On the Centralized Nonlinear Control of HVDC Systems ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 2, APRIL 2013

On the Centralized Nonlinear Control of HVDC Systems Using Lyapunov Theory Robert Eriksson, Member, IEEE

Abstract—The security region of a power system is an important and timely issue; different stability criteria may be limiting. Rotor-angle stability can be improved by modulating active power of installed high-voltage direct current (HVDC) links. This paper proposes a new centralized nonlinear control strategy for coordinating several point-to-point and multiterminal HVDC systems based on Lyapunov theory. The proposed control Lyapunov function is negative semi-definite along the trajectories and uses the internal node representation of the system. The proposed control Lyapunov function increases the domain of attraction and, thus, improves the rotor-angle stability. Nonlinear simulations are performed on the IEEE 10-machine 39-bus system which shows the effectiveness of the controller. In comparison, simulations using the conventional lead-lag controller are also run. Index Terms—Control Lyapunov function, coordinated control, energy function, high-voltage direct current (HVDC), multiterminal dc (MTDC), rotor-angle stability.

I. INTRODUCTION

S

TABILITY phenomena such as power oscillations and transient stability are important in power system stability analysis. The power oscillation phenomenon consists of synchronous generator rotors swinging relative to each other. Under severe disturbances the first swing may be considerably large and endanger the stability of the system. If the first swing is handled and reaches transient stability, there might still be a risk of losing synchronism due to low damping torque or voltage instability. The installed capacity of high-voltage direct-current (HVDC) links is increasing around the world [1]. Improved technology, such as voltage-source converters (VSCs) and multiterminal dc (MTDC) systems, brings more possibilities than conventional point-to-point HVDC links. The power modulation control of HVDC links enhances the security [2]–[4]. This, in turn, may allow more power to be transferred, and has an economic impact [3]. When designing the controller for an HVDC link, the aim is to improve the stability; one also has to ensure that it does not interact negatively with the system in any situation. Higher demand comes with having several controllable devices in the system since one also has to ensure that negative interactions between the devices do not occur, coordination is therefore necessary and highlighted in [5]. Besides one wants to improve the Manuscript received August 20, 2012; accepted December 15, 2012. Date of publication February 04, 2013; date of current version March 21, 2013. This work was supported by the Centre of Excellence in Electrical Engineering at the Royal Institute of Technology. Paper no. TPWRD-00873-2012. The author is with the Department of Electric Power Systems, KTH Royal Institute of Technology, Stockholm 100-44, Sweden (e-mail: robert.eriksson@ee. kth.se). Digital Object Identifier 10.1109/TPWRD.2013.2240021

security region by coordination, utilizing the system in a more efficient way [3]. In addition, this includes wide-area measurement systems (WAMS) put together with different controllable devices [6], [7]. The inherited nature of power systems provides nonlinear dynamics. 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 [8]. In general, control systems may not be LTI systems due to, for example, nonlinear dynamics. These linear design and analysis methods cannot necessarily be applied directly without special consideration. The method of Lyapunov functions plays a central role in the study of control systems, more specifically nonlinear control systems. Lyapunov introduced a criterion for the stability of nonlinear systems, a property where all trajectories of the system tend to the origin. This criterion involves the existence of a certain function, now known as a Lyapunov function. Later, 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 a Lyapunov function for nonlinear systems. Faced with specific systems, we have to use experience, intuition, and physical insights, such as system energy, to search for an appropriate Lyapunov function [9], [10]. The objective of this paper is to develop a new centralized control Lyapunov function to coordinate point-to-point HVDC systems or MTDC systems, and aims to improve the rotor-angle stability margin. Much research has been published concerning the energy function for power systems and single controllable components, and the proposed control Lyapunov function coordinates HVDC systems. Related works are presented in the next section. II. RELATED WORKS Energy or Lyapunov functions for power systems were developed in the late 1970s by Athay et al. [11], [12]. The use of energy functions or similar to estimate regions of attraction for stable equilibrium in electric power systems has a long history. The use of energy-based Lyapunov functions for power systems is well developed in classical literature, such as [13] and the references therein. Energy or energy-like functions are often used as Lyapunov function candidates. A state-of-the-art paper by Fouad and Vittal reviews the transient energy method [14]. In [15], an automatic voltage regulator (AVR) is included in the Lyapunov stability of multimachine power systems.

