Application of the direct Lyapunov method to improve damping of power swings by control of UPFC

MODERN ELECTRIC POWER SYSTEMS Application of the direct Lyapunov method to improve damping of power swings by control of UPFC M. Januszewski, J. Machowski and J.W. Bialek Abstract: Large interconnected power systems often suffer from weakly damped swings between synchronous generators and subsystems. This paper presents an approach, based on the use of the nonlinear system model and application of the direct Lyapunov method, to improve damping of power swings using the unified power flow controller (UPFC). A state-variable control strategy has been derived as well as its implementation using locally available signals of real and reactive power. The results of simulation tests, undertaken using a small multi-machine system model, have been presented. 1 Introduction Weakly damped power swings are one of the main problems endangering secure operation of power systems. Damping can be improved by means of a power system stabiliser (PSS) or by using one of the flexible AC transmission systems (FACTS) devices. Generally speaking, PSS increases damping torque of a generator by affecting the generator excitation control, while FACTS devices improve damping by modulating the equivalent power- angle characteristic of the system [1]. In this paper we consider the stabilising control of the most versatile device in the FACTS family (i.e. the unified power flow controller, UPFC). FACTS devices and their power system applications are described in [2]. The problem of their modelling has been covered by many authors e.g. [3, 4]. From the point of view of power system dynamics, the essential problem is how to control specific FACTS devices, and in particular UPFC. For example, one approach is to apply optimal control [5–8]. The problem with standard optimal control is that it tends to use a linearised system model, which is valid only for a given operating point. This raises the question of robustness, as control based on linearised system model is valid only when the system is in the vicinity of the chosen operating point and one can never be sure whether or not the control will still be satisfactory when the system operating conditions change or when the system model changes due to line or generator outages. Moreover, stressed power systems are known to exhibit nonlinear behaviour. Hence the motivation for the work reported here was to derive a state-variable control using a nonlinear system model in order to take into account the influence of changing operating conditions and changes in the network parameters. To achieve this goal, the direct Lyapunov method was used. The first use of the direct Lyapunov method to design power system stabilising controllers was pioneered by the authors of this paper in the early 1990s by considering the nonlinear control of such shunt FACTS devices as the static var compensator (SVC) or supercondunting magnetic energy storage (SMES) [9, 10]. Later on, this approach was formulated in a more systematic way [1] and was extended to other FACTS devices such as the braking resistor [11], the series static compensator [12] and also to the PSS of a synchronous generator [13, 14]. Lyapunov or energy-based control methods have also been investigated by other authors [15, 16]. In this paper we shall first derive the state-variable control strategy, which improves damping of power swings by maximising the speed with which, following a dis- turbance, the system operating point returns to the equilibrium point. The resulting state-variable control provides damping independent of operating conditions, which is a fundamental advantage of the derived control. However, the problem with state-variable control is that its implementation in a real multi-machine system requires estimation of all the required state variables (rotor angles and speed deviations of all the generators). This is a very complicated