The Inclusion of HVDC Control Modes in a Three-Phase Newton-Raphson Power Flow Algorithm

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1 Downloaded fro orbit.dtu.d on: Oct 17, 2018 The Inclusion of HDC Control Modes in a Three-Phase Newton-Raphson Power Flow Algorith Coffele, Federico; Garcia-alle, Rodrigo; Acha, Enrique Published in: IEEE PowerTech Lin to article, DOI: /PCT Publication date: 2007 Docuent ersion Publisher's PDF, also nown as ersion of record Lin bac to DTU Orbit Citation (APA): Coffele, F., Garcia-alle, R., & Acha, E. (2007). The Inclusion of HDC Control Modes in a Three-Phase Newton-Raphson Power Flow Algorith. In IEEE PowerTech Lausanne, Switzerland: PES - IEEE. DOI: /PCT General rights Copyright and oral rights for the publications ade accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requireents associated with these rights. Users ay download and print one copy of any publication fro the public portal for the purpose of private study or research. You ay not further distribute the aterial or use it for any profit-aing activity or coercial gain You ay freely distribute the URL identifying the publication in the public portal If you believe that this docuent breaches copyright please contact us providing details, and we will reove access to the wor iediately and investigate your clai.

2 1 THE INCLUSION OF HDC CONTROL MODESINATHREE-PHASE NEWTON-RAPHSON POWER FLOW ALGORITHM Federico Coffele, Rodrigo J. Garcia-alle, Meber, IEEE, and Enrique Acha, Senior Meber, IEEE Abstract A three-phase Newton-Raphson power flow algorith with conventional HDC power plant odelling is presented in this paper. Ephasis is placed on the representation of converter control odes. The solution approach taes advantage of the strong nuerical convergence afforded by the Newton-Raphson ethod in polar coordinates to yield reliable nuerical solutions for cobined HAC-HDC systes, where power plant and operational ibalances are explicitly taen into account. Although well-developed algoriths exist for solving fundaental frequency, three-phase power flows or cobined power flows/haronic currents, no HDC converter control ode representation appears to have been addressed in the nuerically reliable threephase Newton-Raphson ethod. Two test cases are solved, one corresponding to a sall syste to enable reproduction of results by interested readers and the other is a large syste containing several HDC lins and operational control odes. Index Ters Three-phase power flows, Newton-Raphson ethod, HDC transission control. I. INTRODUCTION Over any years, interest in HDC technology for bul power transission, and other ore specialised applications, has reained strong. Further to the traditional HDC technology, tered line coutated current (LCC), based on the use of conventional thyristors; two other lines of developent in HDC technology have appeared, capacitor coutated converters (CCC), which is an upgrading of the traditional technology, and voltage source converters (SC) with insulated gate bipolar transistor (IGBT) and pulse width odulation (PWM) control. In each of these applications, anufactures have taen axiu benefits fro the accelerated pace of power electronics technology. It is believed that in a few years tie, HDC-SC-based systes, will tae over a large portion of the traditional HDC aret. However at present, bul power DC transission reains the real of thyristor-based HDC technology 1. The need for bul power exchanges between grids separated by the sea increases the worth of classical HDC, where at present, it has no direct copetitor. A case in point is the 600 MW HDC lin to be licensed in 2007 joining the British and the Dutch power grids; Mr. Coffele is with the Departent of Electrical Engineering, University of Padova, Italy. Mr. Garcia-alle and Dr. Acha are with Departent of Electronics and Electrical Engineering, University of Glasgow, G12 8LT U.K. (e-ail: rgarcia@elec.gla.ac.u) with a doubling of the lin capacity projected by Moreover, interest in HDC transission in the UK goes beyond the niche arets of HDC; owing to the need to preserve the environent and countryside, HDC is becoing attractive in situations where AC high voltage transission was considered the de facto option, such as the 400 transission line to be built between Beauly and Denny, across the Scottish highlands. The HDC option, with its saller footprint s towers, is uch preferred by environentalist presue groups. In large conurbation centres, utility planners are actively considering the viability of reconverting existing AC transission corridors into HDC lines to tae advantage of the higher power density achieved with the latter. To ae infored decisions, however, power systes applications tools with suitable HDC representation are not only desirable but essential. Power flows is the ost basic tool with which to study the syste level perforance of cobined HDC-HAC systes and although a fair aount of wor has been published in the representation of HDC lins in positive sequence power flows, relatively little wor has been published in HDC lin odelling in threephase power flows, where the networ voltage ibalances and their ipact on HDC converter plant play a ey role. In fact, the ain line of developent that has been published in this area belongs to the Fast Decoupled power flow ethod, dealing with either fundaental frequency power flows solutions, 4, or cobined fundaental frequency power flows/haronic currents solutions carried out in a sequential anner 5. However, as stated in 6, the approach ay tae a substantial nuber of iterations to converge; the reasons being the sequential nature of the approach and the linear convergence characteristics, as opposed to quadratic, of the Fast Decoupled ethod. Hence, steps have been taen to solve the cobined fundaental frequency power flows/haronic currents solutions in a unified anner using a full Newton-Raphson ethod in rectangular coordinates to yield quadratic solutions 6. However, the ey issue of HDC converter control ode representation is not addressed and the approach is unduly coplex if the interest is not in power converter haronics but only in fundaental frequency, three-phase power flow solutions. This paper deals with HDC converter control ode representation in three-phase power flows, and a Newton-Raphson polar forulation has been selected for developent as opposed to the rectangular version /07/$ IEEE 419 PowerTech 2007 Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.

