Abstract:
Simultaneous power and voltage control by a d-c-tie between a-c networks is carried out by means of fixed and switchable inductive and capacitive compensation elements including the tap changer of both converter transformers of a d-c-tie for setting the optimum operating point by a transformation ratio of the converter transformer taps matching the primary current ratio, of the converter transformer taps for the maximally permissible stage of the one converter transformer, taking into consideration control, extinction and overlap angles of the d-c-tie.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a combined control method for a d-c-tie connecting two independent three-phase a-c networks. 
     In known control methods for a d-c-tie, the active power of the system is controlled and the reactive power is adapted to the reactive power demand of the respective a-c network with the aid of shunt reactors or capacitors, static compensators (for instance, thyristor-controlled reactors or capacitors) and/or synchronous condensors. For the controlled compensation devices for both a-c networks, a considerable amount of means is required for a necessary fine-step or stepless voltage adaptation to the active power to be transmitted. In addition, it is possible to switch the tap changer of the converter transformer in the wrong direction before the control arrangement can ascertain that compensating devices must be switched, so that the tap position must be switched back in the direction of its starting position after the respective compensation device has been switched by it independently; switching beyond its starting position may even become necessary. The control process with successive switching actions of the tap changer allows only a slow control; the required active and reactive power cannot be maintained during switching operations which take a relatively long time. In addition, the tap changer is stressed unnecessarily by the back and forth stepping operations. 
     In order not be extend the control time further, only the side of the d-c-tie connected to the weaker a-c network is frequently controlled in the sense of meeting the predetermined reactive power conditions of this a-c network, and the other side of the d-c-tie associated with the stronger a-c network is made to approach, as far as reactive power is concerned, the demand only via switchable shunt reactors or capacitors as compensating elements, so that in the second a-c network, the reactive power required there cannot be maintained exactly. This method further cannot recognize a control into undesirable boundary regions so that in the event of changes of the desired power or in the case of voltage changes in the a-c networks, no sufficient control reserve is provided. If in the case of two weak a-c networks an exact control is required also on the side of the second a-c network a further slowing-down of the control is resulted. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a substantially faster control for both a-c sides of the d-c-tie and to determine an operating point which makes it possible to maintain the predetermined reactive power condition for both a-c networks and which, in the case of dynamic processes related to changes, of active power in the d-c-tie allows the fast determination of a new operating point which allows on both a-c sides of the d-c-tie the smallest possible deviations from the respective set-point values of the a-c network voltages and reactive power of the a-c networks, where switchable shunt reactors or capacitors suffice. 
     The above and other objects of the present invention are achieved by a control method for a d-c-tie (high-voltage transmission system) connecting two a-c networks with active power control and combined reactive-power influencing by shunt reactors or shunt capacitors and converter transformers with tap changers for maintaining the voltage of the a-c networks. Simultaneous active and reactive power control of both a-c networks is performed by combinations of fixed and switchable shunt capacitors and/or shunt reactors, including the tap position of both converter transformers for setting the optimum operating point at the maximally permissible tap position for the respective operating voltage of the one converter transformer, wherein the ratio of the primary currents of the two converter transformers and the ratio of the maximally permissible tap position of the one converter transformer to a fitting tap