^>*fd> V*kd'V*d'V V W IR'II' C\u00b0 SV C O S Q r 1dA' iqA' 1dB' V'^c'V'^fS'^dfS' \u00b1qf 5 ?? \u00b1 q f 5 ' \u00b1df 7 ' P l d f 7' ^ 7 ' P lqf7' idfll\u00bb P \u00b1 d f l l ' iaflV P i q f l l ! l d f i 3 ' p i d f l 3 ' \u00b1qfl3' p i q f l 3 ' V The system is of the 39th order. The auxilliary algebraic equations are given in (2.1a), (2.4a), (2.5), (2.9), (2.10), (2.13) and (2.14). The current solutions of equations (2.1) are: - -i q \u2014y Y ^kq yaq \u2022 *q Xkq ~yaq yq\u00a3 % q *d \" ydfk y df y dk *d Akd = \" ydf y df\u00a3 ~ Y d\u00a3 \u2022 *kd ifd \"ydk \" y d\u00a3 y dk\u00a3 (2.15) where \u2014 \u2014 ^ q I X + X \u201e , aq \u00a3kq yaq ~ \\ X aq _ yq^_ x +x\u201e aq \u00a3a A , = (x +x\u201e ) (x +x\u201e. ) - x 1 aq \u00a3a aq \u00a3kq aq ydfk Xad ^Afd^Akd^ + x\u00a3fdX\u00a3kd ydf xad x\u00a3fd ydk 1 A 2 xad X\u00a3kd ydf\u00a3 xad ( x\u00a3fd + x\u00a3a ) + x\u00a3fdX\u00a3a yd\u00a3 Xad X\u00a3a ydk\u00a3 Xad ( x\u00a3kd + X\u00a3a ) + x\u00a3kdX\u00a3a (2.16) 16 A~ = x JX\u201e (x. r, + x\u201e. ,) + x.,, x., , (x , + xn ) 2 ad Jla lid Ikd ifd Ikd ad la\" These equations form the base of the dynamic modelling of hvdc systems which will be developed in the next chapter. 17 . 3. DYNAMIC MODELLING AND ORDER REDUCTION The mathematical model constructed in the previous chapter has 39 state variables which describe the system dynamics in great detail Such complexity presents a very difficult problem in digital computation. In this chapter the order reduction of the system model is achieved throu engineering approximations with the aid of eigenvalue analysis. The validity of the low order models is established through tests on the nonlinear models with disturbances. 3.1 Reduction Techniques Numerical techniques for reducing high order systems to low 45-47 order equivalents are available . But the identity of a l l the para-meters will be lost in the process of reduction. In this thesis the \u2022o reduction will be approached by engineering approximations instead. Again there are two approaches for the approximation. The 47 first is called the state variable grouping technique , which may be called linear approximation technique, by which the nonlinear system equations are linearized first and then the system equations order is reduced in possibly several steps. At each step the state variables x are separated into two groups, x^ and y.^* associated^ with large and small time constants respectively. The linearized system equations are written in a partitioned form as follows: x\u201e A l l A12 A A 21 22 (3.1) Neglecting the small time constants or the fast and short-lived trans-ients by setting equal to zero, is given by 18 = -A, -1 A, (3.2) x 2 22 which is then substituted back into (3.1) to give 1 [ A11 A12 A22 -1 (3.3) The order of the system is thus reduced by the number of state variables in According to this procedure several linear models will be ob-tained, but there is only one high order nonlinear model for the system. technique by which a l l the time derivatives as well as speed deviation terms in the nonlinear differential equations to be eliminated are set equal to zero at each step. This is equivalent to a static representation of the system components for these equations to be eliminated. The resulting algebraic equations are then solved and substituted back into the rest of the nonlinear differential equations before linearization for linear optimal control design or eigenvalue analysis. The advantage of this approach is that for each linearized model there is one original nonlinear model. This is the approach to be followed in this chapter. 3.2 Reduced Models presented. For the procedure, i t is chosen th&t the ac harmonic fi l t e r dynamics shall be eliminated first since i t consists of the largest num-ber of equations. The next group are the ac transmission lines and machine stator windings, and so on. The details are: STEP 1: Neglecting the ac harmonic fi l t e r dynamics of equations (2.11) Another approach may be called the nonlinear approximation In the following the models resulting from reductions will be the model is reduced from the 39th to the 23rd order. The resulting algebraic equations are: 19 \"df qf G f B \u00a3 -Bf G f v where = I 3 ( R f \/ + x f\/) (3.4) \"R f i e fj & Cj.. J J e fj j = 5, 7, 11, 13 (3.5) STEP 2: Neglecting the ac transmission line dynamics the system's order is reduced to 17. Equations (2.8) are replaced by idA _ a i \"a2 a3 a4 v sin6 o V a2 a i ~% a3 v cosS o idB a3 a i \"a2 Vd V = ~a4 a3 a2 a l V _q Vdc X c (a2-a^) x c (ai+a3) \u2022x c ( a2- a4 } X c ( a l + a 3 } V qc -X c ( a l + a 3 ) x c (a 2-a 4) -x c (a 1 +a 3) X c (a 2-a 4) where \"a. 1_ A , 2rx c(x \u00a3 - xc) xc ^ \" x \/ + 2 x i x c > r [ r 2 + (x \u00a3 - x c ) 2 + x c 2] (x\u00a3 ' xc ) + X\u00a32 \" 2 Vc } 4r 2 (x \u00a3 - x c ) 2 + (r 2 - x\/ + Ix^)1 (3.6) (3.7) STEP 3:. Neglecting the transformer voltages and voltages due to speed deviation generated in the armature windings of the synchronous 20 machine a 15th order model is achieved. Equations (2.2d) are replaced by ( 1 + r a ^ q W 2 2 ra ykq ydk ra ykq Ydf ra yaq r y 1 a kq L\" ra ydk \" ra ydf aJ aq df k ra ydfk. , [ * \u00a3 d *kd kq v, v ] d qJ (3.8) STEP 4: Representing the dc transmission line by its static impedance yields a 12th order model. Equations (2.7) become ^ ' k ( VR + V V c - 2

+ 2 v t I r a ? + < Xaq + X*a ) (^ + V t Thus v d can be calculated from (3.32), from (3.31) and i ^ f d from (3.29). Finally the torque angle 6 can be determined from (3.6) to read . ( a 2 a 4 \" a l 3 3 ) v d \" ( a i a 4 + a 2 a 3 ) v q + *1 \u00b1dA + a2 *qA \u201e \u201e, o = arc tan ; r ;\u2014; r \u2014 3 : ; : (3.33; ( a l a 4 + a 2 a 3 ) v d + ( a 2 a 4 \" a l a 3 ) v q \" a2 xdA + \"l\/qA where the ac transmission line currents i , . and i , are determined from dA qA (2.13) after i , and i , i . . , and i I_, I, and I .and i , and i are d q' df qf R d q' cd cq determined from (3.27), (3.4), (3.22), (2.10) and (3.10) respectively. The rectifier and inverter firing angles are determined from (2.5) and (3.13) respectively. 