating energy flow (DEF)-method, which has shown promising results on 1000+ real-world oscillation cases [17].The method, as originally proposed in [12], calculates the average transient energy dissipation (also referred to as, 'oscillation energy dissipation') in a device leveraging the Lyapunov-like energy functions developed in [18].In the DEF method, a source is defined as a system component which aids in sustaining the oscillation by supplying transient energy, either directly -as in the case of forced oscillations, or by deteriorating system damping -as in the case of poorly-damped modes [4].The theoretical deductions leading to the method assume a lossless transmission network and constant power loads [12], [19].In [4], [17], [20], [21] the impact of these assumptions on the accuracy of the method are carefully examined.These studies concluded that, impacts of line resistances on the DEF patterns are marginal [4], [17].The studies further concluded that the deviation of the load characteristics from constant power to constant impedance may obfuscate the energy flow patterns in the network but their impact on the accuracy of the source localization are less likely to be critical [4], [17].In [22], authors identified a few cases where the DEF method might fail.Notwithstanding these limitations, the success of the method is widely recognized, as is evident from its successful adoption in the oscillation management system of ISO-NE [17].In the recent years, extensions and improvements on the existing DEF method have also been studied -notable of these are [4], [23], [24], [25], [26].
The theoretical analysis of the DEF-method and its connection to damping torque analysis is well-studied for synchronous generators [19], [27], [28].However other network components, especially the power electronics-interfaced active transmission devices remains largely unexplored in this context.While it is true that, in most cases the oscillation source is in a generator's control systems (e.g., exciters, stabilizers, governors, or turbineboiler systems) [17], many a time the source has also been detected outside the generator in power electronics-interfaced transmission systems.For example, on May 23, 2018, sustained 0.4 Hz oscillations were observed at multiple locations in the Western Electricity Coordinating Council (WECC) system and the source of these oscillations was identified in the controls of the Pacific DC Intertie HVdc system [17].In [26], a new DEF method is derived for detecting oscillation sources in grid-following inverters.In [21], authors use the passivity theory to offer an insightful interpretation of the DEF.The analysis is then applied to grid-forming converters under droop control with a stiff dc source.In [29], the DEF-based oscillation source/sink behavior of thyristor-controlled series capacitors (TCSCs) and static synchronous compensators (STATCOMs) and their controls are studied.No insights, however, have been developed regarding the DEFs for HVdc transmission systems.In our understanding, compared to a single converter system, the analysis of a HVdc system is more complex due to the coupling between transient energy functions of the rectifier-and inverter-end converters.The coupling stems from the dependence of real powers injected by the two converter terminals due to power balance.In addition, due to the interconnected nature of an ac system corresponding reactive power injections can also be dependent if the ac voltage is controlled at the respective terminals.
Thus motivated, in this article we present the analysis of DEF in VSC-HVdc systems for low-frequency electromechanical oscillations.The operating conditions and controls leading to the source characterization of the device are highlighted.Drawing insights from the notions of passivity and positive realness, the article presents analytical derivations explaining why a VSC-HVdc system with the conventional control strategy (see for example, [30]) involving real power control, reactive power-ac voltage droop, and dc voltage control is a source of oscillation energy flow.
The article is organized as follows.In Section II, the VSC-HVdc system with the rectifier-and inverter-end controls is described.In Section III, the preliminaries on DEF, passivity, and the connections between them are introduced.Thereafter, in Section IV, main result of the article explaining the oscillationsource behaviour of the system in Section II is presented.The case studies on the IEEE test systems are discussed in Section V followed by conclusions in Section VI.