position of the other converter transformer is chosen in accordance with the relationship ##EQU1## taking into consideration the dependence of the prevailing d-c current ##EQU2## in the direction of obtaining a permissible transmission current I dmin  meeting all network requirements, equal to one-half of the sum of the d-c currents of the two converters, wherein u 1 , u 2  are the transformation ratios of the respective tap position; S iV1 , S iV2  are the a-c side currents of the converter transformers; b is the number of the series-connected rectifier bridges; ρ 1 , ρ 2 , χ 2 , χ 1  are variables dependent on control angles α, extinction angles γ, and overlap angles u in accordance with the following equations: ##EQU3## 
     The control method according to the invention starts out from the reactive-power demand of both a-c networks, the a-c network voltage required in each case as well as the required transmission power of the d-c-tie system, while these values are adapted to the actual network conditions. From the permissible voltage range and the fixed and switchable shunt reactors or capacitors provided as well as the converter transformer tap positions and the required voltage-dependent reactive power of each a-c network, a region is obtained in a reactive power, active power diagram for operating of the converters, within which operating points are determined in steady-stage operation which meet all a-c network conditions. Adding and disconnecting capacitive or inductive compensation elements takes place if the range is exceeded or not reached. In the case of dynamic processes, the operating point may lie outside of the mentioned range. 
     For supporting a a-c network, the d-c current can be adjusted as the transmission current determined for the side of the d-c-tie assigned to this a-c network. The reactive power of the other a-c network then deviates from the set value accordingly more. 
     If the compensation elements are designed sufficiently large, the reactive power of one or both converters are changed in the direction of an approximation by switching of shunt capacitors or reactors, the position of the transformer tap changer assigned to them exhibits the smallest possible difference. Finally, the tap positions for each converter transformer as well as the d-c currents of the converters are determined by the given network data and the operating data and preferably the smallest possible common d-c current is determined therefrom which is sufficient for meeting all conditions of both a-c networks. 
     After coarse adjustment by the compensation elements, the optimum operating point is advantageously determined by addressing the combination of the tap position determined in accordance with the method of the invention, so that fine tuning by a static compensator or a synchronous condenser is not necessary. 
     The method of the invention makes possible 
     (a) a targeted addressing of an unambiguous operating point which meets all network requirements, 
     (b) the statement of missing or not required compensating elements in case of structural changes of the a-c networks with changes of the reactive power required by them, 
     (c) the determination of another favorable operating point in the event of a failure or disconnecting of compensating elements for servicing purposes, as well as 
     (d) the adjustment of the most favorable operating point during occurring power or voltage changes (dynamic processes) during the time interval in which no compensation elements and no tap changer can yet operate. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further details will be explained with the aid of an example for a d-c-tie operating in accordance with the control method of the invention, with reference to the drawings, in which: 
     FIG. 1 shows a d-c-tie back to back coupling connecting two a-c networks schematically with fixed filter circuits and switchable compensation elements; 
     FIG. 2 shows an active-reactive power diagram for the d-c-tie with a range of value pairs in which the d-c-tie operates for meeting all operating conditions and specifically as a function of the control or the extinction angle and of the transmitted d-c current and of the transformer tap and a nominal voltage of 230 kV; 
     FIG. 3 shows another embodiment of a d-c-tie connecting two a-c networks; and 
     FIG. 4 are graphs explaining the operation of the d-c-tie of FIG. 3 with voltage conditions of the a-c network N1. 
    