3.4 System Data ^The*sysit-eiiv'sirudied \"has 'the \"fo'l-lowing data, a l l \"in per \"unit except for angles in degrees and inertia constant and time constants in seconds. Most data are taken from reference 20. The rest are assumed. Synchronous machine and voltage regulator r 0.005 r,, 0.00055 r. , 0.02 r, 0.04 a fd kd kq x\u00ab.a \u00b0- 1 x\u00a3fd \u00b0- 1 XJlkd \u00b0' 1 X\u00ab,kq \u00b0'2 x , 1.0 x 0.7 H 3 sec. k 20.0 T 2.0 sec. ad aq r r AC and DC transmission lines r 0.0784 x\u201e 0.52 x 5.556 JI c R 0.2792 X. 18.112 X 20.28 x 0.432 L c co 26 AC harmonic filters, local load, and shunt capacitor r f 5 0.065 r f 0.OB92 r f l l 0.062 r f l 3 0.0704 L,_ 0.00862 L,_ 0.00862 L,,,-- 0.00332 L..- 0.00332 r5 f7 f l l fl3 c\u201e 0.0000325 c,_ 0.00001695 c \u201e . 0.000017625 c,, _ 0.000012575 f5 f7 f l l fl3 Rj^ 6.05 x c 212.0 DC Controllers K_ 30 T 0.002778 sec. KT 30 T_ 0.002778 sec. k 0.9 is. K 1 1 Operating point v. 1.1 v 1.0 P 0.4 P, 0.4 t o ac dc A 5\u00b0 6 51.05\u00b0 i b . , 1.165 \u00b1 . 1.0057 o Tfd kd iK 0.947 -0.497 * -0.568 v J 0.565 d kq q d -V 0;944 -E 1:59 V . 1vl\u00bb8 P 1.-0 q x ref Q 0.15 P 1.004 1. 0.584 i 0.71 m d q *fd ^ l U \u00b0- 1 9 V \u00b0 ' 3 1 - \u00b0 ' 3 4 i- -0.18 v , 0.71 v 0.82 Q 0.0073 qB dc qc ac I R 0.363 I d 0.331 I 0.225 VR 1.65 V 1.55 VT -1.45 O L 8.25\u00b0 a T 141.7\u00b0 c I R I Q, 0.185 i,\u201e 0.09 i 0.16 i , -0.004 dc dl ql cd i 0.003 -0.012 i \u201e 0.007 i,,, -0.006 cq df5 qf5 df7 i 0.004 -0.006 i \u201e, 0.004 -0.0045 qf7 d f l l q f l l dfl3 \\jfl3 0.003 27 3.5 Eigenvalue Analysis Both state variable grouping technique and nonlinear approxi-mation technique described in section 3.1 are employed to calculate the eigenvalues for the different system models developed in section 3.2. The results are tabulated in Tables I and II respectively. There is not much difference in high order models for the two methods. But the difference becomes evident x^ hen the model order becomes low. This is because of two different ways of making approximations. For example the ac harmonic filters equations (2.11) are replaced by (3.4) in non-linear approximation and the filter currents are given in terms of v^ and v^ only. On the other hand when the same equations (2.11) are linearized and then solved the resulting currents are in terms of Ato, ' I ' J T J S & i> \u2022 \u2022 \u2022 > IT,> C O S O L , .... v, and v due to the presence of pv,, fd kd R R d q d \u00bbp.v^ .:and 4>Ato .terms, \/in the-.eq.uations. ..The nonlinear tapproximation ...approach seems to be giving more accurate results. It is especially clear for the second order model. While the nonlinear approximation technique gives a pair of imaginary eigenvalues, which is expected since damping is neglected in the synchronous machine equations, the state variable grouping technique yields a conjugate pair of eigenvalues with'positive real parts indicating an unstable system which contradicts the results from higher order models analysis. The eigenvalues for the 39th order model are plotted in the complex plane in Fig. 3.1. It is noticed that a majority of the eigen-values are clustered in the region between -10 and -200 of the real axis. The dominant eigenvalues are found to be these corresponding to 6, Ato, if i ^ and E^ equations. The largest eigenvalues correspond to v., v , v, and v equations. Based on this information the following d q dc qc n & 2nd Order 0.126+J6.76 4 th Order -0.052+j7.1 - 0 . 3 3 6 i J l . i l 6th Order -0.052+J7.1 -0.336+jl.ll -362, -35156 8th Order -0.052+j7.1 -0.336+jl.ll -362, -35156 10 th Order -0.15+j7.14 -0.33+jl.l -363, -30532 12th Order -0.15+J7.14 -0.33+jl.l -363, -39685 15th Order -0.15+j7.14 -0.33+jl.l -365 17th Order -0.15+j7.14 -0.33+jl.l -365 23rd Order -0.164+J7.14 -0.33+jl.l -365 39 th Order -0.164+J7.14 -0.33+jl.l -365 360,-360 350,-360 -41.9,-26.1 \u2022360,-360 -41.9,-26.1 -18250+j59136 360,-360 -41.9,-26.1 -6623+J64990 \u2022360,-360 -41.8,-26.2 -4415+J10930 \u2022360,-360 -41.9,-26.3 -i039+j5524 \u2022360,-360 -41.9,-26.3 -6577+J15988 Table I Eigenvalues by state v a r i a b l e grouping technique \u2022185+J761 \u202220.5+j 327 \u2022122+J785 \u202215.5+J325 \u2022143+J809 \u202214.8+j 326 \u2022143+j 811 \u202214.8+j 326 847+J1892 \u202273+J1993 \u202276.7+J1977 \u20222620+J4267 \u202259 U +J1216 \u202250+j 376 \u202234339,-7018 \u202261+J1198 \u202250+j 376 -8.95+j1490 -9.5+J2245 -18+J2191 -23.7+J2947 -55+J4358 -73+J5118 - 92+j 3568 -123+J4335 2nd Order 4th Order 6th Order 8th Order 10th Order 12th Order 15th Order 0.0+J6.76 -0.083+J7.1 -0.083+J7.1 -0.083+J7.1 -0.18+J7.13 -0.18+j 7.13 -0.18+j7.13 -0 .34+j l . l l -0 .34+j l . l l -0 .34+j l . l l -0.33+jl . l -0.33+jl . l -0.33+jl . l 23rd Order -0.164+J7.13 -0.33+jl. l -362,-35156 -362,-35156 -360,-360 -362,-30531 -360,-360 -41.9,-26.15 -362,-31455 -360,-360 -41.9,-26.15 -166910+j495377 -360,-360 -41.9,-26.15 -^49562+j488820 -364 17th Order -0.17+J7.13 -0.33+jl . l -364.5 -365 39th Order -0.164+J714 -0.33+jl. l -365 -360,-360 -41.8,-26.2 -37936+j72148 -360,-360 -41.9,-26.? -6572+J13689 -360,-360 -41.9,-26.3 ^-6577+jl5988 . -185+j760 -20.5+j327 -123+J783 -15.5+J325 -143+J806 -14.9+J326 -1116+J1963 -64.3+J2004 -143+J811 -76.7+J1977 -14.8+J326 -36945,-5154 -55+J1227 -50+J376 -34339,-7018 -8.95+J1490 -61+J119S -50+J376 -9.5+J2245 -18+J2191 -23.7+J2947 -55+J4358 -73+J5118 -92+j3558 -123+J4 335 r o Table II Eigenvalues by nonlinear elimination technique REAL JL X X X X X * -X\u2014X L IMAGINARY 4 -105 -JO4 -103 -W2 -10 -1 10~7 Z O - 2 ' 70\" Fig. 3.1 EIGENVALUES FOR 39th ORDER MODEL 31 classification of system models, is suggested: 1. A fourth order model comprising \u00a7, Ato, ip^^ and as state variables is the simplest model one can have for system dynamic studies. 2. A sixth order model including cosa and cosaT in addition to the 4th R 1 order states is fairly accurate and can be used for control design including an hvdc system. 3. If one wants more accurate results the 5th harmonic f i l t e r may be included resulting in a tenth order model. \u2022 4. The thirty-ninth order model is accurate but impractical in digital computation. 3.6 Nonlinear Tests with System Disturbances To compare the four suggested models nonlinear system response tests subjected to different types of disturbances are compared. Figs. 3.2 and 3.3 show the change in rotor angle 6 and speed deviation Ato with time for a 25% step change in the dc reference current. For this disturbance i t is seen from Fig. 3.2 that results are very close but the 6th order model gives the closest response to that of the 39th order model. Figures 3.4 and 3.5 show the system's response with a 25% step reduction in mechanical torque input. The 6th and 10th order models give identical response closer to that of the 39th order model than the 4th order model. Figures 3.6 and 3.7 show the system response to a three phase ground fault at the middle of one ac line for 6 cycles followed by isolating the faulted line at both ends and a successful reclosure after 0.4 second after the fault is removed. In this case only the res-ponses of the 39th and 6th order models are plotted. It is noticed that even for such a severe disturbance the 6th order model results are s t i l l SRAD 32 1.0[ 0.8L 0-6 \\ 0-4 0.21 39 th ORDER 10th ORDER x xxx 6th ORDER \u2022 o \u00ab . 