II. SYSTEM DESCRIPTION
Consider a point-to-point HVdc system connecting two ends of a synchronous ac system, as shown in Fig. 1.In addition to the HVdc link, the rectifier and the inverter bus are also connected via the ac network.The HVdc link, in study, is detailed in Fig. 1.Schematic of an HVdc system routing power between two points of a synchronous AC system.Fig. 2 with all symbols having their usual meanings, as described below.
Notations: V r , I r , P r , Q r , and V i , I i , P i , Q i are respectively, the ac-side voltage, current, and the real and reactive powers flowing into the device at the rectifier-end, and flowing out of the device at the inverter-end.C r , C i , V dc r , and V dc i are respectively, the rectifier-and inverter-end dc-capacitances and dc-bus voltages, i dc is the dc-link current, and R is the dc-link resistance.The steady-state quantities are denoted by superscript 0 and the perturbations from the steady-state by prefix Δ, i.e., any variable x = x 0 + Δx.
The following phasor representations are considered in the rectifier's and inverter's d − q reference frames determined respectively by their phase-locked-loops (PLLs), PLLs ensure that in steady state, the d-axes align with respective bus voltage phasors.Typically, this is achieved by driving the qcomponent of voltage vector to zero using a closed-loop tracking control.Hence, V 0 r = V 0 d r and V 0 q r = 0. Following which, the linearization of (2) Similarly, for the inverter-end we have ΔV i = ΔV d i .
Please note that, in the interest of simplifying the mathematical analysis, the dynamics the PLL is not explicitly modeled in the DEF equations. 1The PLL's effect however, is modeled by considering nonzero ΔV q i and ΔV d i during transients.Also note, such a simplified representation of PLL is only limited to the analytical deductions and the proofs of the theorems.The  numerical verification (see, the case studies in Section (V)) of the theorems consider a detailed model of PLL as shown in Fig. 4.
The real and reactive powers at the rectifier-end are expressed as, [31].In steady state, both rectifier and inverter operates at unity power factor, implying, Q 0 r = Q 0 i = 0. Operation under unity power factor leads to minimum ac current flowing through the HVdc terminals, thereby minimizing losses.Another practical reason behind unity power factor operation is to ensure dynamic reactive power reserve for voltage support under contingencies akin to a static var compensator, as mentioned in [32, pp. 44].The real power deviations, obtained from linearization 2 , can be expressed as, and The inner current control loops reject the measurable disturbances by using feedforward control signals

and
V dc i 2 as shown in Fig. 3.Moreover, the controller K c (s) is designed to be equal to L i τ c , where τ c is the desired time-constant of the closed-loop step response [31].Upon simplification we 2 Recall, V 0 dr = V 0 r and V 0 qr = 0, therefore, can write, In the control of real and reactive powers, following strategies are considered: 1) Rectifier-end real power control (see, Fig. 5): where Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.2) Rectifier-end ac voltage-reactive power droop control (see, Fig. 6): where 3) Inverter-end dc-bus voltage control (see, Fig. 7): 4) Inverter-end ac voltage-reactive power droop control (see, Fig. 8): where, Additionally, the sensor delays in the feedback loops and the resistances in the filter inductors are neglected.

III. PRELIMINARIES ON DEF AND PASSIVITY Definition 1 ([12]):
The DEF into an element E connected to the power network at bus k, denoted by W D E , is expressed as where Δ V k and Δ I inj are respectively, the small-signal phasors of the perturbations in the voltage at bus k and the current injected into E from the network, and (•) denotes the imaginary part of a complex number.Building on this, for the VSC-HVdc system described in Section II, total DEF flowing in (adding the rectifier and inverter contributions), denoted by W D HVdc , is expressed as We define and Following which, we may write Definition 2 ( [12]): Element E is called a sink (source) of oscillation energy if and only if the average slope of the DEF into E over a cycle T , is positive (negative).
In other words, the VSC-HVdc system is a sink of oscillation energy if and a source if Ẇ D HVdc < 0. Next, consider the linearized representation of the system in Section II, denoted by Ω (see, ( 15)), with u and v as the input and output, respectively.
From [33,Def. 6.3], Ω is said to be passive with respect to the supply rate u T y if there exists a continuously differentiable positive semidefinite storage function S(x) such that u T y Ṡ(x).
Passivity of Ω implies, between any two instants, the energy flowing in is greater than the change in system's energy storage (see, ( 16)).Therefore, Since S(x(0)) = 0 and S(x(T )) 0, we may write (16) as, Therefore, passivity of Ω with respect to the supply rate u T y implies, u T y dt 0. Further, from [34, Lemma 10] we know that the system is passive if and only if the corresponding transfer function matrix G(s), between the input-output pair u and y, is positive real.Following which, from Def. 2, it may be said that the positive realness of G(s) implies the underlying system is a sink 3 and its negation indicates the system is a source of oscillation energy.A summary of this is presented in Fig. 9.
To that end, in this article, we derive the expression of G(s) for the VSC-HVdc system described in Section II.Thereafter, we check if G(s) satisfies the conditions for positive realness, as defined in [33,Def. 6.4].This is presented in Section IV next.
At this point it is worth highlighting that, although the analysis in this article focuses on the linearized small-signal representation of the HVdc system, the concepts of passivity and positive realness are stronger and can be applied to the analysis of the nonlinear system.The linearized representation, without much loss of accuracy, adequately captures the ambient dynamics of the system and is reasonable to understand the source/sink behavior in this context.Furthermore, this renders the analysis tractable.

