Coordinated Control of Multiterminal DC Grid Power ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 29, NO. 4, AUGUST 2014

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Coordinated Control of Multiterminal DC Grid Power Injections for Improved Rotor-Angle Stability Based on Lyapunov Theory Robert Eriksson, Member, IEEE

Abstract—The stability of an interconnected ac/dc system is affected by disturbances occurring in the system. Disturbances, such as three-phase faults, may jeopardize the rotor-angle stability and, thus, the generators fall out of synchronism. The possibility of fast change of the injected powers by the multiterminal dc grid can, by proper control action, enhance this stability. This paper proposes a new time optimal control strategy for the injected power of multiterminal dc grids to enhance the rotor-angle stability. The controller is time optimal, since it reduces the impact of a disturbance as fast as possible, and is based on Lyapunov theory considering the nonlinear behavior. The time optimal controller is of a bang-bang type and uses wide-area measurements as feedback signals. Nonlinear simulations are run in the Nordic32 test system implemented in PowerFactory/DIgSILENT with an interface to Matlab where the controller is implemented. Index Terms—Control Lyapunov function, coordinated control, energy function, high-voltage direct current (HVDC), multiterminal dc (MTDC), small-signal stability, transient stability.

I. INTRODUCTION

T

RANSIENT and small-signal stability are the abilities of a power system to maintain synchronism when subjected to a severe or small disturbance, respectively, both referred to as rotor-angle stability. The rotor-angle stability may be improved by controllable devices (e.g., HVDC links), in particular, their control actions. Improved technology, such as voltage-source converters (VSCs), brings more possibilities than conventional HVDC links. There are many proposals, in the future grid expansion, to have multiterminal dc (MTDC) as an overlaid grid. This grid will probably be based on the VSC technology and each converter can control active and reactive power independently. In addition, the electrical power from the massive integration of offshore wind power is likely to be transferred in MTDC grids and has a large impact on overall system stability

Manuscript received May 15, 2013; revised September 16, 2013 and November 20, 2013; accepted November 21, 2013. Date of publication December 23, 2013; date of current version July 21, 2014. This work was supported by the ELEKTRA program at Elforsk AB. Paper no. TPWRD-00585-2013. The author is with the Department of Electric Power Systems, School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm 100 44, Sweden, and with the Department of Electrical Engineering, Center for Electric Power and Energy, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark (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/TPWRD.2013.2293198

[1]. Dynamic stability studies have been performed over many decades and power modulation control of HVDC links enhances the security [2]–[5]. The power injection of the HVDC system has the possibility of counteracting the first swing power imbalance in a multimachine system subjected to a disturbance as well as damp power oscillations [3]. Many well-established analysis and design techniques exist for linear time-invariant (LTI) systems, such as root-locus, Bode plot, Nyquist criterion, state-feedback, and pole placement [6]. These linear design and analysis methods cannot necessarily be applied directly without special consideration since the inherited nature of power systems provides nonlinear dynamics for large disturbances. Lyapunov’s direct method plays a central role in the stability study of nonlinear control systems. This method is based on the existence of a scalar function of the states, that decreases monotonically along the trajectories to the stable equilibrium point (s.e.p.) and the challenge is to find a suitable Lyapunov function. In the classical works of Massera, Barbashin, Krasovskii, and Kurzweil, this sufficient condition for stability was also shown to be necessary under various sets of hypotheses. There is no general way of finding Lyapunov functions for nonlinear systems. Faced with specific systems, one has to use experience, intuition, and physical insights, such as system energy, to search for an appropriate Lyapunov function [7], [8]. The objective of this paper is to develop a new time optimal control Lyapunov function (CLF) to control the MTDC system power injections in order to enhance the rotor-angle stability of the interconnected ac/dc system. Much research has been published in the area of Lyapunov theory as the energy function for power systems and single controllable components. The contribution of this paper is the control Lyapunov function which coordinates the MTDC grid power injections in order to enhance the rotor-angle stability using the wide-area measurement system (WAMS). Related works are presented in the next section. II. RELATED WORKS The use of energy functions or similar to estimate regions of attraction for s.e.ps. in electric power systems has a long history. Energy or Lyapunov functions for power systems were developed in the late 1970s by Athay et al. [9], [10]. The use of energy-based Lyapunov functions for power systems is well developed in classical literature, such as [11] and the references