4th ORDER a o \" \u2014 \" \u2014 i \u2014 L . _ i i i i i i 0.5 W Fig. 3-2 STEP CHANCE IN Iref 1.5 t SEC AGO RAD\/SEC t SEC 39th ORDER V 10th ORDER v xxxx 6th ORDER . . . . 4th ORDER Fig. 3.3 STEP CHANCE IN Iref SRAD 39 th ORDER 7.0 0.5[ 10 th ORDER xxx x 6th ORDER . \u2022 \u2022 \u2022 *4th ORDER _i i i ' \u2022 i i i ' \u2022 0.0 0.5 1.0 15 t SEC Fig. 3.4 STEP CHANGE IN P, m A CO RAD\/SEC 2.01 t SEC 39th ORDER 10th ORDER xxxx 6 th ORDER . . . . 4th ORDER -3.0 Fig. 3.5 STEP CHANGE IN P S RAD 35 close to those of the 39th order model. From these test results, suppor-ted also by eigenvalue analysis, i t is decided that from now on in the suc-ceeding chapters only the 6th order model will be used for the control design. 36 4. STABILIZATION OF DC\/AC PARALLEL SYSTEMS BY LINEAR OPTIMAL CONTROL SIGNALS In this chapter linear optimal control theory is employed to stabilize dc\/ac parallel power systems. Two systems are investigated; the first is to stabilize an existing system, and the second is to stab-ili z e an expanded system by adding a parallel dc link to the existing ac line in order to increase the transmission capacity of the system. Several control schemes are designed and the disturbance test results on nonlinear system models are compared. For a l l designs the sixth order system model, i.e. A6 , Ato, Ait-,, AE , A C O S O L and AcosaT, is fd x R I employed. 4.1 Linear Optimal Regulator Problem The system's linearized equations are written as x = Ax + BU (4.1) It is required to find an optimal control U that minimizes the quad-ratic cost function oo J = Y J (xfc Ox + T\/ RU) dt (4.2) o subject to (4.1), where Q and R are positive definite matrices. The 48 required optimal control is given by U = -R_1 Bt Kx (4.3) where the Riccati matrix K is obtained from the solution of the non-linear algebraic matrix equation t -1 t KA + A K - KBR B K + Q = 0 (4.4) Several computation techniques for the solution are available utilizing 49 50 the properties of the state and costate composite system matrix M ' , 37 A M = (A.5) the matrix K may be computed from K X. (4.6) where X. T X. I l l X = (4.7) XII XIV is the eigenvector matrix of M and the eigenvectors X^ and X^ correspond tb the stable eigenvalues of M or the eigenvalues of the controlled system. 4.2 Control Signals for an Existing System The system considered here is the same as described in Chapter 3, which was shown in Figure 2.3. Three different stabilization schemes are investigated; the first with an optimal excitation control u^ on the synchronous machine alone without any stabilization signal on dc, the second with optimal current control u^ on dc but without excitation stabilization on the synchronous machine, and the third with both u^ and Up controls designed together. The control signals designed for the system's linearized model are tested on the original nonlinear system and the system's response to the disturbances and also the system's eigenvalues are compared. For the data given in Chapter 3 the linearized state equations for the system are 38 1 \u2014 A6 0.0 1.0 0.0 0.0 0.0 \u2022 -52.1 0.0 -112.6 0.0 -92.1 -0.14 0.0 -0.46 0.21 -0.2 AE -0.24 0.0 -6.92 -0.5 -1.11 X Acosct R -278.2 0.0 -8042.0 0.0 -14197 Acosct^ . 5.09 0.0 147.1 0.0 259.7 0.0 -46.7 -0.2 0.92 -10672.0 -164.8 A6 Aw Aifj AE fd A cos a R + BU The B matrices for the three controls are given by t Acosa, (4.8) Ug! B = [0 0 0 k r \/ T r 0 B = [0 0 0 0 0]' V T R V t \u00b0 ] t (4.9) Ug and u^rB 0 0 0 k \/T r r 0 0 0 0 V T R V t \u00b0 In a l l three cases the matrices Q and R of equation (4.2) are' taken as unit matrices. The system's eigenvalues for a l l cases are listed in Table III and the corresponding control laws are given in (4.10). Table III Eigenvalues of the existing power system for various controls Control Used Eigenvalues no control -0.084 + J7.093 -0.339 + jl.108 -365, -13997 \"E -0.728 + j7.154 -0.403, -9.915 -365, -13997 UD -1.512, -43.89 -0.349 + jl.13 -348, -17097 \" E A N D % -1.56, -43.89 -0.383, -9.87 -348, -17097 39 UE ~ -5.79 -0.36 -7.32 -1.09 -0.002 -0.025 ~ \"D 0.415 1.004 -0.17 -0.0004 -0.319 -0.082 UE -0.159 -0.0017 -0.495 -0.961 0.8xl0~5 -0.0045 _UD_ 0.402 1.005 -0.532 0.008 -0.319 -0.082 [A6 Aw Aii_, AE A C O S O L AcosaT] fd x R I (4.10) It is noticed in Table III that the dominant pair of eigenvalues in a l l three cases have been shifted to the left of the complex plane indicating the stabilizing effect of these controls. The damping ratio for the dominant eigenvalues is improved in the case of u^ , control. For the other cases the dominant pair is decoupled into two real eigenvalues. For Up and u^ controls a l l the eigenvalues are real indicating a non-oscillatory system. 4.3 Nonlinear Tests The system disturbance considered in a l l cases is a three phase to ground fault at the middle of one circuit of the ac transmission line for 0.1 sec. The faulted line is then isolated by disconnecting i t from both ends followed by a successful reclosure at 0.5 sec. after the fault is completely removed. The system's response is summarized in Figures, 4.1 to 4.6. It is noticed from these figures that the excitation con-trol signal Ug by itself does not improve the system's stability effec-tively despite the fact that the control effort reaches maximum a l l the time as shown in Fig. 4.6. The limits set for the control signals are +0.12 p.u. for Ug and +0.3 p.u. for u^. The limits for u g were chosen as +0.12 p.u. after several other values were tried. It is shown in Figure 4.7 that the higher the Ug limits, the higher will be the overshoot in machine I Par P\"-42 Fig. 4.5 Variation of AC Power for Existing System u 4 1 NO CONTROL 2 u\u00a3 CONTROL 3 uD CONTROL CONTROL Fig. 4.6 Control Effort for Existing System \/ rad ;.0| Up p.u. 44 swing. The machine goes unstable when u^ , is not restricted. Figure 4 . 