3 2 II. HDC MODELLING The onopolar HDC transission syste schee, shown in Fig. 1, is used in this paper to explain the odel developed and its inclusion in a three-phase power flow using the full Newton-Raphson ethod. However, the approach is quite general and it can easily be expanded to include soe of the ost coon HDC configurations, such as bipolar HDC systes. Fig. 1. a a I a a b b b b I c c I c c Converter Station PdR + - dr Id Monopolar HDC syste. R d P di + - di Converter Station a a I a b b I b c c I c a b c In the onopolar HDC lin, two converter stations are joined together by a line conductor, and the earth (sea) is used as the return path, requiring two electrodes capable of carrying the full current. In addition to the six pulse bridge, each converter station coprises a three-phase tapchanging transforer (LTC). A. Three-Phase Power Equations To derive the power flow Newton-Rapshon odel, the three-phase power consued by the converters is written as a function of AC voltages and DC variables. ( ) P = f a,b,c,θ a,b,c,x (1) ( P = f a,b,c ),X,θ a,b,c where P and P are active powers at buses and, respectively; and θ and and θ are the voltage agnitudes and phase angles, respectively, at the sending and receiving buses and ; denotes the a, b, and c phases of the syste. X represents, in generic for at this stage, the DC variables. The reactive powers are (1 Q (S = )2 (P A )2 = I ) 2 d (P )2 () (1 Q (S = ) 2 (P) 2 A ) 2 I d = (P) 2 (4) where S and Q and S and Q are the apparent and reactive powers at buses and respectively; 1 is a constant associated with the coutation overlap, A and A are the tap-changers positions at buses and, respectively; and I d is the current in the dc-lin. The full derivation is given in Appendix I. III. LINEARISED POWER EQUATIONS A three-phase power networ with n-buses is described by 6 (n 1) non linear equations. The inclusion of one HDC lin in the networ odel augents the nuber of (2) equations by six. The solution of the cobined syste of non-linear equations is carried out by iteration using the full Newton Raphson ethod. The power through the DC lin can be controlled by varying the DC-voltages, as indicated by the following two basic relations: P dr = dr I d (5) P di = di I d (6) where P dr and P di are the DC powers at the inverter and rectifier ends, respectively; and dr and di are the DC voltages, at the rectifier and inverter, respectively. The DC-voltages can be controlled either with the transforers tap ratios or with the converters control angles. At the rectifier, the control angle is the firing angle α; and at the inverter, the control angle is the extinction angle, γ. An increase in the control angle gives a decrease in DC-voltage, an increase in reactive power consuption (since the current will lag the voltage) and an increase in haronics generation. The latter is clearly undesirable and there is thus soe benefit in eeping the control angles at low values 7. There are several odes of operation in converter control. The first one, when the firing angle α, the extinction angle γ, di and P di are specified, is called control ode A. The converter transforer taps are varied in order to eet these specifications. In control ode B, the extinction angle γ, is ept at its lower liit at the inverter and the transforer taps and the power P di are specified. The power is controlled by varying the firing angle α at the rectifier. The firing angle rather than the tap transforer is preferred for control, since the tap changer control is too slow copared to the firing angle. In control ode C, α is fixed to its iniu value and γ is regulated to aintain constant power. If the current I d is ept constant instead of the power P di, three other control odes are defined. In this paper only the ode A, B and C with constant power are addressed. The Jacobian used in conventional power flows is suitably extended to tae account of the new eleents contributed by the HDC lin. The set of linearised power flow equations for the coplete syste is, ΔP Δθ ΔQ J 11 J 12 Δ ΔP dr = + (7) J 21 J 22 ΔX 1 ΔP di ΔX 2 where ΔP =ΔP 1,, ΔP n t ΔQ =ΔQ 1,, ΔQ n t Δθ =Δθ 1,, Δθ n t Δ = Δ 1 1,, Δ n n t ΔP and ΔQ are the noral power isatch equations, 420 Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.