    
     DETAILED DESCRIPTION 
     The operating data associated with the a-c networks N1 and N2 are designated with appropriate subscripts and furthermore with additional subscripts N and n, of which the subscript N indicates network variables and the subscript n indicates nominal values. 
     In the following description, the following designations are made: ##EQU4## 
     For the example shown in the following of d-c-tie with nominal active power P n  =200 MW, P ref  =20 MW and a nominal network voltage U Nn  =230 kV, the requirements for the a-c networks N1 and N2 given by the utility are as follows: 
     
         ______________________________________Q.sub.N1 = 33 MVAr (capacitive)Q.sub.N2 = 40 MVAr (capacitive)U.sub.N1 = 241.5 kV corresponding to 241.5/230 = 1.05 per unitU.sub.N2 = 234.6 kV corresponding to 234.6/230 = 1.02 per unit______________________________________ 
    
     From this, the necessary compensation elements keeping in mind the filter circuits FK can be calculated with the values preferably intended for filtering harmonics: 
     FK 1  ≡Q FK1  =105 MVAr; in addition, a shunt capacitor C 1  ≡Q cap1  =35 MVAr 
     FK 2  ≡Q FK2  =105 MVAr: additionally a shunt capacitor C 2  ≡Q cap2  =35 MVAr 
     For this transmission in the range from full to partial loads, with the available compensation elements, the transformer data for the converter transformers as well as the voltage-dependent reactive power requirements of the a-c networks, there is determined an operating range of the converter reactive power between 90 and 155 MVAr, which assures reliable operation for all operating requirements and which has sufficient control reserves in both power flow directions for dynamic processes. 
     The following is obtained for: 
     Network N1 
     Reactive power demand Q N1  =33.0 MVAr (capacitive) for U N1  =241.5 kV; (1.05 p.u.) 
     Filter circuit FK 1  with Q FK1  =105×1.05 2  =115.76 MVAr 
     C 1  with Q cap1  =35×1.05 2  =38.59 MVAr 
     no shunt reactors required, i.e., Q ind1  =0 
     Network N2 
     Reactive power demand Q N2  =40.0 MVAr (capacitive) for U N2  234.6 kV; (1.02 p.u.) 
     Filter circuit FK 2  with Q FK2  =105×1.02 2  =109.24 MVAr 
     C 2  with Q cap2  =35×1.02 2  =36.41 MVAr 
     no shunt reactors necessary, i.e., Q ind2  =0 
     From this is obtained: 
     
         Q.sub.V1 =Q.sub.FK1 +Q.sub.N1 +Q.sub.cap1 -Q.sub.ind1 =115.76-33.0+38.59-0=121.35 MVAr 
    
     
         Q.sub.V2 =Q.sub.FK2 +Q.sub.N2 +Q.sub.cap2 -Q.sub.ind2 =109.24-40.0+36.41-0=105.66 MVAr 
    
     The calculated reactive power of the converters Q V1  =121.35 MVAr and Q V2  =105.66 MVAr are in the said operating range between 90 and 155 MVAr. 
     With the a-c network conditions given by the utility, the active power at the inverter and the required compensation elements for this example, the tap position for each converter transformer of the d-c-tie and the transmission current I dmin  can be determined in the following manner from ##EQU5## φ is set equal to α and γ, respectively, in first approximation: ##EQU6## (with b=number of converter bridges connected in series). 
     If concrete values are substituted, the following is obtained: 
     
         ______________________________________cos ρ.sub.1 = 0.1626 and thus              ρ.sub.1 = 80.64°              ρ.sub.2 = 79.28°s.sub.ivl = 0.2940 kA and              S.sub.iV2 = 0.2646 kAu.sub.1 = 9.20° and              u.sub.2 = 9.23°ε.sub.1 = 0.08 and              ε.sub.2 = 0.08ρ.sub.1 = 0.0835 and              ρ.sub.2 = 0.1060 .sub.1 = 0.9954 and               .sub.2 = 0.9933______________________________________ 
    