problem in itself, which additionally requires reliable wide-area measurements and extended communica- tion links. To avoid those problems, a much simpler local control (i.e. control based on local measurements) will be derived, which emulates well the state-variable nonlinear control. This is achieved by using local measurements of real and reactive power and by carefully choosing gains in the supplementary control loops. Simulation tests have validated the analysis by confirming that the state-variable control results in excellent damping, while the stabilising control based on local signals is easier to execute and results in a good damping, although weaker than that obtained for the state-variable control. 2 Unified power flow controller The unified power flow controller (UPFC) belongs to the power electronics-based family of FACTS devices. It can be M. Januszewski and J. Machowski are with the Instytut Elektroenergetyki, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland J.W. Bialek is with the School of Engineering and Electronics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, UK r IEE, 2004 IEE Proceedings online no. 20040054 doi:10.1049/ip-gtd:20040054 Paper first received 30th January 2003 and in revised form 19th November 2003 252 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 connected in series with a transmission line inside a system or in a tie-line connecting subsystems in a large inter- connected system. As shown in Fig. 1, the UPFC consists of the booster transformer TB and the excitation transformer TE linked by back-to-back converters GTO1 and GTO2. The aim of the booster transformer TB is to inject into the transmission line additional voltage DU controlled both in magnitude and phase. The main task of the UPFC is to control the flow of power in steady-state conditions. However, high speed of operation of thyristor devices makes it possible to also use UPFC dynamically, to improve damping of power swings. That area of application is the main subject of this paper. 3 Power system model with UPFC 3.1 Assumptions To generalise the considerations, it is assumed that the UPFC is installed at point b located anywhere inside the transmission link (Fig. 2). From the point of view of influencing the steady-state and the transient state, the thyristor controlled UPFC can be treated as a lag element with a very small time constant or, approximately, as a proportional element [3]. Hence the series part of the UPFC (booster) can be modelled by the series reactance included (Fig. 2(b)) in the reactance of the left-hand side of the transmission link Xa and by the ideal complex transformation ratio: Z ¼ Ua Ub ¼ jZjejy; Ib Ia ¼ Z� ¼ jZje�jy ð1Þ The shunt part of the UPFC can be modelled as a controlled shunt susceptance Br (Fig. 2(b)). To derive the UPFC control strategy, a simplified system model will be used, which is described below. The derived control strategy will be then validated by simulation using a more detailed system model: one normally used for transient stability studies. The network resistance in Fig. 2(b) is neglected in order to simplify equations. The synchronous generator is modelled by EMF Eg¼E0 behind the transient reactance X 0d. This reactance is included in the left-hand side reactance Xa of the transmission link. In this simplified generator model the angle of the EMF, d¼ arg Eg, is also the power angle of the system and the rotor angle (i.e. the angle between the rotor and the synchronously rotating reference axis defined by the infinite busbar voltage [1]). Voltage DU injected by the booster transformer can be resolved (Fig. 2(c)) into two orthogonal components: direct DUQ and quadrature DUP. Both are proportional to the voltage Ub feeding the excitation transformer: DUQ ¼ bjUbj; DUP ¼ gjUbj ð2Þ sin y ¼ DUPjUaj ¼ gjUbj jUaj ¼ g jZj ð3aÞ cos y ¼ jUbj þ DUQjUaj ¼ jUbj þ bjUbj jUaj ¼ 1þ b jZj ð3bÞ ð1þ bÞ2 þ g2 ¼ jZj2 ð4Þ The direct component DUQ influences the reactive power flow, hence subscript Q. Similarly, the quadrature compo- nent DUP influences the real power flow, hence subscript P. 3.2 Nodal network equations The branch g–b in the left-hand side of Fig. 2(b) can be treated as the transformer with complex transformation ratio Z and admittance Ya¼ 1/Za, where Za¼ jXa. As shown in Chapter 3.2 of [1], it can be proved that such a branch, together with Z, can be transformed into the p- equivalent passive circuit, without the transformation ratio Z but with all branches depending on value of Z. Using this equivalent circuit, the nodal equations describing the whole network (Fig. 2(b)) can be written as: Ig Is 0 2 4 3 5 ¼ Ya 0 �ZYa 0 Yb �Yb �Z�Ya �Yb jZj2Ya þ Yb þ Yr 2 4 3 5 EgUs Ub 2 4 3 5 ð5Þ The first and the second row correspond to the generating nodes (generator and infinite busbar), while the third node corresponds to the network node b. Eliminating this node by means of partial inversion [1] results in: Ig Is � � ¼ Y �bb ðYaYb þ YaYrÞ �ZYaYb �Z�YaYb ðjZj2YaYb þ YbYrÞ � � Eg Us � � ð6Þ Ub ¼ Z�YaY �1bb Eg þ YbY �1bb Us ð7Þ Ybb ¼ jZj2Ya þ Yb þ Yr ð8Þ GTO 1 GTO 2 ∆U TE TB Fig. 1 Schematic diagram of UPFC g a b sI g X a η bI b X sI sUbUaUgE r g a b s bU aU ∆U ∆ U ∆U P Q a b c B Θ Fig. 2 Generator-infinite busbar system with UPFC a Schematic diagram, b Single-phase diagram, c Phasor diagram IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 253 Admittance Ybb is the self-admittance of the node b, where UPFC is connected (Fig. 2). In this nodal approach all the currents and voltages are in the same network reference frame, where: Eg ¼ jEgjejd; Us ¼ jUsj ð9Þ and d is the above defined power angle [1]. 3.3 Generator power Apparent power of the generator is given by the well-known formula Sg ¼ EgI�g , where generator current Ig is deter- mined by (6). Hence: Sg ¼EgI�g ¼ EgðY �1bb Þ� ðY �a Y �bE�g þ Y �a Y �r E�g � Z�Y �a Y �bU �s Þ ð10Þ After substituting into this equation the exponential expressions from (1) and (9), it is possible to derive apparent power Sg¼Pg+jQg as the function of the power angle d and the angle y of the complex transformation ratio Z. Then, neglecting the network resistance and using some simple but arduous algebra [17], one can calculate Pg¼ReSg as: Pg ¼jUsjjEgjB�1bb BaBbjZj ½sin d cos y� cos d sin y� ð11Þ where Ba¼ 1/Xa, Bb¼ 1/Xb and similarly as in (8): Bbb ¼ jZj2Ba þ Bb þ Br ð12Þ Equation (11), after taking into account (3), can be written as: Pg ¼ jUsjjEgjB�1bb BaBb½ð1þ bÞ sin d� g cos d� ð13Þ In (13) the real power of the generator is an explicit function of the transformation ratio g and b. The factor B�1bb BaBb, relating the generator real power to the network parameters and the shunt branch Br, can be transformed in the following way: B�1bb BaBb ¼ BaBb jZj2Ba þ Bb þ Br ¼ Xr X�Xr þ XaXb ð14Þ where X� ¼ Xa þ jZj2Xb ð15Þ is the equivalent reactance of the transmission link seen from the generator node to the infinite busbar without taking into account the shunt branch (Fig. 2(b)). Equation (14) can be shown as a sum in the following way: B�1bb BaBb ¼ 1 X� � 1 X� XSHC Xr þ XSHC ð16Þ where XSHC ¼ XaXbX� ¼ XaXb Xa þ jZj2Xb ð17Þ is a quantity corresponding to the short-circuit reactance at the node b where the booster transformer is connected (Fig. 2). As the rating of the shunt branch of UPFC is at least an order of magnitude smaller than the short-circuit power in a power system, one can assume that XrcXSHC. This can be used to simplify (16) to the following form: B�1bb BaBb ffi 1 X� � 1 X� XSHC Xr ¼ B� � B�BrXSHC ð18Þ Substituting (18) into (13) gives, after some simple algebra: Pg ¼b� sin d� XSHCBrb� sin dþ ð1� XSHCBrÞ ðbb� sin d� gb� cos dÞ ð19Þ where b� ¼ jUsjjEgjB� is the amplitude of the power-angle characteristic without the shunt branch. 