4 and Δθ and Δ are voltage increents of phase angle and agnitude, respectively. There are two additional state variables X 1 and X 2, (i.e. X 1,2 ), and two power isatch equations introduced by the HDC lin, ΔP dr = PdR sch PdR cal (8) ΔP di = P sch di P cal di (9) P sch and P cal are the scheduled and calculated powers for the HDC lin. The power isatches for buses and, where the HDC lin is connected to becoe: ΔP = P G P L P N (θ, ) P dr (,,X 1,2 ) (10) ΔP = P G P L P N (θ, ) P di (,,X 1,2 ) (11) ΔQ = Q G Q L Q N (θ, ) Q dr (,,X 1,2 ) (12) ΔQ = Q G Q L Q N (θ, ) Q di (,,X 1,2 )(1) where P G, Q G and P G and Q G represent the active and reactive powers injected by the generator at buses and, respectively; P L, Q L and P L and Q L represent the active and reactive powers drawn by the load at buses and, respectively; The J 11 atrix corresponding to the AC networ at buses and P θ Q θ P θ Q θ P Q P Q P θ Q θ P θ Q θ P Q P Q (14) is augented with the partial derivatives of the active and reactive powers contributions by the HDC lin. Matrix J 12 includes the partial derivatives of the threephase powers P, and Q,n with respect to the HDC lin state variables and J 21 represents the partial derivatives of P dr and P di with respect to the three-phase voltage angles and agnitudes θ, and,. Moreover, J 22 represents the self partial derivatives of the HDC lin. A. Control ode A In control ode A, the variables α, γ, di and P di are specified, aing it possible to calculate the current I d, the voltage di and the power P dr : I d = P di di (15) dr = di + R DC I d (16) P dr = dr I d (17) The new state variables for this control ode are the tap changers, A and A, with the Jacobian eleents J 12, J 21 and J 22 having the following structure: J 21 = θ θ J 12 = J 22 = P A Q A P A Q A A A P A Q A P A Q A θ θ A A θ θ (18) (19) (20) It should be noted that a single tap-changer is used to regulate voltage agnitude at all three phases. B. Control ode B In control ode B, the variables a, a, γ and P di are specified and the new state variables are the current I d and the firing angle α. For this control ode, J 21 has the sae structure as in control ode A. The other two ters of the Jacobian are: P P Q Q J 12 = P P (21) J 22 = Q Q (22) C. Control ode C In control ode C, the variables a, a, α and P di are specified and the new state variables are the current I d and the extinction angle γ. Matrix J 21 has the sae structure as in the previous control odes, whereas atrices J 12 and J 22 tae the following structure, P P Q Q J 12 = P P (2) J 22 = Q Q (24) It is iportant to rear that this is a three-phase forulation, and that all eleents in the above equations are vectors of order 1, except for J 11 which are vectors of order and J 22 which are single ters. 421 Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.