     From the relationship according to the idea of the invention, I d1  =I d2 , the relationship of the transformation ratios ##EQU7## can be calculated as 0.9. 
     Accordingly, the transformation ratio u 2  of the converter transformer associated with the a-c network N2 is 0.9-times as large as the transformation ratio u 1  of the converter transformer associated with the network N1, and the possible tap combinations are fixed thereby. 
     In the numerical example, both converter transformers are equal and each have 35 taps. For calculating the smallest permissible d-c current I dmin  it is assumed that one of the two converter transformers is always set to the highest tap permissible for this a-c network voltage, in the case of the tap 35 of the converter transformer T1 of the a-c network N1, since u 2  is smaller than u 1 . 
     At the converter transformer T2 associated with the a-c network N2, the tap coming closest to the calculated value u 2  =u 1 (35) ×0.9 is set. 
     For an actual value u 1 (35) =0.110444, a numerical value of u 2  =0.09939 is obtained. 
     This is approached closest by the tap 28 of the second converter transformer T2 with u 2 (28) =0.09924. 
     The d-c current for the 35th and 28th tap calculated therefrom is obtained as ##EQU8## is thus obtained with 1.7107 kA as the smallest possible direct current for meeting all specified network conditions and within the operation range of the converters. 
     For minimal deviations from the required data of both sides, also other tap ratios such as 34/37, 33/26 or 32/25 can be used, where I dmin  has an increasingly larger magnitude. In this connection it must be noted that the current I dmin  which increases with decreasing tap combinations must increase only to such an extent that it does not exceed the given nominal value I dn  or the control angle α, the extinction angle γ, and the overlap angle u stay within their permissible limits. 
     For acceptable operation of the d-c-tie, meeting the specified conditions, it must be assumed that I dmin  can flow in connection with the permissible tap position, where smaller deviations of the set-point values for the active and reactive power can possibly be corrected by the control or extinction angle. 
     The control method according to the invention can proceed in an analog as well as a digital manner and allows, as explained, the determination of the optimum operating point for steady-state operating cases as well as of the operating point with the lowest deviation from the desired values for both networks in case of dynamic processes in the networks. 
     An application of the invention is possible in the manner described in the following not only for two given a-c networks N1 and N2 coupled by the d-c-tie according to FIG. 1, but also for the cooperation of a-c networks of which at least the one a-c network can have different stages of development and the requirement is to be met that the converter transformers, the reactive power control for the d-c-tie and the determined compensation elements are sufficient for all a-c network configurations. 
     As an example in FIG. 3, the a-c network N1 is to remain unchanged and the a-c network N2 is to be developed in stages to become the network N2a and finally, in a further stage, the network N2b. 
     Each a-c network is connected according to FIG. 3, similar as in FIG. 1, permanently as a not switchable filter circuit FK1 and FK2 for the elimination of harmonics. 
     By the utility, network voltage limits permitted for each of the a-c networks N1, N2, N2a and N2b for active power P ref  and P n  in the two directions of power flow within voltage ranges A for steady-state operation and ranges B permitted only for dynamic processes according to FIG. 4 are set as well as the corresponding reactive powers Q N  according to Tables Ia to IVa. From these Tables Ia to IVa, maximum and minimum inductive and capacitive reactive-power demand of a-c networks are given. 
     In addition, the average values for inductive or capacitive reactive network power demand can be calculated from the sum of the highest permissible individual active power P ref , P n  of the range A, divided by the number of active-power steps per power flow direction and a-c network (noting the minus sign for inductive values) can be summed for the networks N2, N2a, N2b and divided by 3. The respective values for Q NL  max, Q NC  max ; Q NL  mean and Q NC  mean (for the a-c networks N2, N2a, N2b in common) are entered into the Tables Ia to IVa. 
     Into the Tables Ib, c to IVb, c, the largest and smallest values of the reactive converter power, Q V  for the different a-c networks and power flow directions in the d-c-tie under favorable and unfavorable a-c network conditions are entered which are calculated in the manner discussed below. 
     Under favorable a-c network conditions, the minimum values Q V  with the highest permissible network voltage for the respective active power P ref , the corresponding reactive a-c network demand Q N  and all inductive compensation elements L are calculated. In this case, the action of the reactors is aided by the respective a-c network. 
     The maximum values Q V  on the other hand are determined with the smallest permissible network voltage for the respective active power P ref  (P n ), the corresponding a-c network reactive-power demand Q N  and all capacitive compensation elements C. 
     Under unfavorable a-c network conditions, the inner upper limit of Q V  over of each a-c network with the maximally permissible network voltage is calculated for the corresponding active power P ref , the corresponding reactive power demands Q N  and all associated capacitive compensation elements. 
     The inner lower limit Q V  under of each a-c network is calculated with the minimally permissible respective network voltage (range A) for the respective active power P ref , the corresponding Q N  and all inductive compensation elements provided. 
     The reactive-power demand Q N  of the a-c network must be covered by the capacitors as well as the filter circuits FK to the extent that Q V  is within the operating range. 
     For determining the required minimum and maximum transformation ratio u of the converter transformers T1, T2 and for checking existing converter transformers for applicability (Table XII), use is made of the equations ##EQU9## as well as ##EQU10## with 
     