A. Deriving the Transfer Function Matrix
Consider the VSC-HVdc system desribed in Section II.Linearizing (5), and thereafter, substituting (3) we obtain Defining α 2 (s) := V 0 r + (1 + sτ c )/K r (s), (18) may be rewritten as Similarly, linearizing 1 (6), we obtain Next, for the dc-side, the following power balance equations can be written at the rectifier-and inverter-end dc-buses, 3 or neither a sink nor a source, i.e., lossless, for the u T y dt = 0 case Linearizing 4 ( 21)-( 23) and taking Laplace transform, we obtain 5 Further, substituting (25) in (26), we arrive at Next, we linearize the control law in (8) to obtain Substituting these expressions of ΔV dc r , ΔV dc i , and Δi dc into (24), we get For the detailed expressions of f 1 (s), f 2 (s), and f 3 (s), see Appendix A. The expression in ( 29) can be simplified by substituting the linearized expressions of ΔP r and ΔP i (from ( 3) and ( 4)) into it.Doing so, we get where, α 1 (s) := (f 2 (s) + f 3 (s))/f 1 (s).
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Also, note that Substituting this, (32) can be written as where, The detailed derivation of G(s) is presented in Appendix B. Observe that from the expression of G(s), the structure of L(jω) can be inferred as

B. Positive Realness of the Transfer Function Matrix
The characteristic equation of L(jω) is Since, in accordance with our assumption, L(jω) is positive semidefinite, all four eigenvalues (λ i , for i = 1, . . .4) of L(jω) should be non-negative.As seen from (40), one of the eigenvalues (say, λ 1 ) of L(jω) is zero.For the three other eigenvalues, the following ((i)-(iii)) should hold true: i) Their product should be non-negative.Observe from (40), Therefore, this imposes the necessary condition that 44 0. ii) Also, the sum of the products of λ i taken two at a time should be non-negative.From (40), Observe that, for a nonzero real power transfer through the HVdc-link in the steady-state, I 0 d r = 0. Therefore, from proposition 1 (see, Appendix C), | 23 | 2 > 0. Further, from (i) we have 44 0. These together 6 , imposes the necessary condition that, 22 < 0 and 44 < 0. iii) Finally, their sum should also be non-negative.From (40), this imposes the necessary condition that λ 2 + λ 3 + λ 4 = 22 + 44 0. Clearly, the necessary conditions from (i) and (ii) contradict that in (iii).Therefore, L(jω) is not positive semidefinite, and as a consequence, G(s) is not positive real.This concludes the proof.
Theorem 1 (Main Result): A VSC-HVdc system operating at unity power factor in steady state, with constant real power control, constant dc-link voltage control, and ac voltage-reactive power droop control is a source of oscillation energy.
Proof: Observe from Lemma 1, the transfer function matrix G(s) between the input-output pair u and y is not positive real.Following which, from [34, Lemma 10], it can be said that the system Ω (linearized representation of the VSC-HVdc system with these controls) is not passive with respect to the supply rate u T y.This implies, the integral T 0 u T y dt < 0. And therefore, the system is a source of oscillation energy.This concludes the proof.
Remark 1: Constant reactive power control at the rectifier and the inverter (i.e., ΔQ ref r = ΔQ ref i = 0 or in other words, K q r = K q i = 0) is a special case of the system description in Section II and Theorem 6.1.Therefore, a VSC-HVdc system operating at unity power factor in steady state, with constant real and reactive power control and constant dc-link voltage control is also a source of oscillation energy.
Remark 2: Please note that in this article the oscillation source behavior of VSC-HVdc is studied only for low-frequency electromechanical oscillations (0.2−2 Hz).The averaged model discussed in deriving the mathematical results and the simplifications made may not hold true for analyzing oscillations outside this range.Therefore, it should be highlighted that the applicability of the Theorem 1 is restricted to electromechanical oscillations and should not be extended to subsynchronous oscillations (SSOs).The analysis of DEF for higher frequency oscillations (e.g.SSOs) requires detailed electromagnetic transients (EMT) modeling of the system.EMT modeling is outside the scope of this article.Interested readers may take a look at the following research papers [26], [35], [36], and [37].
Remark 3: Inclusion of an additional modulating control on real power set point P ref r , for example -a feedback control for power oscillation damping, may transform the VSC-HVdc from a source to a sink of oscillation energy.In that case, with a suitable choice of feedback control parameters, G(s) can be made positive real.Although intuitive, a detailed analytical proof may be necessary to strengthen the claim.This may be a potential topic for future research.