5 4 I. NUMERICAL EXAMPLES A. Test Case 1 To show the applicability of the proposed approach, the five-bus networ depicted in Fig. 2 is used. All necessary data to carry out a balanced three-phase AC power flow is taen fro 8. TABLE III POWERS FROM GENERATOR BUSES AND HDC CONERTERS. HDC converter Generator buses Main Lae North South P Q P Q P Q P Q Phase A Phase B Phase C j0.15 North Lae j0.05 Main B. Test Case 2 To show the wider applicability of this technique, the New England test syste 9, which consists of 68 nodes, 16 generators and 86 transission lines, is used. The syste has been odified to include three HDC lins, each operating with a different operational control ode, as described in Table I Fig. 2. South 0.2+j El 0.6+j0.1 Five-bus test networ for three-phase balanced conditions. The five-bus networ is odified to include one HDC lin connected between Lae and Main buses. In this contrived exaple, the AC networ and HDC converters are assued to wor under three-phase balanced conditions. The HDC lin is ade to operate in control ode A. The data used for the HDC lin is given at Table I and the three-phase nodal voltage agnitudes and phase angles are given in Table II. Notice that voltage values are given in per unit and angles are given in degrees. TABLE I DATA FOR THE HDC LINK. Transforer reactances D.C. lin resistance Firing angle for the rectifier Extinction angle for the inverter Transitted power pu 0.10 pu 15 deg 18 deg 0.4 pu per phase TABLE II BUS OLTAGES AND GENERATION RESULTS. TABLE I HDC LINK CONTROL MODES. fro to Control ode Lin 1 bus 41 bus 42 C Lin 2 bus 1 bus 27 A Lin bus 2 bus 24 B Ibalances are introduced into the networ by changing the active and reactive power loads at the buses where the HDC stations connect, by ±10% with respect to the balanced case. Phase A is subtracted 10%, Phase B is added 10% and Phase C is ept unchanged. Tables, I and II show the unbalanced three-phase voltage profiles and the active and reactive powers for each HDC converter. The solution was achieved in 6 iterations to a power isatch tolerance of 1e 12. TABLE PARAMETERS OF LINK 1. Sending Node Receiving Node A B C A B C θ P Q Bus nae Phase A Phase B Phase C oltage Angle oltage Angle oltage Angle North South Lae Main El Table III shows the generated powers in the North and South buses, and the powers in both the rectifier and the inverter, which are connected to Lae and Main, respectively. All other active and reactive power flows are shown in Fig. 2, for phase A only. TABLE I PARAMETERS OF LINK 2. Sending Node Receiving Node A B C A B C θ P Q Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.

6 5 TABLE II PARAMETERS OF LINK. Sending Node Receiving Node A B C A B C θ P Q The three HDC lins are set to operate with different control schees and the state variables are A and A (control ode A) for lin 1; α and I d (control ode B) for lin 2; and γ and I d (control ode C) for lin. Figures, 4 and 5 show the behaviour towards the convergence for each state variable during the power flow solution. Tap position, pu HDC 2 A HDC 2 A iterations Fig.. Tap position of lin 2, Control Mode A. Firign and extinction angles, degrees HDC alpha HDC 1 gaa iterations Fig. 4. Firing and extinction angles of lins and 1, Control Mode B and C, respectively. Direct current, pu HDC 1 Id HDC Id iterations Fig. 5. Direct current of lins 1 and, Control Mode B and C, respectively. It should be noticed that the state variables A, A and I d have initial values of 1 per unit, and upon convergence, settle to final values. For the reaining state variables, α and γ, are given values of 15 and 18 degrees, respectively, which are typical specified control angles 10,.. CONCLUSIONS This paper presents a three-phase power flow ethod to include conventional HDC lin odels where several control odes are applied. The approach taes advantage of the strong convergence characteristics of the full Newton-Raphson technique to ensure reliable iterative solutions. Two non-linear equations per HDC lin, in linearised for, are cobined with the linearised equations of the rest of the power syste for unified, iterative solutions that yield quadratic convergence. Two test cases have been presented, one correspond to a siple networ with one HDC lin and operating under balanced conditions. This should enable interested readers to reproduce results with ease. The other is a large networ including unbalanced loading and three HDC lins; each operating with its own control ode. The ipleentation of the proposed technique can be expanded to include other HDC configurations, such as bipolar and ulti-terinal HDC systes. ACKNOWLEDGEMENTS The authors gratefully acnowledge the financial assistance given to Mr. Coffele by the University of Padova, Italy to carry out his specialisation degree thesis and to Mr. Garcia-alle by the Consejo Nacional de Ciencia y Tecnologia, Mexico to carry out Ph.D. studies at the University of Glasgow, UK. REFERENCES 1 L. Carlsson, Classical HDC: Still continuing to evolve, Modern Power Systes, MPS Review: Transission & Distribution, pp , June National Grid Copany plc, Interi Great Britain seven year stateent. Online. Available: J. Arrillaga and C. P. Arnold, Coputer Analysis of Power Systes. John Wiley & Sons, Inc., B. J. Harer and J. Arrillaga, -phase ac-dc load flow, IEE Proceedings, vol. 126, pp , Deceber J. Arrillaga and C. D. Callaghan, Three phase ac-dc load and haronics flows, IEEE Transactions on Power Delivery, vol. 6, pp , January J. Arrillaga, N. Watson, and G. Bathrust, A ultifrequency power flow of general applicability, IEEE Transactions on Power Delivery, vol. 19, pp , January J. Arrillaga, High oltage Direct Current Transission. IEE Power and Energy Series, E. Acha, C. R. Fuerte-Esquivel, H. Abriz-Perez, and C. Angeles- Caacho, FACTS Modelling and Siulation in Power Networs. John Wiley & Sons, Inc., J. H. Chow, Tie-Scale Modeling of Dynaic Networs with Applications to Power Systes. Springer-erlag, E. T. Sed, A new approach to ac/dc power flow, MSc. Thesis, Auburn University, Deceber Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.