8 shows the control efforts for different cases. The dc control limits are so chosen that i t will not overload the dc line or operate i t under very light load. It is found that the stabilization is very effective either with the dc control signal u^ alone or combined with the excitation control signal u^ ,. The angular deviation is very small, and the dc line picks up the load originally carried by the faulty line very quickly. It is also noted that the system's responses with u^ control alone or with Ug and u^ controls are very close except that the terminal voltage changes are smaller in the later case. 4.4 A Fourth Order Model Study Due to the presence of two large eigenvalues on the sixth order model corresponding to the dc firing circuits dynamics the step size in numerical integration is very small thus requiring longer compu-tation time. It was thought that i t would be interesting to find out i f these two small time constants associated with the firing circuits can be neglected in order to save the computation time. The resulting system's linearized equations of the fourth order become A6 n Ato A * f d AE - A6 \" - 0 .0 1.0 0 . 0 o .o -\u2022 Ato - 5 0 . 3 0 .0 - 6 0 . 4 0 .0 A * f d - 0 . 1 3 5 0 .0 - 0 . 3 4 5 0 .207 AE _ x _ - 0 . 2 1 7 0 .0 - 6 . 3 - 0 . 5 + BU ( 4 . 1 1 ) The matrix B is given by 1 ^ : 6 = [0 .0 0 .0 0 .0 1 0 . 0 ] u D: B = [ 0 . 0 - 5 2 . 5 - 0 . 1 6 3 0 .108 ] 45 \" E and u^: B 0.0 0.0 0.0 10.0 0.0 -52.5 -0.163 0.10.8 _ t (4.12) The system's eigenvalues are listed in Table IV and the corresponding control laws are given in (4.13) Table IV Eigenvalues for a fourth order model control used eigenvalues no control -0.084 + j7.093 -0.339 + jl.108 \" E -0.728 + J7.154 -0.388, -9.915 -1.39,-51.49 -0.349 +jl.13 u E and u D -1.42, -51.49 -0.35, -9.87 \\~ ~-5.78 -0.359 -7.24 -1.09 \" ~ A6 %<\u2022 0.444 1.005 -0.023 -0.02 Aco UE -0.109 -0.002 -0.286 -0.957 A^ 0.43 1.006 -0.45 0.004 AE fd (4.13) The same nonlinear test with system disturbance is applied to this model and the corresponding swing curves are plotted in Figure 4.9. It is found that there is considerable deviation' between the fourth and sixth, order models and some accuracy is sacrificed by neglecting the two small time constants for faster computation. It is decided that the dc firing circuit dynamics will be retained from now on in the dc system studies for the rest of this thesis. 4.5 Expanded System The system investigated in this section had an ac system con-sisting of a synchronous generator transmitting power to an infinite system over a double circuit ac transmission line, Figure 4.10. The 46 Fig. 4.10 AC System to be Expanded 47 generator is equipped with a voltage regulator and also has a local load at the terminal bus. This system is expanded by adding a dc link in parallel after the generating capacity is doubled. The system was unstable at the outset and a conventional excitation control signal is designed. The ac system's per unit data was as follows: Synchronous machine and voltage regulator: x, 0.973 x 0.55 d q do 7.76 sec. x' 0.19 d x\u201e 0.04 H 4.63 sec. \u00a3a K 130 r T 0.05 sec. r AC transmission line and local load G r -0.034 x \u00a3 0.997 Operating point: 0.249 B \u00a3 0.262 v 1.05 v 1.02 o fd 0.02 1.22 0.952 E 1.34 x v 65.2' 0.45 0.95 0.4 0.81 From the data given the linearized fourth order system's model, equations (2.2a) and (2.2b) were \u2014 1 i A6 Aco Aiji A6 0.0 1.0 0.0 0.0 \u2022 Aw -22.4 0.0 -39.7 0.0 A $ f d -0.1 0.0 -0.2 0.15 AE x _ 223.6 0.0 1794.2 -20.0 fd AE X _ i ( 4 . 14 ) She system's eigenvalues were 0.18+J5.16, -10.3+J13.3 which indicates an unstable system. A conventional excitation control 48 signal u^ is designed''\"'\", u k S e x 1 + T S e . 0.04 S . Au = - \u2014 : \u2014 \u201e ,\u201e Au (4.15) \u2014 \u2014 A6 \u00ab Aw A * f d = AE X u X A6 Aw Aip AE fd 1 + 0.5Sand is added to the system resulting in 0.0 1.0 0.0 0.0 0.0 -22.4 0.0 -39.7 0.0 0.0 -0.1 0.0 ~0.2 0.15 0.0 223.6 0.0 1794.2 -20.0 2600.0 -1.8 0.0 -3.2 0.0 -2.0 The corresponding eigenvalues become -2.5+J3.5, -4.6, -6.3+J11.9 Thus the system is stabilized. Next the system is expanded by doubling its capacity. With the base MVA of the synchronous machine doubled, the p.u.. ac line and local load data become r -0.068 x^ 1.994 0.1245 B 0.131 The new dc line and harmonic filters' data are R 0.066 x 0.2 G. 0.22xl0~3 ' B, -0.14 co f f It happened that the reactive power can be sufficiently provided by the harmonic filters and no special condenser for power factor correc-tion is needed. The system's new operating point is given as v\u201e 1.05 v 1.02 P 0.9765 Q 0.0622 t o P 0.504 Q 0.187 P. 0.335 Q, 0.135 ac ^ac dc dc 6 81.15\u00b0 A 15\u00b0 v , 0.449 v 0.949 o d q (4.16) 49 1.233 E- 1.388 V ref 1.061 V 1.61 R VT -1.57 cosa_ 0.962 cosaT -0.868 I R 0.312 For the given data the system's linearized equations are A6 0.0 Aw Aifi fd AE A cos a R Acosa, u X 1.0 0.0 0.0 0.0 0.0 0.0 0.0 -119.4 0.0 -173.3 -120.3 0.0 0.0 -0.38 0.15 -0.35 -0.33 0.0 -0.04 183.0 0.0 -1860.0 -20.0 -443.8 134.0 2600.0 2171.1 0.0 -22068.0 0.0 -40020.0 -32172.4 0.0 -17.7 0.0 . 179.5 0.03 0.0 -9.55 0.0 318.2 0.0 -13.9 -105.4 -9.62 [A6 Aa> AiJ;,., AE Acosa\u201e Acosa,. fd x R I u ] x 0.0 -2.0 (4.17) and the corresponding eigenvalues are -0.87 +j2.4, -4.76, -7.85 + j l l , -363.6, -39762.6 The system is s t i l l stable. However, because of the new situation, the excitation control signal originally designed for the ac system is modified to become 0.09 S Ux \" 1 + S Aw (4.18) and the system's eigenvalues for this case are -1.94+J2.04, -2.06, -7.63+J10.8, ~363.6, -39762.6 The system is more stable with the new adjustment. In addition to the above two cases where the system is stabilized by excitation control signals u x and u^ respectively, three new controls are designed and compared with old schemes J 50 1. No modification of the excitation control u and no optimal control x signals. 2. With only the modified excitation control signal u^. 3. An optimal dc current control is designed with excitation control removed. 4. A dc control u^ is designed in conjunction with u^. 