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therein. Energy or energy-like functions are often used as Lyapunov function candidates. A state-of-the-art paper by A. A. Fouad and V. Vittal reviews the transient energy method [12]. In [13], an automatic voltage regulator (AVR) is included in the Lyapunov stability of the multimachine power systems. In [14], control Lyapunov functions are presented for a single HVDC link to improve transient stability. It is based on the structure-preserving model and uses the frequency deviation of the HVDC connecting buses amplified by a gain as the input for the active power modulation. This frequency deviation as the feedback signal is directly connected to the observability and does not reflect the controllability. The drawback is that this signal could be zero at maximum angle deviation and could decrease closer to the stability boundary since it only depends on the bus frequency difference. This often implies less power modulation in the HVDC link; thus, less stability enhancement is closer to the boundary. In [8] and [11], transient stability analysis is performed via the energy function method using internal node representation including one HVDC link. To find a rigorous energy function transfer, conductances are ignored. This is developed for a single HVDC link, and the speed difference of the closest generators is used as the input for the HVDC controller and the overall transient stability is not captured. In [15], the structure preserving energy function is presented for ac/dc systems. Through approximations, the energy function was formulated as path independent and the proposed auxiliary and emergency controllers use inverter and rectifier bus measurements as feedback signals. A common approximation, used in the derivation of the transient energy functions for ac/dc systems, is to neglect the HVDC system dynamics. Reference [16] discussed the inclusion of the terminal effect of the line-commutated converter (LCC) HVDC and SVC systems in the Northern state power network into the transient energy function. A similar approach was proposed in [17]; however, this representation of the LCC HVDC system into the transient energy function is not sufficient to account for its behavior. Reference [18] presented an improved model for the transient energy of integrated ac/dc power systems that incorporated these dynamics. The transient energy function incorporates the algebraic and differential equations that represent the ac and dc systems, including the rectifier angle controller, the inverter angle controller, the dc current phase compensator, the voltage-dependent current order limiters (VDCOLs), and the converter control mode reversals. In [19], a method is presented to integrate the dynamics of dc power flow into calculations of transient energy functions of ac/dc power systems. This method treats the generator input power as a function of the parameters and the input and output variables of the dc damping power controller. By using this transient energy function, the accuracy of the stability prediction of ac/dc power systems is improved.

a time optimal control Lyapunov function is developed for the MTDC grid’s power injections to enhance the rotor-angle stability of the power system. In [21], a centralized nonlinear control strategy is developed for HVDC systems using Lyapunov theory, which is not directly applicable for MTDC grids. The controllability in the disturbance directions, that is, fault trajectory, is analyzed using singular value decomposition based on this internal model. The size of the singular values in the disturbance direction indicates the transient stability enhancement performed by a change of power injections of the MTDC system. The proposed CLF does not only rely on small changes around an equilibrium point, instead it is valid for large disturbances. Remote signals are used to sense the state in the system and the controller derives the active power modulation of the MTDC grid as the control signal. The developed CLF makes the time-derivative negative along the trajectories to the origin. To the author’s best knowledge, no publications present controllers which coordinate MTDC systems based on the control Lyapunov theory and analyze the controllability in different disturbance directions/fault trajectories. IV. PROBLEM DESCRIPTION USING CONTROL LYAPUNOV THEORY In control theory, a CLF is a generalization of the notion of Lyapunov function used in stability analysis [22]. The system energy is replaced by this scalar function . The ordinary Lyapunov function is used to evaluate whether a dynamical system is stable (more restrictively, asymptotically stable), thart is, whether the system starting in a state in some domain will remain in , or for asymptotic stability will eventually return to 0. The time derivative of the Lyapunov function of the uncontrolled system is (1) where stable if

is the Lie-derivative. The uncontrolled system is (2)

The CLF is used to evaluate whether a system is feedback stabilizable, that is, whether for any state there exists a control such that the nonlinear system can be brought to the zero state by applying the control . For the controlled system, the time derivative is given by (3) (4) To make a CLF, one needs to find the control vector such that is negative definite (or at least negative semidefinite).