A. IEEE 2−area 4−machine Test System
The IEEE 2−area 4−machine test system from [1] is modified to include a VSC-HVdc link connecting buses 7 and 9 (see, Fig. 10) with the controls described in Section II.The transfer functions of the proportional intergal (PI) controllers are as follows: K r (s) = K v (s) = 1 + 10/s.The detailed model of the PLL in Fig. 4 is used in the simulation with K p PLL = 20 and K i PLL = 200.For synchronuous generators, their sixth-order subtransient models are considered along with DC1A exciters and automatic voltage regulators (AVRs).
In this study, we consider two operating cases: (i) when P ref r = 200 MW, and (ii) when P ref r = 400 MW.For each of these two cases, we consider two subcases: (a) constant reactive power control at both ends 7 i.e., K q r = K q i = 0, and (b) ac voltagereactive power droop control 8 with K q r = 2.5 and K q i = −2.5.Note that, in (b), the signs of the droop coefficients are opposite because in the system description (see, Section II) the directions of Q r and Q i are taken opposite -inward in the rectifier and outward in the inverter.
1) Verification Under Small Disturbance: A 0.1 s duration pulse disturbance is applied to the mechanical power input (P m ) of generators.The measurement signals P r , P i , Q r , Q i , V r , V i , θ r , and θ i thus obtained, are detrended and filtered for the 0.65 Hz inter-area mode.Thereafter, W D HVdc -the total DEF into the HVdc system adding both the rectifier and inverter injections, is calculated (see, (49) and Fig. 20).