5. A dc control is designed in conjunction with the modified exci-tation control u'. x The system's eigenvalues for the last three cases are listed in Table V. Table V Eigenvalues for expanded system Control Eigenvalues -0.93, -38 .5 -10.75 + J13.1 -379, -41071.2 u and u^ x D -1 .06 + jl.5 -55.35 + J66.7 -0.8 -384.5 , -41071.2 u' and u^ x D -0.85 + jl.01 -58.4 + j71 -0.667 -385.2 , -41071.2 The control laws for different cases are given by \" V \" 2.05 1.13 -13.2 -0.045 -0.133 -0.27 0.0 \" ux and u^ UD = 1.37 0.92 -7.73 0.75 -0.14 0.88 19.75 ux a n d \"D _UD_ 1.52 0.99 -9.3 0.76 -0.14 0.9 18.8 [A6 Au) Ail),., AE A C O S O L AcosaT u or u'] (4.19) fd x R I x x The nonlinear test of section 4.3 with the same disturbance is applied to this system but with a fault duration reduced to 0.05 sec. and line reclosure at 0.25 sec. after the fault occurrence. The nonlinear system response is summarized in Figures 4.11 to 4.16. Figure 4.11 in-dicates that in the first two cases, where u and u' respectively are applied, the system is unstable despite the fact that the corresponding 51 eigenvalues are stable. This clearly demonstrates the necessity of nonlinear tests and that eigenvalue analysis, although necessary, is not sufficient. Although the dc control signal u^ is more effective than the excitation controls, i t is not sufficient by itself to maintain stability. The system's response with combined excitation and dc controls is much better and the two responses with u or u' controls are very x x close whether the excitation control signal is modified or not. The terminal voltage variations, Figure 4.13, are more pronounced in cases 1 and 2 and the voltage oscillation continues after the line is restored whereas the terminal voltage approaches its steady state value asymp-totically after 0.35 sec. in cases 3,4 and 5. Figures 4.14 and 4.15 show the variation-of transmitted power over the dc and ac lines res-pectively. In cases 1-and 2 the dc link is operating under constant current control and the dc power is nearly constant about, the original value, 0.335 p.u. while the ac power is changing widely after line reclosure. In cases 4 and 5 the dc line picks up the power lost by the ac line and returns to rated value when the line is reclosed. In the mean time the ac line is transmitting the rated power. A similar behav-iour is noted in case 3 up to a point where the ac power starts to drop again and the dc line picks up the difference. The dc control effort in cases 4 and 5 is smaller than case 3 as shown in Figure 4.16. The control stays at limit values, + 0.25, for about 0.3 sec. then decreases rapidly and approaches zero asymptotically. It is concluded from the above discussion that a combined dc and excitation control scheme is the best for the dc\/ac parallel system with slightly better results for u^ and than and u^ controls. S RAD < i i l i i i i i i i L. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 t S E C Fig. 4.11 Swing Curves for Expanded System t S E C Fig. 4.12 Speed Deviation for Expanded System 53 0] 02 03 t l 05 OS 07~~ ts 05 Fig. 4.14 Variation of DC Power for Expanded System 54 55 5. MULTITERMINAL HVDC SYSTEMS Al l hvdc systems designed so far have been restricted to the two-terminal operation in which power is transmitted over an hvdc link from system A to system B. However, hvdc need not be so limited and can be readily extended to the multi-terminal operation. In this and the next Chapter the operation and control of a three terminal dc link in conjunction with a delta connected ac system are studied. Two operation modes are considered^ two rectifiers and one inverter or one rectifier and two inverters. The ac system is either a two-machine in-finite system or a three-machine system. The equations describing system dynamics and the determination of the operating point are given in this chapter. 5.1 Ope-rati-on.antd-Conitrnl<,sf ;Multl-fce-rminal..-DC.Lines As partly explained in Chapter 2 the normal operation mode of a two-terminal dc line is to control the currents of both the rec-t i f i e r and inverter with identical current orders but giving the inver-ter a current margin signal of a polarity opposite to the current order. The inverter is normally operating on constant extinction angle control which maintains a minimum safe angle of advance and a maximum dc line voltage while the line current is regulated by the delay angle a R. The current control will be taken over by the inverter only i f the rectifier is unable to provide the ordered current at its minimum delay angle. The same principles of operation can be adopted without change for a tapped dc line with two rectifiers and one inverter. The inverter current order is the algebraic sum of the two rectifiers' current orders plus the current margin. However, the same policy can not be applied 56 to the case of one rectifier and two inverters. If both inverters were on constant extinction angle control there would only be one ratio of power flow between the two receiving stations. For this reason one of the inverters and the rectifier are given specific current orders 37 while the second inverter regulates the dc line voltage .: The difference between these two current orders minus the current margin is applied to the second inverter's current override control. Changing the current order of the first inverter without changing the rectifier' s current order would result in current transfer between the two inverters. An alter-40 native control policy was also proposed where constant extinction angle control is applied at more than one inverter. It is argued that the first control method has the disadvantage that reactive power com-pensation must be provided for constant current operation at a l l but one of the converter stations whereas the second method improves this situation. In the second method the desired power flow can be achieved by adjusting the ac voltages at the inverter stations by means of tap changer control. In this chapter the first control method is adopted. 5.2 System Equations The system to be studied is given in Figure 5.1, consisting of a tapped dc line superimposed on a delta connected ac network. Bus number 3 is an infinite system which will be replaced by a small gen-erator for later studies. There are in general three converter stations where converter station 1 is always operated as a rectifier and that of station 2 always as an inverter. As for converter station 3 i t is operated either as a rectifier or as an inverter depending on which cases are studied. The system is modelled in general as follows; the Fig. 5.2 TRANSFORMATION BETWEEN COMMON AND INDIVIDUAL MACHINE FRAMES 58 synchronous machines are represented by third order models with single time lag voltage regulators. Converter controls are also represented by single time lags. The transmission network is represented by static impedances and the ac harmonic filters are not considered but the con-densers supplying the required reactive power are included. The system state equations are given by p~~\u00a3^' The reason for giving weight on ty^^ is mainly be-cause of the dc lines involved and it was not necessary for ac lines alone. As for the weighing matrix R i t is always taken as a unit matrix. 