III. CONTRIBUTIONS

V. TRANSIENT STABILITY ASSESSMENT

This paper extends the theory in [11] as in [20] to derive the internal node representation for the ac system with a controllable overlaid MTDC grid. Furthermore, this paper analyzes the stability enhancement performed by the change of power injection of the MTDC system. Based on the energy function theory,

The potential energy is often envisioned as a bowl in the multidimensional space of machine angles in the center of inertia reference frame. The s.e.p. is located at the bottom of the bowl and the stable region inside the bowl, commonly called the Region of Attraction of the s.e.p. The unstable equilibrium

ERIKSSON: COORDINATED CONTROL OF MULTI-TERMINAL DC GRID POWER INJECTIONS

points (u.e.p.) surrounding the s.e.p. form the so-called potential energy boundary surface (p.e.b.s.). The trajectory of a point representing the evolution of the transient state of the postfault system in a marginally unstable case passes near an u.e.p., called the controlling u.e.p., since it goes over the stability boundary toward instability. Let be the s.e.p. (equilibrium point) and be the u.e.p. for , is the stability boundary of the s.e.p. and is the stability manifold of the u.e.p. Then

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An MTDC system is said to have control variables if it has converters, . In (8), converter is assumed to be the slack converter. In addition, the maximum change of the MTDC powers is assumed to be for . B. AC Power System Modeling The ac power system can be described by a set of differential and algebraic equations

(5) where the unstable trajectory crosses the stability The point boundary , that is, is the exit point of this trajectory. If the point is close to , then is the true critical energy [12]. assumes its minimum on the stable manifold, implying the unstable trajectory passing through the constant energy surface (6) To find whether the system is stable under a contingency, one calculates the value of when the disturbance is terminated to the critical value . The assessment is made by computing the energy margin given by

(9) (10) contains the state variables and where contains the algebraic variables. The classical generator model is used; thus, where and are the rotor angle and speed, respectively. The number of generators are ; thus, and , where and are the bus angle and voltage, respectively. The number of network nodes are ; thus, . It is common to use the COI reference frame which is defined as follows:

(7) and the positive value implies the system is stable [12]. Using the system fault trajectory, disturbance direction, along the state trajectory subsequent to the contingency one, can derive the improvement of using the proposed CLF. To securely operate the system, the operators should keep an adequate margin . The controllability depends on the fault trajectory, implying the stability enhancement will vary for different contingencies. VI. POWER SYSTEM MODELING This section describes the MTDC and power system modeling. A. MTDC System Modeling The time frame of the dynamics of the MTDC system is much faster than for the generator swing equation. This can be assumed if there exists an appropriate solution in the dc voltage–current characteristic diagram. Thus, the MTDC grid dynamics are ignored and developing the control law and the control variables, active power injections are given as (8a) (8b) .. . (8c) (8d)

where is the synchronous inertia of machine . Using the internal node representation in the COI frame, a power system can be described by a set of first-order differential equations [8] as (11) C. Controllable AC/DC Power System Modeling The effect of a control variable on generator can be expressed by a term as [3], [8], and [20]. The entire system can therefore be described by (12) . This describes the general where power system and MTDC system with any number of converters and various topologies, such as radial or meshed. This modeling does not include the field flux and its control, which, if included, would add more state variables for the generator. VII. ENERGY FUNCTION The system energy by the Lyapunov function

is treated mathematically .