For details on how W D
HVdc is computed from these terminal measurements, please refer to Appendix D. The time domain plots of W D HVdc for different values of P ref r and droop coefficients K q r and K q i are shown in Fig. 11(a) and (b).Observe, in each case, the average slope of W D HVdc is negative, implying that the 7 with unity power factor in steady-state, i.e., 8 again, unity power factor in steady-state, i.e., HVdc system with these controls is injecting oscillation energy into the ac network.This supports our claim in Theorem 1.
The eigenvalues of L(jω) = G(jω) + G T (−jω) for the case P ref r = 200 MW, Q 0 r = 0, and K q r = 2.5 and K q i = −2.5, in the electromechanical oscillation range 0.2 − 2 Hz, is shown in Fig. 12.Clearly, one eigenvalue is always negative.This confirms that L(jω) is not positive semi-definite and G(jω) is not positive real, as claimed in Lemma 1.
2) Verification Under Large Disturbance: Next, we verify the claim in Theorem 1 for large disturbances.To that end, a three-phase self-clearing fault is simulated near bus 8.The timedomain plots of the measurement signals from the rectifier and inverter terminals are shown in Fig. 21 of Appendix D. These signals are detrended and filtered for the inter-area mode and thereafter, the DEF is computed (see, Fig. 20).The DEF plots for two different droop settings are presented in Fig. 13(a) and (b).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Observe that the average slope of W D HVdc is negative supporting the claim that the VSC-HVdc is source of oscillation energy.The DEF into the generators G1-G4 for the case (b) are also shown in Fig. 14 for comparison.The average slopes of W D Gen for all 4 generators are positive indicating that the generators, in this case, are acting as sinks to absorb the transient oscillation energy emanating from the VSC-HVdc source.
3) Consistency With Eigenvalue Analysis: To support the conclusions of DEF-based source characterization, the following approach using eigenvalue analysis is adopted.First, we consider the base case -i.e., the IEEE 4-machine system without the HVdc.The eigenvalues and the damping ratio for the inter-area mode are calculated.Next, keeping the total system load and the set points of the generators same as before, the VSC-HVdc is introduced (as shown in Fig. 10) with the controls described in Section II.The eigenvalues for the modified system are recalculated.We know, if the VSC-HVdc is an oscillation source then the addition of it into the system should have a  15.Rectifier-end real power control with a supplementary damping controller modulating the power reference.Fig. 16.Total DEF into the VSC-HVdc system following a three-phase fault in the 4−machine system in presence of a damping controller modulating the reference to the real power control.
negative contribution to the damping, i.e., the eigenvalues for inter-area mode after inclusion of the HVdc, as compared to the base case, should shift to the right reducing the overall margin of stability.The eigenvalues in Table I supports the claim that the inclusion of the VSC-HVdc deteriorates modal damping, thereby indicating that the added device is a source of transient energy aiding in sustaining the oscillation.
4) Addition of a Damping Controller in the Rectifier-End Real Power Control: In Remark 3, it was hypothesized that the inclusion of a power oscillation damping control modulating the real power reference could transform the behavior of the VSC-HVdc from source to sink of oscillation energy.To confirm this, the following study is performed.The real power control in Fig. 5 is modified to include a damping controller as shown in Fig. 15.The phase angle difference between the rectifier and inverter buses is the input to the controller.In this study, the   VSC-HVdc is operating with P ref r = 200 MW, Q 0 r = Q 0 i = 0, and droop coefficients K q r = 5 and K q i = −5.The output of the damping controller can be seen as a modulation of the real power setpoint over over its steady-state value P ref r .The damping controller parameters: the gain K PSS = 13043, the lead-lag compensator time-constants T d 1 = 0.0133 s and T d 2 = 4.479 s, and the washout filter time-constant T w = 10 s, are designed [1] to achieve the desired pole-placement for enhancing the damping of the inter-area mode.The eigenvalues of the system before and after addition of the damping controller are shown in the Tables I and II respectively.
To study the DEF of the modified system, a three-phase self-clearing fault is simulated near bus 8, as before.The measurement signals obtained from the rectifier and inverter-end buses are detrended and filtered.The total DEF flow into the VSC-HVdc computed from these measurements is shown in the Fig. 16.The positive slope of the W HVdc confirms that the HVdc with the added damping control is acting a sink of oscillation energy.This characterization is also consistent with improvement in the damping ratio obtained from eigenvalue analysis (see, Tables I and II).

B. IEEE 16−machine 68−bus NY-NE Test System
Next, for confirming our claims, the IEEE 16-machine 68-bus NY-NE Test System [38], as shown in Fig. 17, is studied.The fundamental-frequency phasor model of the system with subtransient synchronous generator models, exciters (DC1A at G1-G8; ST1A at G9; manual excitation at G10-G16), automatic voltage regulators, and a power system stabilizer (at G9) is considered.The test system is modified to include a VSC-HVdc link connecting buses 60 and 61, with the controls described in Section II (see, Fig. 17).The PLL as shown in Fig. 4 is included.Bus 61 is operating as the inverter-station and bus 60 as the rectifier-station with , and droop coefficients-in case (a): K q r = K q i = 0 and case (b): K q r = 5 and K q i = −5.
A 0.2 s pulse disturbance is applied to the P ref r .The output variables are filtered for the 0.54 Hz inter-area mode.The total DEF into the HVdc system for a system mode computed for both cases (a) and (b) are shown in Fig. 18(a) and (b).Observe, for both cases (a) and (b), the average slope of W D HVdc is negative, confirming our claim in Theorem 1 that the system can be source of oscillation energy.The DEFs into the generators for the case (b) are shown in Fig. 19 for comparison.In this case, G14, G15, and G16 are the sinks for the oscillation energy produced by the VSC-HVdc and other generators.Further observe, the negative Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

slope of W D
HVdc in Fig. 18(b) is greater than any of the generators G1 -G13.