68 6.1 Two Machine-Infinite Bus System with Two Rectifiers and One Inverter Converter stations 1 and 3 are operated as rectifiers while converter station 2 is operated as an inverter in this case. For the data given in Chapter 5 the linearized state equations for the system are given in eqn. (6.1). The diagonal matrices Q and the nonzero elements of matrices B for different control schemes are given by u E 1,u E 2 ; Q = diag {10, 100, 1, 1, 1, 10, 100, 1, 1, 1, 1} b^ 1 = 10.0, bg 2 = 1 0 , 0 U j ^ . u ^ . Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, l ) b_ . = 10485.44, b . . _ = 10588.23 n. D , 1 11, I (.0 . 1 ) . Q = diag {10, 1, 100, 1, 1, 10, 1, 100, 1, 1, 1> b c , = 10485.44, b Q . = 10.0, b . . . = 10588.23 -> > -L \" \u00bb ^ J . J . , J u E 1,u D l,u E 2,u D 3. Q = diag {10, 1, 10, 1, 1, 10, 1, 10, 1, 1, 1> b. . = 10.0, bK _ = 10485.44, b Q = 10.0, b . . , = 10588.23 H , i D , z y , J i ^ - > ^ and the corresponding control laws are given in eqn. (6.3). The sys-tem's eigenvalues are listed in Table VI. A three-phase ground fault at the middle of one circuit of ac line 1 is used as a disturbance for the nonlinear tests to compare the different control schemes designed. The fault duration is 0.07 sec. followed by isolating the faulted line from both ends. The system responses are shown in Figures 6.1 to 6.7 for the following cases case 0 case 1 case 2 no optimal control whatsoever. optimal excitation controls for both machines. optimal dc controls for rectifier stations 1 and 3. A6\\ Aip fdl A * x l \u2022 Acosa AS, Aw, Rl Aip fd2 A Ex2 Acosa * Acosa 12 R3 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -26.0 0.0 -114.0 0.0 -123.4 7.225 0.0 -0.72 0.0 -38.8 -0.35 0.0 -1.5 0.45 -1.26 0.142 0.0 0.145 0.0 -0.393 0.405 0.0 -6.58 -0.5 -0.434 -0.369 0.0 -0.87 0.0 -0.145 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 20.98 0.0 22.42 0.0 17.07 -53.9 0.0 -151.4 0.0 43.89 0.11 0.0 0.388 0.0 0.409 0.617 0.0 -0.834 0.0 -1.26 -1.099 0.0 -6.55 -0.5 -2.89 0.0 63.6 0.82 -0.769 501.6 0.0 -18991.0 0.0 -33213.0 177'.0 0.0 5075.5 0.0 -11125.0 16942.0 0.0 68.0 -0.372 0.0 -1.902 0.447 0.988 0.81 -0.501 16.43 0.0 72.54 0.0 133.0 -34.8 0.0 -228.0 0.0 -78.9 -6.43 1657.4 0.0 12689.0 0.0 21752.4 2339.4 0.0 11649.4 0.0 -13772.0 -35626.0 \u2022[A6, Ato, Aip.,, AE , Acosa,,, A6\u201e Au. Aij i - , - AE \u201e AcosaT\u201e Acosa_~] + BU 1 1 fdl xl Rl 2 2 fd2 x2 12 RJ (6. \"El -38.4 6.37 -60.4 -2.48 -0.023 21.4 -1.01 -3.53 -0.057 0.285 -0.003 \"E2 5.66 3.38 -3.53 -0.057 -0.004 -40.6 6.84 -88.4 -2.93 -1.04 0.015 UDl 3.1 1.01 -0.98 0.75 -0.22 -1.69 -0.22 -5.0 0.75 \u2022 -0.101 -0.108 UD3 -0.39 -0.35 0.49 0.79 -0.11 -2.58 -0.97 -1.02 0.67 0.15 -0.195 UD1 2.63 1.05 -3.3 0.41 -0.22 -1.09 -0.26 -2.41 -0.04 -0.086 -0.108 \"E2 0.51 0.02 -1.42 0.51 -0.4x10\" -4 -0.77 0.12 -9.81 -1.32 -0.004 -0.3x10\" \u20225 \"D3 -0.76 -0.31 -1.4 0.53 -0.109 -2.11 -0.998 0.93 -0.003 0.16 -0.195 \"El -0.06 0.03 -2.16 -1.04 -0.8x10\" -5 -0.005 -0.003 -0.04 -0.005 -0.002 0.1x10\" \u20224 \"Dl 2.4 1.02 -1.3 -0.009 -0.22 -0.85 -0.27' -1.12 -0.013 -0.087 -0.108 \"E2 0.16 0.016 -0.25 -0.005 -0.1x10\" -4 -0.6 0.034 -3.83 -1.11 0.004 0.1x10\" -4 -0.93 -0.33 0.037 0.012 -0.11 -1.94 -1.01 2.06 0.01 0.16 -0.19 . [A6 Ao) Aif\/ AE . Acosa A6 Au). Aip AE Acosa 0 Acosa Q ] t (6.3) Table VI Eigenvalues for 2 Machine-Infinite Bus, 2 Rectifiers System Control used Eigenvalues no control -53657.4, -15084.8, -177.3, -0.605+jl.35 , -0. 342+J4.77, -0.293+j6.81, -0.261+J1.83 UE1 ' \"E2 -53657.4, -15084.8, -177.3, -15.74,-13.5, -7.45+J12.9, -6.16+jlO.74, -0.35, -0.32 \"DI ' UD3 -54675, -18423, -162.1, -39.8, -22.95 ,-3.36, -3.16 , -1.17y2.1, -0.5+J1.58 \"DI ' \"E2 ' \"D3 -54675, -18423, -162 , -39.83,-22.96 -6.98,-5.95, -4.39, -3.37, -0.717 + jl.7 \" E I \u00bb \"DI ' \"E2 \u00bb \"D3 -54675, -18423, -162, -39.44,-22.8,-9.6,-9.2, -3.76,-3.4, -2.39,-1.81 72 case 3: case 2 plus excitation control for the machine connected to inverter station 2. case 4: case 1 plus case 2. Figures 6.1 and 6.2 show that machine 1 loses stability but machine 2 remains stable when there is no optimal control signals, case 0. Next when excitation control signals are applied to both mach-ines the stability of machine 1 is getting worse and machine 2 that was originally stable goes out of step, case 1. S t i l l next where optimal dc .controls are applied to rectifiers 1 and 3 the system is stable, case 2. Finally the system is stable and the responses are very close for both cases 3 and 4. The angle and speed deviations are less in case 2 than cases 3.and 4. The changes in terminal voltages are shown in Figures 6.3 and 6.4. It is noted\" that' at ac bus- I \"the deviations are small for cases 2 and 3 with larger changes for cases 4, 0, and 1 respectively. For machine 2 the terminal voltage deviations, Figure 6.4, are smallest for case 4 followed by cases 2, 1, 3 and 0 respectively. Figures 6.5 (a, b, c) show the changes in power transmitted over the dc transmission network. Consider case 0 fi r s t . The dc power drops for the first 0.07 sec. during the fault because of ac terminal voltages drops, and then returns to rated values. After 0.69 sec. the power carried by dc line 1 drops sharply for 0.1 sec, between approxi-mately 0.69 and 0.79 sec, then continues to drop at a slower rate because of the deterioration of v ^ while the power carried by line 3 increases in a similar fashion because of drops in v 2 with vfc^ remain-ing constant. But the total dc power is decreased. Consider case 1 next. The same pattern is noticed but with sharper drops in P , .. 73 \u2014\u00ab 1 1 1 I I I I , 01 0.2 03 Q.4 05 05 07 OS 09 1.0 t sec Fig. 6.4 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Rectifier System 76 Fig. 6.7 Excitation Control Effort 'for 2 Machine, 2 Rectifier System 77 because of larger drops in v^. For cases. 