         U.sub.Nmin =U.sub.Nn ·0.85 and U.sub.Nmax =U.sub.Nn ·1.05 
    
     (where 0.85 is a value lower than the normal operation data). 
     The currents on the primary side of the converter transformers are obtained as ##EQU11## From this, the required reactive converter powers 
     
         Q.sub.Vmin =√3·U.sub.Nn ·S.sub.iVmin ·sin γ.sub.min                                           (14) 
    
     
         Q.sub.Vn =√3 U.sub.Nn ·S.sub.iVn ·sin α.sub.n                                             (15) 
    
     and, from the difference 
     
         Q.sub.Vn -Q.sub.Vmin =ΔQ.sub.V,                      (16) 
    
     the reactive power excursion ΔQ V  of the converter transformers can be determined. 
     The reactive power demand of the cooperating two a-c networks must be compensated to the extent that the reactive converter powers for both sides are within the operating range, and therefor their difference is smaller than ΔQ V . By appropriate combination of the taps of the two converter transformers, a mutual approximation of the reactive converter power can then be accomplished. 
     The transmission of lowest converter d-c currents I d  and thereby minimizing of the transmission losses of the d-c-tie is achieved if the reactive power of both converters UR1, UR2 are at the lower limit of the respective operating range. So the limits of the operating range are given. The compensation elements are therefor determined as 
     
         Q.sub.Vn =±1/2ΔQ.sub.v                            (17) 
    
     where Q Vn  =Q Vref , and form the desired operating range for the converter transformers used and the given voltage ranges A. 
     For converter transformers with: 
     
         ______________________________________a nominal short-circuit voltage                 u.sub.Kn = 0.17nominal converter d-c voltage                 U.sub.dn = 24.4 kVnominal control angle αn = 30°extinction angle      γmin = 18°one obtains  Q.sub.Vmin = 65 MVAr        Q.sub.Vn = 115 MVAr = Q.sub.Vref        ΔQ.sub.V = 50 MVAr______________________________________ 
    
     and therefrom the underlimit 90 MVar and the upper limit 740 Mvar of operating range is decided in accordance with Q Vn  ±1/2ΔQ V . 
     From a comparison of the values Q NLmax  and Q NCmax  as well as Q NLmean , Q NCmean  for the a-c network N1 from Table Ia and corresponding values for the a-c network N2b from Table IVa, with Q V , the following compensation units are available for the networks N1 and N2, N2a, N2b. 
     For the network N1: 
     
         Q.sub.NLmax +Q.sub.FK -Q.sub.Vref -Q.sub.indmax =69 (L) MVAr 
    
     
         Q.sub.NCmax -Q.sub.FK +Q.sub.Vref -Q.sub.capmax =69 (C) MVAr 
    
     
         Q.sub.NLmean +Q.sub.FK +Q.sub.Vref ↑Q.sub.indmean =42.6 (L) MVAr 
    
     
         Q.sub.NCmean -Q.sub.FK +Q.sub.Vref =Q.sub.capmean =37.6 (C) MVAr 
    
     For the a-c network N1, Q L  =70 MVAr and Q C  =35 MVAr are sufficient, where Q L  is subdivided into two units of 35 MVAr each so that it is smaller than ΔQ V  =50 MVAr. 
     A similar consideration for the a-c networks N2, N2a, N2b yields three capacitors with 35 MVAr each as well as a choke with 35 MVAr, i.e., for both a-c networks N1, N2 (N2a, N2b) applies: 
     C 1  =35 MVAr, L 11  and L 12  are 35 MVAr each; 
     C 21 , C 22 , C 23  are 35 MVAr and L 2  each likewise 35 MVAr. 
     From the quantities Q FK , Q N , Q L , Q C  result the values Q V  contained in Tables Ib, c to IVb, c for the corresponding highest and smallest voltages of the respective ranges A according to the equations: 
     