7 6 APPENDIX I HDC POWER EQUATIONS A standard six-pulse converter brindge is used to derive the atheatical odel for the three-phase HDC station. The three-phase supply voltages to the converter transforer v a, v b and v c are taen to be unbalanced but sinusoidal, as shown in Fig. 6. The coutation is assued to be delayed by an angle α with equidistant firing angle control 7, i.e. firing instants occur at successive intervals of 60 following the first voltage crossing. B a = ϕ ab =0, B b = ϕ bc 4 and Bc = ϕ ca 2. Taing into account the voltage drop Δ, the DC voltages of the rectifier and the inverter are: dr = A 2 cos (α + A ) Δ (2) di = A 2 cos (γ + B ) Δ () where Δ is defined as, Δ = XcId (4) Fig. 6. Unbalanced converter voltage wavefor. The powers of the inverter and the rectifier are given by the following expressions, P dr = dri d = P di = dii d = =a,b,c =a,b,c drid (5) diid (6) The relations between the currents on the priary side of the three-phase transforers and the direct current I d, are given by, Fro Fig. 6, the rectified voltage dr is obtained by integration, α α+ 2 dr = 1 v ca (θ) dθ + 1 α α+ v bc (θ) dθ + 1 α+ v ab (θ) dθ α+ 2 The voltage dr is ade up of three ters, where dr a = A 2 ab dr b = A 2 bc dr c = A 2 ca giving (25) dr = a dr + b dr + c dr (26) ( cos α + ) ( cos α + 2 ) ( ) ( cos α + θ bc cos α + θ bc 2 ) ( cos α + θ ca ) cos (α + θ ca ) dr A 2 (27) (28) (29) cos (α + A ) (0) where A a = θ ab =0, A b = θ bc 4 and Ac = θ ca 2. A siilar expression is developed for the inverter, where di = A 2 cos (γ + B ) (1) I 6 = A Id = AId I 6 = A Id = AId (7) where is a constant associated with coutation overlap. For power flow analysis, it can be taen to be = The apparent powers are given by, where 1 = 6 S = I S = I = 1A Id (8) = 1A I d (9) Federico Coffele was born in Italy. He was an Erasus-Socrates student at the University of Glasgow, UK, in He graduated fro the electrical engineering departent at the University of Padova, Italy, in Rodrigo Garcia-alle was born in Mexico. He received the electrical engineering degree fro ESIME, IPN, Mexico city, Mexico, in 2001 and the MSc degree fro CINESTA, Guadalajara, Mexico, in 200. Currently he is pursuing the PhD degree in electrical power systes at the University of Glasgow, UK. Enrique Acha (SM 02) was born in Mexico. He graduated fro Universidad Michoacana de San Nicolas de Hidalgo in 1979 and obtained his PhD degree fro the University of Canterbury, Christchurch, New Zealand, in He was a postdoctoral Fellow at the University of Toronto, Canada, and the University of Durha, UK. He is the Professor of Electrical Power Systes at the University of Glasgow, UK. He is an IEEE PES distinguished lecturer. 424 Authorized licensed use liited to: Danars Tenise Inforationscenter. Downloaded on February 25, 2009 at 10:8 fro IEEE Xplore. Restrictions apply.