3.4 Generator equations Generator dynamics is described by the well-known swing equation [1]: M dDo dt ¼ Pm � PgðdÞ � D dd dt ð20Þ where d is the earlier defined power (rotor) angle and Do¼ dd/dt is the rotor speed deviation, M is the inertia coefficient, Pm is the mechanical power, Pg is the electrical real power, D is the damping coefficient. Substituting (19) into (20) gives: dd dt ¼ Do ð21aÞ M dDo dt ¼Pm � b� sin d� D dd dt þ XSHCBrb� sin d � ð1� XSHCBrÞ½bb� sin d� gb� cos d� ð21bÞ The above equations form nonlinear state equations _xx ¼ f ðx; uÞ, in which x¼ (d, Do) are the state variables, while u¼ (g, b, Br) are the control variables. The equilibrium point x^x of the state equations has the following coordinates: x^x ¼ ðd^d; Do^o ¼ 0Þ. 3.5 Power in the transmission link The current Ib entering the booster transformer can be calculated from Ohm’s Law for branch Yb as Ib¼Yb(Us�Ub). Using this equation and (7), it is possible to derive the following equation for the apparent power flowing through the booster transformer: Sb ¼UbI�b ¼ Y �1bb ðZ�YaEg þ YbUsÞ bY �bU�s ð1� Y �1�bb Y �b Þ � ZY �1�bb Y �b Y �a E�gc ð22Þ After substituting into this equation the exponential components from (1) and (9), it is possible to derive an equation for Sb¼Pb+jQb as the function of the power angle d and the angle y of the complex transformation ratio Z. Again, neglecting network resistance and using simple but arduous algebra, gives [17]: Pb ¼� jEgjjUsjB�1bb BaBbjZjðsin d cos y � cos d sin yÞ ð23Þ Qb ¼� jEgjjUsjB�1bb BaBbjZjð2BbB�1bb � 1Þ cos d cos yþ sin d sin y½ � � B�1bb Bb ½B�1bb jEgj2B2ajZj2 � jUsj2Bbð1� B�1bb BbÞ� ð24Þ Equation (23) is identical to (11) apart from the sign, as Pb¼�Pg. This is due to neglecting the network resistance and therefore the power loss. Taking into account (3) and (18), one can rewrite equation (23) in the form similar to (19): Pb ¼� b� sin dþ XSHCBrb� sin d� ð1� XSHCBrÞ bb� sin d� gb� cos d½ � ð25Þ 254 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 Substituting for sine and cosine in (24) expressions resulting from (3), one obtains: Qb ¼� jEgjjUsjB�1bb BaBbð2BbB�1bb � 1Þ ½ð1þ bÞ cos dþ g sin d� � B�1bb Bb ½B�1bb jEgj2B2ajZj2 � jUsj2Bbð1� B�1bb BbÞ� ð26Þ Factor jEgjjUsjBaBbB�1bb is identical to (13) describing the real power. Hence, taking into account notation from (18) and (19), one can write: jEgjjUsjBaBbB�1bb ¼ b�ð1� XSHCBrÞ ð27Þ The following coefficients can be introduced: KX ¼ ð2BbB�1bb � 1Þ ¼ Xa � jZj2Xb � XaXbXr Xa þ jZj2Xb þ XaXbXr ð28Þ Qb0 ¼B�1bb Bb½jZj2jEgj2B�1bb B2a � jUsj2Bbð1� B�1bb BbÞ� ð29Þ Substituting (27)–(29) into (26) gives: Qb ¼� KXb� cos dþ KXXSHCBrb� cos d� KXð1� XSHCBrÞ½bb� cos dþ gb� sin d� � Qb0 ð30Þ The structure of this equation is very similar to the structure of (25) expressing the real power. It is worth noting that both real power (25) and reactive power (30) depend on the power angle d and control variables g, b, Br. 4 Control strategy based on the direct Lyapunov method The way UPFC is controlled in order to enhance damping of power swings in power system will be referred to as the control strategy. 4.1 Direct Lyapunov method Let V(x) be a Lyapunov function defined for the power system model described by (21). Any disturbance in power system involves a power imbalance that moves the system trajectory from the pre-fault stable equilibrium point to a transient point x(t) that has a higher energy level than the post-fault equilibrium point #x. If ’V¼ dV/dt is negative, Lyapunov function V(x) decreases with time and tends towards its minimum value, which appears at the post-fault equilibrium point #x. The more negative the value of ’V, the faster the system returns to the equilibrium point #x (i.e. the better the damping in the system). Consequently, any given control is optimal (in the Lyapunov sense) if it maximises the negative value of ’V at each instant of the transient state. 