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ERIKSSON: ON THE CENTRALIZED NONLINEAR CONTROL OF HVDC SYSTEMS

In [16], Lyapunov control 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 gain as the input for the active power modulation. One drawback with this is that the speed deviation is zero at maximum angle deviation. Also, the HVDC link is modeled by a simple model which neglects the dynamics. In [10] and [13], 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. The dynamics of the HVDC link consist of one differential equation. Only one HVDC link is included, and the speed difference of the closest generators is used as the input for the HVDC controller. In [13] and [17]–[20], the internal HVDC links dynamics are incorporated in the transient energy function. The HVDC damping controller is included in the transient energy function [17], [20]. III. CONTRIBUTIONS This paper extends the theory in [13] by developing a new centralized control Lyapunov function for several point-to-point HVDC or MTDC systems. This nonlinear centralized control strategy coordinates the HVDC systems and improves the rotorangle stability margin. The proposed control Lyapunov function 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 setpoints change of the HVDC systems as the control signal. The developed control Lyapunov function makes the time-derivative negative along the trajectories to the origin. The control strategy can be applied to single and coordination of several point-to-point HVDC links or multi-terminal HVDC links in multi-machine systems. To the best knowledge of the author, no publications present controllers which coordinate several HVDC links based control Lyapunov function method. IV. REMOTE SIGNALS Remote signals utilized by the wide-area monitoring and control (WAMC) system applications increase the observability of rotor-angle stability compared with local measurements [21]. The WAMC comes along with how to effectively utilize the WAMS for power state estimation and stability enhancement by power system stabilizer (PSS), flexible ac transmission system (FACTS) devices and HVDC systems [21]–[24]. The feedback signals should have high observability to identify stability issues enabling appropriate control actions. In [25], advantages are shown by using remote measurements as feedback signals compared with using local signals for stability enhancement. For optimal placement of phasor measurement units (PMUs) one needs to find the optimal locations by looking at the observability matrix, this matrix can be constructed in different ways. Observability factors based on residues have the drawback of scaling problem when

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different measurement signals are involved [26]. Instead observability factors based on geometrical factors are more robust and accurate [27]. The higher observability is observed using rotor-angle speed of the generators. Furthermore, the controllability is the measure of the impact seen from the input to the output. The controllability is inherited by the location of the installed controllable device in the power system. The impact of an input signal to the system’s states is described later on. V. PROBLEM DESCRIPTION USING CONTROL LYAPUNOV THEORY In control theory, a control Lyapunov function is a generalization of the notion of Lyapunov function used in stability analysis [28]. The system energy is replaced by this scalar function . The ordinary Lyapunov function is used to test whether a dynamical system is stable (more restrictively, asymptotically stable). That is, whether the system starting in a state in some domain will remain in , or for asymptotic stability will eventually return to . The control Lyapunov function is used to test 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 . The time derivative of the Lyapunov function is given by (1) In a stable power systems, the time derivative of the uncontrolled system is (2) For the controlled system, the time derivative is given by (3) To make a control Lyapunov function one needs to find the conis negative definite (or at least negtrol vector such that ative semi-definite). VI. TRANSIENT STABILITY ASSESSMENT Let be the stable e.p. (equilibrium point) and be the unstable e.p. for is the stability boundary of the stable e.p. and be the stability manifold of the unstable e.p. Then (4) The point where the unstable trajectory crosses the stability boundary (i.e., ) is the exit point of this trajectory. If the point is close to then is the true critical energy [14]. assumes its minimum on the stable manifold, implying the unstable trajectory passing through the constant energy surface (5)

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Fig. 1. Internal control system in the HVDC link.

To find whether a contingency is stable 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 (6) and positive value implies the system is stable [14]. Using the system trajectory along the contingency one can derive the improvement of using the proposed control Lyapunov function. To securely operate the system the system operators should keep an adequate margin .