2, 3 and 4 the dc reference currents are modified by optimal control signals and therefore the total power transmitted over the dc network is significantly increased to absorb some power from the ac system, which may cause large acceleration of the synchronous machines, and consequently resulting in a more stable system. The control efforts are shown in Figures 6.6 and 6.7. It is interesting to notice that the dc control effort at rectifier station 1, IL^> is larger in case 4 than that of cases 2 and 3. This suggests that the presence of an excitation control signal at machine 1 is work-ing against the dc control at the same bus. This may be explained by the fact that the voltage drop at bus 1 is larger in case 4 than cases 2 and 3. It is also noticed that in general the dc control efforts at rectifier station 1 are much larger than the dc control efforts at rectifier station 3 since v ^ is constant and machine 1 is also closer to the fault location. The excitation control efforts are greatly reduced in the presence of dc controls. 6.2 Two Machine-Infinite Bus System with One Rectifier and Two Inverters The study in this section is similar to that of section 6.1 except that converter station 3 is now operated as an inverter instead of a rectifier. So the only change in the cases listed before will be the fact that station 3 is an inverter. The operating point was given in Chapter 5 and the linearized system state equations are A 6 Ai|> f d l A E x l Acosa A 6 , Aa), R l Aifi fd2 AE x2 Acosa \u2022 Acosa 12 13. 0.0 1.0 -33 0.0 -0.38 0.0 0.056 0.0 -286 0.0 0.0 0.0 19.4 0.0 0.09 0.0 0.37 0.0 12.6 0.0 726 0.0 0.0 -117.6 -1.58 -6.51 -19722 0.0 22.1 0.36 -0.8 77.9 -13434 0.0 0.0 0.46 -0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -140.6 -1.36 -1.33 -36163 0.0 20.1 0.41 -1.14 152.4 -23864 0.0 0.0 11.3 0.0 0.16 0.0 0.2 0.0 291 0.0 0.0 1.0 50 0.0 0.28 0.0 0.74 0.0 23.9 0.0 1633 0.0 0.0 0.042 0.17 -1.01 4652 0.0 -139 -1.78 -6.24 -212.3 -12456 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.45 -0.5 0.0 0.0 0.0 -31.5 -0.38 0.14 -11241 0.0 . 0.7 1.01 -1.91 -46.8 15930 0.0 -62.3 -0.84 0.82 -16153 0.0 -67.9 -0.7 0.49' 4.45 -35443 A 6 . A I D , A^ fdl AE: , xl Acosa. A 6 R Aw, Rl Alp fd2 A Ex2 Acosa Acosa + BU 12 I3-1 (6 4) oo 79 The weighing matrices Q and the corresponding nonzero elements of matrices B are V L ^ and ^ ; Q \u00bb diag. {100, 10, 1, 1, 1, 100, 10, 1, 1, 1, 1} b 4 , l = 1 0 ' b9,2 = 1 0 and u D 3 . Q = diag. {100, 10, 1000, 1, 1, 100, 10, 1000, 1, 1, 1} b. . = 10383.8, b.. 0 = 10799.1 ->, J - 1-L, 2. Qo. D) ^ , 1 ^ 2 and . Q = diag. {100, 10, 1, 1, 1, 100, 10, 1, 1, 1, 1} b 5 \u00b1 = 10383.8, b 9 2 = 10, b u 3 = 10799.1 \"El* \"Dl' UE2 a n d UD3- Q = d l a g ' 1 > 1 0 ' l j 1' 1 0 ' 1\u00bb 1 0 ' 1' 1 } b. ' = 10, b_ \u201e = 10383.8, b. . = 10, b.. . = 10799.1 H , J - D , y , J \u00b1\u00b1,4 The control laws are given in equation (6.6) and the system's eigen-values are listed in Table VII. The same disturbance described in section 6.1 is applied to this system for the nonlinear tests. The system responses are summari-zed in Figures 6.8 to 6.14. Figures 6.8 and 6.9 show the angle and speed deviations res-pectively. The system is more stable without optimal control, case 0, than with excitation control, case 1. Cases 2 and 3 give the smallest angular deviations and i t is also close for case 4. The terminal voltage variations at bus 1, Figure 6.10, are largest in case 1 followed by cases 4, 0, 2 and 3 in that order. At bus 2 a l l the voltage deviations are small. In Figures 6.12 (a, b, c) the variations of power transmitted over the dc network are shown. In cases 0 and 1 with the dc network operating on constant current control the dc power goes back to original values immediately after the faulted line is isolated and then decreases \" E l -13 1.6 -25.6 -1.75 -0.004 12.15 -0.66 -12 -0.34 0.52 0.0013 ~ \"E2 -4.4 2.74 -14.7 -0.34 -0.01 -12 1.11 -23.3 -1.67 -0.36 -0.0016 UD1 10.8 3.1 -0.63 1.99 -0.22 -3.25 -0.25 -11.6 0.27 -0.12 0.102 \" D3 0.17 0.57 1.28 -2.81 0.11 10.4 3.1 7.76 -0.49 -0.93 -0.2 UD1 9.04 3.21 -1.39 0.06 -0.21 -1.17 -0.45 -0.72 0.0003 -0.025 0.102 \"E2 0.21 0.02 -1.08 0.21 0.3xl0~6 -0.41 0.01 -1.3 -1.0 0.0026 -O.SxlO - 5 UD3 = 1.73 0.48 0.42 -0.08 0.11 8.48 3.24 -1.75 -0.009 -1.0 -0.2 \" E l -0.15 0.03 -2.16 -1.05 -0.5xl0~6 0.18 -0.003 -0.006 -0.005 -0.005 -0.15xl0~ 4 \"Dl 2.43 1.06 -1.07 -0.0005 -0.21 -0.08 -0.04 -1.01 0.005 -0.12 0.1 \"E2 0.09 0.009 -0.24 -0.005 0.5xl0~5 -0.2 0.01 -1.32 -1.01 -0.0014 -0.2xl0~ 4 0.52 0.07 0.49 -0.02 0.11 1.94 1.07 -0.72 -0.02 -0.48 -0.19 \u2022[A61 A W 1 A * f d l AEx^ Acosa -KJL A6 2 Ao 2 A*fd2 A E x 2 Acosaj 2 Acosa.^] t (6.6) co o Table VII Eigenvalues for 2 Machine-Infinite Bus, 2 Inverters System Control used Eigenvalues no control -55443, -16056, -155, -0.785+j1.49, -0.405+J1.82, -0.201+J4.76, -0.168+J6.52 u E l a n d \"E2 -55443, -16056, -155, -10.75,-9.91, -3.66+J7.91, -2.92, 2.15+J6.23, -2.11 UD1 a n d \"D3 -56428, -19286, -191, -94.45,-48.4, -3.16, -1.82 +j2.2, -1.81, -0.78 + jl.41 UD1'UE2&UD3 -56428, -19286, -191, -93.9,-46.8, -9.64, -3.26, -3.18, -1.59, -0.874+J1.68 UE1'UD1>UE2 and u D 3 -56428, -19286, -149, -32.3, -14.38, -9.55, -9.49, -4.18, -3.39, -2.66, -2.17 82 t sec Fig. 6.8 Swing Curves for 2 Machine, 2 Inverter System Au rad\/sec '3 t sec J.O 0.6^ ^ ^ ^ I r r . Fig. 6.9 Speed Deviation for 2 Machine, 2 Inverter System 7.7 7.05 7-0 0.35 .9 0.85 0.8 v., A \" 83 2 0 . . \u2022 \u2022\u2022 *\u2022 0.0 0.1 02 0.3 04 0-5 06 07 0-8 0.9 LO t sec Fig. 6.10 Terminal Voltage Variations at Bus 1 for 2 Machine, 2 Inverter Svstem 7.7 7.05 7.0 0-95 0.90 0.85 0.80 vt2 A\"-0-7 0-2 03 0.4 05 0.6 0.7 08 0.9 7.0 t sec Fig. 6.11 Terminal Voltage Variations at Bus 2 for 2 Machine, 2 Inverter System I I I I I I I I I I l_ 0.0 0-7 0.2 0-3 0.4 0.5 0-6 0.7 0-8 0.9 1.0 0.0 0.7 0.2 0.3 0-4 0.5 0.6 0.7 0-8 0.9 10 (c) \" f sec Fig. 6.12 DC Power Variations for 2 Machine. 