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A. Uncontrolled The energy function of the power system without HVDC is given by [8]

space of . The null space of and the gain in different directions may be analyzed by singular value decomposition (SVD). Clearly, the energy function depends on the controllability of the system which can be analyzed through and its feedback input directions . The matrix can be written as

.. . .. .

(13) ..

(14) is a constant implying uncontrolled system is given by

(14)

.. .

0. The time derivative of the

.. .

.

.. . (18)

(15) This modeling is for the classical generator model which represents the generator by a voltage source with constant magnitude behind a reactance. The field flux modeling would add extra terms in the Lyapunov function as described in [23]. The field flux and active power injections are very weakly connected, which implies that it will not affect the control law that is to be developed. B. Controlled The time derivative of the energy function along the trajectories of system (12) is given by

where , , and . The gains to are the gains for the feedback input directions to . The feedback input directions to span the null space. The trajectory following after a contingency cannot be affected by the MTDC system if the fault trajectory lies in the null space of . Furthermore, the gain of the feedback input directions can be adjusted by the feedback gain according to stability preferences. This can be analyzed through the potential energy boundary surface (PEBS) or similar methods to find the stability boundary in different directions. Therefore, one can place more gain in a direction where the stability boundary is closer, needing lower energy to pass. Based on this the feedback gain, can be derived as

.. .. .

(16) Thus

(17) To make (17) a Lyapunov candidate, it needs to satisfy the asymptotic stability condition. For this function, the time derivative needs to be negative along the trajectories. Clearly, if the dc power injections are uncontrolled, 0, then 0. Formulating a control Lyapunov function that implies a negative part of the first term in (17) is a CLF. This function is proposed in the next section. VIII. CONTROL STRATEGY The control law should be determined to make (17) negative definite or negative semidefinite. It may not be negative definite because 0 for 0 and in the null

.

.. . .. .

(19)

The nonzero elements in , that is, to are positive and decide the gain in different disturbance directions. To obtain equal gain in all directions, except the null directions, the gains should be chosen as . This means that the time derivative of the energy function is equal in all directions except for the null space. This does not take the control effort into account, as low gain (small singular value) in a direction takes more control effort. A. Time-Optimal Control Now, the control problem can be formulated as a time optimal problem since for a disturbance, one wants to control the power of the converters to take the system back to its s.e.p. 0 as fast as possible. In other words, improve the transient stability as much as possible and damp oscillations as fast as possible to make the system less vulnerable if another disturbance occurs. It is assumed that there is a limit on the control signals to consider the converter power limits. Limitations of the MTDC


system, such as line flows, can be included as an additional condition in the optimization problem formulation. Consider the time optimal problem written as

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The time derivative of the energy function is with the proposed control given by (29)

(20) (21) (22) where and are positive definite functions and in the time optimal problem 1 and 0. In general, the time optimal control problem formulated as the Hamilton–Jacobi–Bellman equation is not straightforward to solve for nonlinear systems. In the special case of the linear relation in the control variable, the problem has a solution of a bang-bang type coming from the direct derivation of the control law described below. The control law that solves the optimization problem (20) subjected to (21) and (22) is [24] (23) First, solving this for the conditions (21) and (22) gives the following equation: (24) else is the switching function and is small to make where the controller not act for small deviations. The parameter is the deadband parameter in the controller which is to be set by the transmission system operator (TSO). The time optimal control law depends on this switching function. The true control law also needs to fulfill the power limitations in (22), and the additional optimization problem can be written as (25) (26) (27)