VI. CONCLUSION
In this article, analytical insights were developed into the dissipating energy based source characterization of a VSC-HVdc system for low-frequency electromechanical oscillations.The results from passivity and positive realness were used to mathematically explain why a VSC-HVdc system operating at unity power factor with constant real power control, constant dc-link voltage control, and ac voltage-reactive power droop control is a source of oscillation energy.Through this rigorous mathematical justification, the article strengthens the understanding of oscillation energy flow in converter-interfaced transmission systems.

A. Calculation of DEF From Terminal Measurements
In the simulations of Section V, DEF flowing into the VSC-HVdc is calculated using the terminal measurements from its rectifier-and inverter-end buses.
Please refer to [12, Eq. ( 16)] where it is shown that the transient energy flowing into a device E connected to the power network at bus can be expressed as, P inj and Q inj are respectively the real and reactive power injected into the device, and θ is the bus voltage angle at node .Following which, the dissipative part W D E , can be expressed as [12,Eq. (17)].
Extending this to the VSC-HVdc system in Section II, we can write Discretizing (48) we obtain, where k is the sample number.The time-domain plots of the DEFs in Section V are obtained following the process described in the schematic of Fig. 20.Equation ( 49) is used for the DEF calculation.For initialization, we choose W D HVdc [1] = 0.The Δ variables in (49) are the detrended measurements bandpassfiltered for the modal frequency of interest.

B. Simulation Results for Three-Phase Fault in Section V
Time-domain plots of the measurement signals for the largedisturbance case in Section V-A are shown in Fig. 21.
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Fig. 3 .
Fig. 3. Inner current loops (only shown for the inverter-end) and their simplification.

Fig. 4 .Fig. 5 .
Fig. 4. The model of the PLL used in the case studies in Section V.

Fig. 9 .
Fig. 9. Equivalence of positive realness, passivity, and the oscillation-sink behavior with respect to DEF.

Fig. 10 .
Fig. 10.Single-line diagram of the modified 2−area 4−machine system with a VSC-HVdc line between buses 7 and 9.

Fig. 11 .Fig. 12 .
Fig. 11.Total DEF into the VSC-HVdc system for pulse disturbance in the 4−machine system with (a) constant Q ref and (b) AC voltage-reactive power droop, and different P ref and unity power factor in steady-state.

Fig. 13 .
Fig. 13.Total DEF into the VSC-HVdc system following a three-phase fault in the 4−machine with (a) constant Q ref and (b) AC voltage-reactive power droop, and unity power factor in steady-state.

Fig. 17
Fig. 17.Single-line diagram of the modified IEEE 16−machine 68−bus test system with a VSC-HVdc line between buses 60 and 61.
Fig. 17.Single-line diagram of the modified IEEE 16−machine 68−bus test system with a VSC-HVdc line between buses 60 and 61.

Fig. 18 .
Fig. 18.Total DEF into the VSC-HVdc system for a pulse disturbance in the 16−machine system with (a) constant Q ref and (b) AC voltage-reactive power droop, and unity power factor in steady-state.

Fig. 21 .
Fig. 21.Measurement signals (a) bus voltage magnitude, (b) relative bus voltage angle, (c) real, and (d) reactive powers at rectifier-and inverter-end buses (7 and 9 respectively) for a large disturbance (three-phase self-clearing fault near bus 8) in the IEEE 4-machine system.Detrended signals are shown in inset.

TABLE I EIGEN
-ANALYSIS OF THE IEEE 4−MACHINE SYSTEM Fig.