2 Inverter Svsfem 0.15 0.05 -0.05 -0.1 -0.151 CASE 2 CASE 3 x xCASE 4 Fig. 6.13 DC Control Effort for 2 Machine, 2 Inverter System uF p.u. 0.151 0.05 -0.05 -0.151 Fig. 6.14 Excitation Control Effort for 2 Machine, 2 Inverter System CASE 1 CASE 3 x xCASE 4 86 because of the ac voltage drop at the rectifier bus. In cases 2, 3 and 4 where the dc reference currents are modified by optimal control sig-nals, the rectifier power is increased even with the fault on. The power is even increased further after the fault is removed and stays almost constant at about 0.48 p.u. then starts to go down at 0.7 sec. The control efforts are shown in Figures 6.13 and 6.14. Again i t is seen that the excitation control efforts, Figure 6.14, are greatly reduced in the presence of dc controls. It is also noticed that the dc control at converter station 3 jumps to high positive values for the fault duration and then changes sign after the faulted line is iso-lated. The reason for this is that during the fault inverter 3 draws more dc power since inverter 2 is incapable of absorbing this power due to the ac voltage drop.' After the faulted line is removed inverter 2 starts to pick up more dc power to compensate for the ac line power loss. 6.3 Three Machine System with Two Rectifiers and One Inverter The system studied here is the same as that of section 6.1 except that the infinite bus 3 is replaced by a small synchronous machine having one third the rating of either machines 1 or 2. The sys-tem's operating point is the same as section 6.1. The system's linearized state equations are given in eqn. (6.7). A l l 0.0 1.0 0.0 0.0 0.0 0.0 0.0 ,0.0 Au^ -23.2 0.0 -99 0.0 -105 13.4 0.0 7.85 A * f d l -0.36 0.0 -1.23 0.45 -0.9 0.24 0.0 0.35 0.63 0.0 -7.19 -0-5 -1.28 -0.52 0.0 -1.5 Acosa^ -14690 0.0 -14935 0.0 -27366 1069 0.0 9644 ti2 tm 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 27.6 0.0 44.4 0.0 43.9 -44.1 0.0 -141 A*fd2 0.11 0.0 0.7 0.0 0.82 -0.26 0.0 -1.67 A*x2 1.03 0.0 -1.28 0.0 -1.95 -1.14 0.0 -7.21 Acosaj2 2.0 0.0 129 0.0 211 -18.45 0.0 -175 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Au^ 49.6 0.0 42.8 0.0 42.3 29.8 0.0 -9.89 A*fd3 0.52 0.0 0.97 0.0 1.12 0.5 0.0 0.27 A*x3 1.27 0.0 -3.13 0.0 -4.45 -0.76 0.0 -3.33 Acoaa^ 1718 0.0 3593 0.0 8926 36.9 0.0 2264 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -52.6 9.77 0.0 10.7 0.0 31.2 0.0 -0.57 0.125 0.0 0.21 0.0 0.35 0.0 0.03 -0.11 0.0 -0.55 0.0 -0.1 0.0 -11826 400 0.0 3849 0.0 13269 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.7 . 16.4 0.0 14.75 0.0 15.8 0.48 0.77 0.15 0.0 0.24 0.0 0.25 0.5 -3.03 0.11 0.0 -0.49 0.0 -0.46 0.0 -103 16.45 0.0 49.3 0.0 140 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 -86 -79.4 0.0 -129 0.0 -170 0.0 -1.27 -1.02 0.0 -2.25 0.49 -2.73 0.0 0.82 -0.51 0.0 -2.89 -0.5 -3.3 0.0 -11034 -17554 0.0 -8268 0.0 -25443 A5, Ai|> fdl AE , xl Acosa A52 Au. Rl Ai(i fd2 A Ex2 Acosa AS, Au. 12 A^ fd3 AE , x3 L A c o s o R 3 j + BU (6.7) oo \u2014i 88 The matrices Q and the nonzero elements of B are given by ^.u^.^.-Q = diag {10,. 100, 1, 1, 1, 10, 100, 1, 1, 1, 10, 100, 1, 1,1} b 4 , l * 1 0 ' b9,2 \" 1 0 ' b14,3 = 1 0 \"DI* UD3 : Q = d i a g { 1 0 ' 1\u00bb 1 0 0> 1\u00bb 1> 10> 1\u00bb 1 0 0> 1\u00bb X> 1 0\u00bb 1' 1 0 0 \u00bb 1' 1 } b 5 j l = 10484.6, b 1 5 > 2 = 10587.4 ( 6 > g ) \" D l ' ^ ' ^ : Q = d i a g { 1 0 ' 1\u00bb 1 0 0 ' 1' 1' 1 0 ' 1\u00bb 1 0 0 ' l j 1\u00bb 1 0 ' l j 1 0 0 ' 1' 1 } b 5 , l = 1 0 4 8 4- 6> b 9 2 = 1 0 , b15 3 = 1 0 5 8 7 ' 4 \" E l ^ l ' ^ ' ^ E a ' ^ : Q = d \u00b1 a g { 1 0 ' 1 0 , 10-' 1 > 1 > 1 0 > 1 0 ' 1 0> 1\u00bb 1\u00bb 1 0 ' 1 0\u00bb 10, 1, 1} b 4 , l = 1 0 ' b5,2 = 1 0 4 8 4-6\u00bb b 9 j 3 = 10, b U ) 4 = 10, b 1 5 ) 5 = 10587.4 The corresponding control laws are given in (6.9). The system's eigenvalues for different controls are listed in Table VIII. The same disturbance applied to the nonlinear tests of section 6.1 is applied to this three machine system. Figures 6.15 to 6.22 show the system responses for different control schemes. Again four cases are studied as listed in section 6.1 except that in cases land 4 excitation controls are applied to a l l three machines in this case. Since this three machine system is unstable without optimal controls as indicated by the eigenvalues in Table VIII, the nonlinear response for this case is not shown. \u2022 The angle and speed deviation curves, Figures 6.15 and 6.16 respectively, show that in case 1 machine 1 is unstable and that in general the variations in angle and speed deviations are the largest in this case. Case 2 results in the smallest deviations followed by cases 3 and 4 in that order. Figures 6.17, 6.18 and 6.19 show the ac terminal voltages variations. At bus 1, Figure 6.17, the voltage changes most in case 1 \" U E l \" ~37.4 6.37 -63.4 -2.53 -0.023 21.3 -1.04 -0.26 -0.002 0.19 17.3 -1.19 10.7 0.17 0.002 \"E2 26.3 5.24 0.69 -0.002 -0.016 -23.9 7.81 -129 -3.49 -0.4 2.26 2.36 -9.19 -0.19 -0.015 UE3 25 2.88 8.5 0.17 -0.015 40 0.6 -16.. 5 -0.19 0.06 -62.5 4.89 -69.7 -2.74 -0.03 \" D I 7.14 1.26 0.47 0.19 -0.21 -3.94 0.64 -21.6 -0.61 -0.14 1.08 0.05 -0.54 1.09 -0.09 \"D3 1.26 0.13 0.65 -0.34 -0.09 -1.67 0.31 -8.73 -0.17 -0.12 3.82 1.03 0.3 ' -1.62 -0.25 . \" D I 6.24 1.23 -0.001 0.19 -0.21 -2.77 0.49 -15.2 -0.32 -0.13 0.69 0.05 -0.62 1.1 -0.09 UE2 2.83 0.13 1.88 0.03 -0.0003 -2.91 0.61 -20.6 -1.6 -0.03 1.11 0.03 0.25 0.13 -0.0001 \"D3 0.89 0.12 0.44 -0.3 -0.09 -1.11 0.25 -6.05 -0113 -0.12 3.64 1.03 0.22 -1.61 -0.25 \" E I a -0.73 0.007 -2.41 -1.05 0.2xl0~4 0.78 -0.12 2.92 0.08 0.005 -0.14 -0.006 0.04 0.001 0.8x10\" -5 \" D I 6.99 3.37 0.17 0.03 -0.22 -5.32 0.48 -21.3 -0.43 -0.16 1.93 0.001 0.27 0.008 -0.09 \"E2 9.87 0.44 3.16 0.08 -0.0004 -11.8 2.17 -48.3 -2.21 -0.13 4.13 0.1 0.9 0.02 -0.0002 UE3 0.13 0.02 -0.01 0.001 0.8xl0~5 0.34 -0.001 0.69 0.02 -0.003 -0.43 0.02 -2.48 -1.07 0.3x10' -5 _\"l>3_ 2.32 0.14 0.92 0.01 -0.09 -1.92 0.28 -9.06 -0.18 -0.16 3.1 3.19 -0.63 0.004 -0.26 .[Afi1 A* f < J 1 AExx Aco Sc R 1 Afi 2 A* f d 2 AE^ Acosa^ A\u00ab 3 A ^ A* f d 3 AE^ Acosa^]' (6.9 Table VIII Eigenvalues for 3 Machine, 2 Rectifier System. Control used Eigenvalues no control -37332, -15211, -372, -0.585 + , -0.526 + , -0.501 + , -0.483 + , -0.314 + , +0.88 xlO - 3 jl.95 jl.16 jl.58 J8.92 J6.88 UE1 , UE2 , UE3 -37332, -15211, -372, -18.5, -15.4, -13.26, -0.9 + , -7.05 + , -6.1 + , -0.35, -0.34, -0.32 J14.9 J14.2 jlO.26 UD1> ^3 -38764, -18557, -342, -70.25, -38.24, -3.28, -3.18, -2.8 + , -1.44, -1.2 + , -0.526, -0.4 + J5.82 jl.98 jl.68 \" D l ' ^ ' V , -38764, -18557, -342, -70.2, -38.2, -9.57, -4.27, -4.19 + , -3.25, -3.2, -1.19 + , -0.41 + J6.88 jl.97 jl.67 \"E3' \"D3 -38764, -18556, -341, -225, -122, -13.06, -9.6, -9.57, -6.1 + , -1.73 + , -1.13, -1.1, -1.0 J9.89 jO.07 O 2.0, Srad 91 7.5 i.ol -x X' \u00a312 0.51 H3 0-0 J 1 1 L I I I I i \u201ej 0-8 0.9 10 t sec 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1.0 S23 Fig. 6.15 Swing Curves for 3 Machine, 2 Rectifier System ACJ rad\/sec AOJI ^Aco3 ~~