and depends on the disturbance trajectory. IX. SIMULATION Nonlinear simulations are run in the Nordic32 test system extended by a three-terminal MTDC grid in PowerFactory/DIgSILENT, incorporating the nonlinear dynamics of the ac/dc power system simulated through the positive sequence. The generator and dc system models are of high precision, and further details can be found in [25]. An overlaid three-terminal MTDC system connects the buses 4012, 4044, and 4062, as shown in Fig. 1. The power transfer is from the north to the central area which makes the connecting tie lines between these areas vulnerable to disturbances. The controllable power flow in the MTDC system comes in handy to enhance the stability of the system as a disturbance occurs. Each converter that is assumed to have a range within the power can be changed that is 100 MW. The converter in the south is assumed to be the slack converter implying the control variables to be the power setpoints in the converters in the center and north, respectively. The dynamic model of the MTDC system includes the converter dynamics and its current control in the common manner. The dc voltage is controlled through the current by an internal control loop to the reference value and this is handled by the slack converter. The other two converters are in P-mode, thus, controlling the current to achieve the active power setpoint. The time optimal Lyapunov controller derives the active power setpoints based on the signals from the WAMS. This is simulated in PowerFactory using an interface to Matlab where the controller is implemented, as depicted in Fig. 2. A. Case Study Critical faults in this system are, among others, three-phase short circuits and disconnection of the tie lines around the border between the north and central areas. During such faults, the power transfer from the north to the south is reduced and, therefore, the rotor-angle deviation increases between the groups of generators in the north and central/south areas. The result of the proposed time optimal CLF strategy is compared to the cases of no power modulation and local CLF control. The local CLF controller is given by the frequency deviation of the HVDC connecting buses amplified by a gain given by (30)

which is a linear optimization problem and is straightforward to solve at each time step. The problem is NP-hard but is only of dimension , since there are converters, which makes it possible to solve it in real time. The constraint in (26) is equivalent to the constraint in (22). Finally, the control law is given by (28) else

The gain is tuned to not go over the limits of the power changes in the converters. B. Contingency I A solid three-phase-to-ground fault occurs close to bus 4022 at one of the lines connecting buses 4022 and 4031, and the fault is cleared: 1) automatically without disconnection of the line;

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Fig. 3. Rotor-angle separation for generators G1043 and G1021 in the COI frame for Contingency Ia. To the right is a zoom of the first second.

Fig. 4. Rotor-angle separation for generators G4041 and G1022 in the COI frame for Contingency Ia. To the right is a zoom of the first second.

Fig. 1. Nordic32 power system with an MTDC system.

Fig. 5. Power modulation from north (Conv. 1) to south (Conv. 3) converters for Contingency Ia.

Fig. 2. Control interface between PowerFactory and Matlab.

2) by disconnection of the line. The fault duration is 230 ms and 180 ms in cases a and b, respectively. In Figs. 3 and 4, two generators’ rotor angles in the center of the inertia (COI) reference frame, which are representative of the system’s behavior, are plotted for Contingency Ia. Clearly, the rotor-angle separation is smaller using optimal Lyapunov control. After the first swing, the concern is to damp the power oscillations. The damping performance of the two controllers—local and time optimal CLF— is not off by much, but the time-optimal CLF adds better damping. Fig. 5 shows

the control signal for the two controllers. As can be seen, the switching points are not similar and the time-optimal CLF uses a maximum control signal, either plus or minus (i.e., bang-bang). The behavior is similar for Contingency Ib and is therefore not plotted. The transient stability enhancement is clearly indicated in Table I for both contingencies. C. Contingency II A solid three-phase-to-ground fault occurs close to bus 4032 at the line connecting buses 4021 and 4032, and the fault is cleared: 1) automatically without disconnection of the line; 2) by disconnection of the line. The fault duration is 230 and 220 ms in cases a and b, respectively. Figs. 6 and 7 show the result of Contingency IIb disconnecting the line. The figures indicate less rotor-angle separation


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TABLE I CRITICAL CLEARING TIME (in milliseconds)

Fig. 9. Rotor-angle separation for Generators G1043 and G1021 in the COI frame for Contingency IIIa. To the right is a zoom of the first second.

Fig. 6. Rotor-angle separation for generators G1043 and G1021 in the COI frame for Contingency IIb. To the right is a zoom of the first second.

Fig. 10. Rotor–angle separation for generators G4041 and G1022 in the COI frame for Contingency IIIa. To the right is a zoom of the first second.