4.2 Application to the stabilising control of UPFC For the considered generator-infinite busbar system and when the network resistance has been neglected, the Lyapunov function V can be defined [1] as the total system energy: V ¼ EK þ EP ð31Þ where EK ¼ 1 2 MðDo� Do^oÞ2 ¼ 1 2 MDo2 ð32aÞ EP ¼ �½Pmðd� d^dÞ þ b�ðcos d� cos d^dÞ� ð32bÞ are kinetic and potential energy, respectively, and d^d; Do^o ¼ 0 are the co-ordinates of the post-fault equilibrium point. At the post-fault equilibrium point (d^d; Do^o ¼ 0), the total energy (31) is equal to zero, V¼ 0. Any disturbance causes rotor acceleration or deceleration and therefore nonzero speed deviation Do and a change in the power angle d. This moves the current transient point from the equilibrium point, causing V40. The goal of UPFC control strategy is to enforce such changes of control variables g(t), b(t), Br(t), which make the system return as quickly as possible to the post-disturbance equilibrium point (d^d; Do^o ¼ 0) at which V¼ 0. To achieve this goal, the control strategy must, at any instant of the transient state, maximise the negative value of derivative ’V¼ dV/dt calculated along the trajectory of (21). It can be easily proved for (32) that: dEK dt ¼ @EK @o dDo dt ¼ M dDo dt Do ð33Þ dEP dt ¼ @EP @d dd dt ¼ @EP @d Do ¼ � Pm � b� sin d½ �Do ð34Þ Substituting into (33) the expression resulting from (21b) one obtains: dEK dt ¼þ ½Pm � b� sin d�Do� DDo2 þ Brb�XSHC sin dDo� ð1� XSHCBrÞ ½bb� sin d� gb� cos d�Do ð35Þ Adding both sides of (34) and (35) results in: _VV ¼ dEP dt þ dEK dt ¼� DDo2 � bð1� XSHCBrÞb� sin dDo þ gð1� XSHCBrÞb� cos dDo þ BrXSHCb� sin dDo ð36Þ This equation shows that each control variable g, b, Br can contribute to the power system damping by increasing the negative value of ’V¼ dV/dt. It should be noticed that the product XSHCBr is very small as XrcXSHC (where Xr¼ 1/Br). Consequently, the factor (1�XSHCBr)D1 has no influence on the sign of the second and third component in (36). For each of the control variables g, b, Br to contribute to the negative value of ’V¼ dV/dt, the control strategy should ensure that all the components in (36) are negative, independently of the sign of power angle d and speed deviation Do. This can be achieved if: gðtÞ ¼ �Kg½b� cos d�Do ð37Þ bðtÞ ¼ þKb½b� sin d�Do ð38Þ BrðtÞ ¼ �KB½b� sin d�Do ð39Þ where Kg, Kb, KB are positive coefficients. For the control strategy (37), (38), (39) assuming approximately that (1�XSHCBr)D1, one obtains: _VV ¼� DDo2 � ½Kgðcos dÞ2 þ Kbðsin dÞ2 þ KBXSHCðsin dÞ2�b2�Do2 � 0 ð40Þ Each component in the square bracket is always positive, no matter what is the value of power angle d. That means that each of the controls (37)–(39) increases negative value of ’V hence improving damping of power swings in the system. The damping influence of the controls can easily be confirmed by substituting (37)–(39) into (21b). After simple transformations, and assuming as above that IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 255 (1�XSHCBr)D1, one obtains: M dDo dt ¼ Pm � b� sin d� DDo� DPcontrolðdoÞ ð41Þ where DPcontrolðdoÞ ¼ DcontrolðdoÞDo ð42Þ is the damping power introduced by the state-variable control of UPFC, and DcontrolðdoÞ ¼ðXSHCKB þ KbÞðb� sin dÞ2 þ Kgðb� cos dÞ2 � 0 ð43Þ is the positive damping coefficient. Inspection of (43) reveals that, if the gains of individual controls satisfy the following condition Kg ¼ ðXSHCKB þ KbÞ ¼ K ð44Þ DcontrolðdoÞ ¼ Kb2� (i.e. the damping coefficient introduced by UPFC) is constant and independent on power angle d. This is an important and beneficial conclusion, as it shows that damping introduced by UPFC control does not depend on the loading of the transmission link. The above strategies (37)–(39) use state variables d, Do and will be referred to as the state-variable control. However, the problem with state-variable control is that its implementation in a real multi-machine system requires estimation of all the required state variables (rotor angles and speed deviations of all the generators). This is a very complicated problem in itself, which additionally requires reliable wide-area measurements and extended communica- tion links. Consequently, the approa