B. HVDC Model The control method to be developed can be used for both conventional-type line commutated converter (LCC) and voltage source converter (VSC). One may control several point-to-point HVDC links and MTDC systems. 1) LCC HVDC: For the LCC HVDC, the current through the HVDC link is controllable, and the LCC HVDC link is modeled as follows: (11) where DC current through the HVDC link;

VII. POWER SYSTEM MODELING This section follows in general the internal node description as in [10]. In [10] the effect of a single HVDC link on each generator is derived, this is extended to include several HVDC systems.

current set-point through the HVDC link; current change time constant.

where contains the state variables and contains the algebraic variables. Using the internal node representation the power system modeling can be written as only a set of first-order differential equations [10]. The dynamics of generator is described by

For each HVDC link, the setpoint signal is placed in the input vector which is the actual control signal. Measuring the bus voltage, one can instead control the active power, similar to VSC. The power through an LCC HVDC link is controlled by controlling the firing angle . An overview of the internal control system is displayed in Fig. 1. In the figure, the control Lyapunov function (CLF) and he power oscillation damping (POD ) controller also could be installed. 2) VSC HVDC: For the VSC, one may control the active and reactive powers independently of each other. Transient stability is mainly affected by active power modulation. Reactive power control focus on locally controlling the connected bus voltage. One may instead model the VSC HVDC link, similar to LCC HVDC, as follows:

(9)

(12)

A. AC Power System Modeling A power system can be described by a set of differential and algebraic equations. (7) (8)

To model the VSC HVDC, one basically sets the active power setpoints as the control variable following the same procedure as described in the next section. C. Internal Node Description With HVDC (10) where is the mechanical power, is the moment of inertia, is the internal voltage, and and are the rotor angle and speed of the generator , respectively. is the bus admittance matrix, , and is the damping. is the impact of the HVDC links on generator [10], [29], described in next subsection.

The effect of the HVDC link be expressed as [10] and [29]

given as

can

(13)


(14)

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it has converters, one of the converters acts as the slack. The time derivative of the energy function along the trajectories of system (21) is given by

(15) is element in B, derived for all HVDC links and where generators [10], [29]. Including the HVDC system dynamics, the following is achieved: (16) (22)

(17) (18) where (18) is a vector containing the first-order differential equations of the HVDC links. and contain the actual and setpoints values, respectively, and is the vector of time constants. The system is now described by first-order differential equations. VIII. ENERGY FUNCTION The system energy is treated mathematically by the scalar function . To formulate the energy function one needs to use the approach of center of inertia. It is given as

The following equation is achieved:

(23) To make (23) a Lyapunov candidate, it needs to satisfy the asymptotic stability condition. For this function, the time derivative needs to be negative along the trajectories. If the HVDC links are not controlled, (23) is identically zero. Formulating a control Lyapunov function that implies a negative part of the first term in (23) is a control Lyapunov function. This function is proposed in next section. IX. CONTROL STRATEGY To make (23) negative definite, the new control Lyapunov function can now be formulated as

The energy function for the power system without HVDC is given by [10]

.. .

.. .

.. .

(19) (20) Including the dynamics of the HVDC links, the proposed energy function is given by

(24) This implies the time derivative of the energy function to be negative semi-definite along the trajectories. It is not negative definite because for , where and are positive matrices. The time derivative is given by (25)

(21) where is the number of point-to-point HVDC links or MTDC system. An MTDC system is said to have controllable links if

needs to Naturally, to decrease the second term in (25), be different from zero. To improve transient stability it is reasonable to make the input set-points dependent on angle deviation from steady-state. Using only speed deviation would have less impact. In a stable first swing at the largest angle separation, the speed deviation is zero, using only speed deviation would then not apply any

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Fig. 2. Overview of the CLF controller.

power change of the HVDC links. By including the angle deviation power change is applied along the first swing until it goes back. Once the first swing is over one may only use speed deviation as input to dampen the remaining oscillations. An overview of the Lyapunov control is depicted in Fig. 2 where the sign of decides by the sign of . X. NONLINEAR SIMULATIONS Simulations are performed on the IEEE 10-machine 39-bus system, data for this system are given in [10], [11]. The 39-bus system is well-known and represents a greatly reduced model of the power system in New England. It has been used by numerous researchers to study both static and dynamic problems in power systems. The system has 10 generators, 19 loads, 36 transmission lines, and 12 transformers. Generator 1 represents the aggregation of a large number of generators. In addition, three HVDC systems are installed, and the system is shown in Fig. 3. The power change during the contingencies of the HVDC links is limited to 200 MW. The new control Lyapunov function is compared to no control and lead-lag control using local inputs, frequency deviation of the HVDC connecting buses.

Fig. 3. IEEE 10-machine 39-bus system including three HVDC systems.