Fig. 7. Rotor-angle separation for generators G4041 and G1022 in the COI frame for Contingency IIb. To the right is a zoom of the first second.

Fig. 11. Power modulation from north (Conv. 1) to south (Conv. 3) converters for Contingency IIIa.

D. Contingency III

Fig. 8. Power modulation from north (Conv. 1) to south (Conv. 3) converters for Contingency IIb.

in the first swing as a result of the proposed time-optimal control action. In addition, the power oscillations are damped well for the controllers. No plots are given for Contingency IIa since the behavior is similar to the previous case IIb, but the critical clearing time is given in Table I.

A solid three-phase-to-ground fault occurs close to bus 4031. At the line connecting buses 4031 and 4041, the fault is cleared: 1) automatically without disconnection of the line; 2) by disconnection of the line. The fault duration is 180 and 160 ms in cases a and b, respectively. Figs. 9 and 10 show the result of Contingency IIIa. The figures indicate less rotor-angle separation in the first swing as a result of the time-optimal CLF control action. The control signals are plotted in Fig. 11. The time-optimal CLF enhances the first swing stability is both cases which is also clear from the result in Table I. c) Contingency IIIc.

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improves the damping of power oscillations in the ac system, similar to local CLF. Nonlinear simulations are performed in the Nordic32 test system with a three-terminal MTDC grid which shows promising results. REFERENCES

Fig. 12. Rotor-angle separation for generators G1043 and G1021 in the COI frame for Contingency IIIc. To the right is a zoom of the first second.

Fig. 13. Rotor-angle separation for generators G1041 and G1022 in the COI frame for Contingency IIIc. To the right is a zoom of the first second.

In this contingency, the same fault as in contingency III occurs which is cleared automatically without disconnection of the line after 180 ms. The difference is the increased possibility to change the power in the converter in north part, converter 1. The converter limit gives room to change the power of 200 MW. The performance differs more by comparing the result of the time optimal and local CLF, shown in Figs. 12 and 13. E. Critical Clearing Time and the Effect of Time Delays in Contingencies I–III The critical clearing time for these contingencies is shown in Table I. It clearly indicates that the transient stability is enhanced using the time-optimal CLF. The time-optimal CLF increases the critical clearing time in each of the simulated contingencies. The effect of data latency has also been simulated for each contingency, simply by delaying the feedback signal 50 ms. The result is shown in Table I where a small negative effect, compared to the cases with no data latency, can be identified. X. CONCLUSION This paper presents a time-optimal power injection modulation control strategy for MTDC grids to enhance the rotor-angle stability of the ac system. The control strategy is based on Lyapunov control theory, formulated and solved as a time-optimal problem ending up as bang-bang control. Wide-area measurements are used as feedback signals to the controller deriving the active power injections of the MTDC grid. The controller also