A. Contingency 1 A solid three-phase-to-ground fault occurs at the middle of the line between Buses 16 and 17. The fault is cleared by disconnection of the faulted line after 150 ms. The result is shown in Fig. 4 for Generator 5, which is representative for the system’s behavior. The result clearly shows the effectiveness of the developed controller. Especially, large reduction of the transient angle separation is seen and one can conclude that the transient stability is highly improved. Increased damping can also be seen which is also an important aspect. Nevertheless, the damping is also improved using lead-lag power modulation control. The main difference is the reduction of the transient which hardly is affected by lead-lag control. Fig. 5 plots the normalized energy function comparing different controls. The effectiveness can clearly be seen of the proposed controller since it is in the lower peak value and it decays much faster.

Fig. 4. Contingency 1—rotor angle Generator 5.

B. Contingency 2 In this contingency, a solid three-phase-to-ground fault occurs at the middle of the line between Buses 2 and 25, and the fault is clear after 130 ms by disconnecting the faulted line. Fig. 6 shows the angle versus time for Generator 5. The rotorangle stability is significantly improved for this contingency too. C. Contingency 3 A similar fault occurs as in Contingency 2; however, the fault is cleared after 130 ms without disconnecting the line. In Fig. 7, the phase plane for Generators 9 and 7 is plotted. Clearly, the transient stability is improved and the damping is increased.


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Fig. 5. Normalized energy function.

Fig. 6. Contingency 2—rotor-angle Generator 5.

The figure shows the phase portrait which gives an overview of speed deviation versus angle.

Fig. 7. Phase plane for Contingency 4.

D. Contingency 4 For this contingency, a solid three-phase-to-ground fault occurs at the middle of the line between Buses 16 and 21; furthermore, the fault is clear after 200 ms. In Fig. 8, the result applying this fault is shown, comparing Lyapunov control, lead-lag control, and no power modulation. The coordinating controller also improves the stability margin for this contingency. E. Critical Clearing Time The stability margin indicates the margin to the minimum boundary of instability. To visualize the transient stability improvement of the proposed controller the critical clearing time is found through simulations for the applied contingencies. The critical clearing times for each contingency, obtained from nonlinear simulations, are shown in Table I. It clearly shows improved results and improved transient stability. F. Loadability Surface Instead of looking at the stability improvement, loadability limits are studied. We look at how much we can increase the power transfers but keep the rotor-angle stability margin as in

Fig. 8. Fault III—rotor angle Generator 5.

lead-lag control. The loadability surface is made up of all loadability limit points in parameter space. We let (26) is the load at the present operating point. We increase where in one direction in load space until we reach , which

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TABLE I CRITICAL CLEARING TIME

rotor-angle stability for different contingencies and loadings. The influence of signal latency is also demonstrated and is handled well by the controller.

REFERENCES TABLE II INCREASED POWER TRANSFER

TABLE III CRITICAL CLEARING TIME

is the margin in the uncontrolled case. Then, we apply the developed controller in nonlinear simulations, increasing the load until reaching the new . Table II shows the amount of power we may increase in direction . This, in turn, means that we can transfer more power between different regions in the system. By keeping a certain stability margin (e.g., a certain value of ), we may rate the economic impact of the proposed control Lyapunov function. The direction in load space has an impact of . G. Signal Latency Signal processing and remote measurements contribute to signal latency which should be kept to a minimum. Large signal latency has an impact on the control action during transients. In power oscillation damping, one may apply proper phase compensation to compensate for reasonably small-signal latency. Signal latency has a minor impact on power oscillation damping if it is compensated correctly [30]. In this case, we apply signal latency of 50 ms to see the impact on the transient behavior; an overview of the result is shown in Table III. It can be seen that the controller performance is slightly reduced compared to no signal latency; however, the controller performance is promising. XI. CONCLUSION This paper derives a new control Lyapunov function to coordinate the power modulation of MTDC or several point-to-point HVDC systems in order to increase the domain of attraction. The proposed centralized controller uses wide-area measurements to derive the coordinated action of the HVDC systems. The proposed controller decreases the postdisturbance energy which, in turn, improves the rotor-angle stability margin. Nonlinear simulations are run in the IEEE 10-machine 39-bus system; the centralized control Lyapunov function improves the

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[29] 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. [30] N. Chaudhuri, B. Chaudhuri, S. Ray, and R. Majumder, “Wide-area phasor power oscillation damping controller: A new approach to handling time-varying signal latency,” IET Gen., Transm. Distrib., vol. 4, no. 5, pp. 620–630, 2010. Robert Eriksson 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 a Postdoctoral Researcher in the Division 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.