[1] R. Eriksson, J. Beerten, M. Ghandhari, and R. Belmans, “Optimising dc voltage droop settings for ac/dc system interactions,” IEEE Trans. Power Del., 2014, to be published. [2] R. Eriksson and L. Söder, “Wide-area measurement system-based subspace identification for obtaining linear models to centrally coordinate controllable devices,” IEEE Trans. Power Del., vol. 26, no. 2, pp. 988–997, Apr. 2011. [3] R. Eriksson, “Coordinated control of HVDC links in transmission systems,” Ph.D. dissertation, Dept. Elect. Power Syst., School Elect. Eng., KTH Royal Inst. Technol., Stockholm, Sweden, Mar. 2011. [4] R. Eriksson and L. Söder, “Optimal coordinated control of multiple HVDC links for power oscillation damping based on model identification,” Eur. Trans. Elect. Power, vol. 22, no. 2, pp. 188–205, 2012. [5] S. Azad, R. Iravani, and J. Tate, “Damping inter-area oscillations based on a model predictive control (mpc) hvdc supplementary controller,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3174–3183, Aug. 2013. [6] T. Kailath, Linear Systems. Englewood Cliffs, NJ, USA: PrenticeHall, 1980. [7] M. Malisoff and F. Mazenc, Construction of Strict Lyapunov Functions. Berlin, Germany: Springer-Verlag, 2009. [8] M. A. Pai, Energy Function Analysis for Power System Stability. Norwell, MA, USA: Kluwer, 1989. [9] T. Athay, R. Podmore, and S. Virmani, “A practical method for the direct analysis of transient stability,” IEEE Trans. Power App. Syst., vol. PAS-98, no. 2, pp. 573–584, Mar. 1979. [10] T. Athay, V. R. Sherket, R. Podmore, S. Virmani, and C. Puech, “Transient energy analysis,” presented at the Syst. Eng. Power, Emergency Operating State Control, Sec. IV, U.S. Dept. Energy, NTIS, Springfield, VA, USA, 1979, Publ. no. CONF-790904-PI. [11] M. A. Pai, K. R. Padiyar, and C. Radhakrishna, “Transient stability analysis of multi-machine AC/DC power systems via energy-function method,” IEEE Trans. Power App. Syst., vol. PAS-100, no. 12, pp. 5027–5035, Dec. 1981. [12] A. A. Fouad and V. Vittal, “The transient energy function method,” Int. J. Elect. Power Energy Syst., vol. 10, no. 4, pp. 233–246, 1988. [13] H. Shaaban, “Lyapunov stability of multimachine power systems considering the voltage regulator,” Int. J. Elect. Power Energy Syst., vol. 24, no. 2, pp. 141–147, 2002. [14] H. Latorre, M. Ghandhari, and L. Söder, “Active and reactive power control of a VSC-HVdc,” Elect. Power Syst. Res., vol. 78, no. 10, pp. 1756–1763, 2008. [15] K. Padiyar and H. Sastry, “A structure-preserving energy function for stability analysis of AC/DC systems,” Sadhana, vol. 18, pp. 787–799, 1993. [16] C. Jing, V. Vittal, G. Ejebe, and G. Irisarri, “Incorporation of HVDC and SVC models in the northern state power co. (NSP) network; for on-line implementation of direct transient stability assessment,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 898–906, May 1995. [17] C. A. Canizares, “Voltage collapse and transient energy function analysis of ac/dc systems,” Ph.D. dissertation, Univ. Wisconsin-Madison, Madison, WI, USA, 1991. [18] N. Fernandopulle and R. Alden, “Incorporation of detailed HVDC dynamics into transient energy functions,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 1043–1052, May 2005. [19] N. Fernandopulle and R. Alden, “Improved dynamic security assessment for AC/DC power systems using energy functions,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1470–1477, Nov. 2003. [20] R. Eriksson and L. Söder, “On the coordinated control of multiple HVDC links using input-output exact linearization in large power systems,” Int. J. Elect. Power Energy Syst., vol. 43, pp. 118–125, Dec. 2012. [21] R. Eriksson, “On the centralized nonlinear control of HVDC systems using lyapunov theory,” IEEE Trans. Power Del., vol. 28, no. 2, pp. 1156–1163, Apr. 2013.


[22] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ, USA: Pearson Education, 2000. [23] N. Kakimoto, Y. Ohsawa, and M. Hayashi, “Transient stability analysis of multimachine power system with field flux decays via Lyapunov’s direct method,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 5, pp. 1819–1827, Sep. 1980. [24] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. New York: Springer, 1998. [25] DIgSILENT, Information About PowerFactory. Apr. 2013. [Online]. Available: http://www.digsilent.com

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Robert Eriksson (M’11) received the M.Sc. and Ph.D. degrees in electrical engineering from the KTH Royal Institute of Technology, Stockholm, Sweden, in 2005 and 2011 respectively. Currently, he is Associate Professor with the Center for Electric Power and Energy, Technical University of Denmark. He is also Part-Time Researcher at the Department of Electric Power Systems, KTH Royal Institute of Technology. His research interests include power system dynamics and stability, HVDC systems, dc grids, and automatic control.