         Q.sub.V =Q.sub.FK +Q.sub.NL +Q.sub.C -Q.sub.L              (18) 
    
     and 
     
         Q.sub.V =Q.sub.FK -Q.sub.NC +Q.sub.C -Q.sub.L, respectively, (19) 
    
     and more specifically referred to the respective p.u. voltages. 
     The calculated values in Tables Ib, c to IVb, c are presented in Tables V to X in such a way that for the individual a-c networks N1, N2; N2a, N2b, common regions are obtained from which, according to Table XI, common valid operating ranges for all a-c networks are taken which show that a common operating range exists for the d-c-tie determined according to the invention with reactive power control while maintaining a sufficient control reserve in the event of a-c network disturbances, dynamic processes of the a-c network and fast power changes of the generators in the a-c networks (power ramps) is assured with the converter transformers used and the compensation elements selected. 
     The values contained in the above-mentioned Table XI show that the operating range 90 to 140 MVAr can be started up with the determined compensation elements. 
     In the foregoing specification, the invention has been described with reference to specific exemplary embodiments thereof. It will, however, be evident that various modifications and changes may be made thereunto without departing from the broader spirit and scope of the invention as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. 
     
                                           TABLE I__________________________________________________________________________Network N1RANGE A Powerflow direction N1 to N2                      Powerflow direction N2 to N1Voltage P (MW)             P (MW)in p.u. 200      150 100 50 20 0  20 50 100                                150                                   200__________________________________________________________________________1,05  Q.sub.N   B  59 C          49 C              42 C                 37 C  34 C                          33 C                             31 C                                29 C                                   B  (a)1,025 Q.sub.N   1 C        28 L                 31,5 L                       35 L                          36 L     39 L1,016 Q.sub.N  51 L              B  B     B  B  66 L1,01  Q.sub.N      53 L          B   B  B     B  B  B  79 L                                   79 L1,0   Q.sub.N   68 L      B   B   B  B     B  B  B  B  B__________________________________________________________________________   Q.sub.NLmax = 79 MVAr                     Q.sub.NLmean = 52,6 MVAr   Q.sub.NCmax = 59 MVAr                     Q.sub.NCmean = 27,6 MVAr__________________________________________________________________________1,05  Q.sub.V   B  95,4          105,4              112,4                 117,4 120,4                          121,4                             123,4                                125,4 (b)1,025 Q.sub.V   146,1      175,1                 178,6 182,1                          183,1    186,11,016 Q.sub.V  195,5              B  B     B  B  210,51,01  Q.sub.V      195,8          B   B  B     B  B  B  221,8                                   221,81,0   Q.sub.V   208,0      B   B   B  B     B  B  B  B  B1,05  Q.sub.V   B  -20,4          -10,4              -3,4                 1,6   4,6                          5,6                             7,6                                9,6                                   B  (c)1,025 Q.sub.V   35,8       64,8                 68,3  71,8                          72,8     75,81,016 Q.sub.V  87,13              B  B     B  B  102,11,01  Q.sub.V      88,7          B   B  B     B  B  B  114,7                                   114,71,0   Q.sub.V   103,0      B   B   B  B     B  B  B  B  B__________________________________________________________________________ 
    
     
                                           TABLE II__________________________________________________________________________Network N2RANGE A Powerflow direction N1 to N2                    Powerflow direction N2 to N1Voltage P (MW)           P (MW)in p.u. 200      150         100            50 20 0  20 50  100 150 200__________________________________________________________________________1,05  Q.sub.N   51 C      54 C         57 C            65 C               71 C  77 C                        87 C                            100 C                                118 C                                    B   (a)1,025 Q.sub.N   22 C      25 C         27 C            34 C               40 C  46 C                        55 C                            68 C                                85 C                                    103 C1,0   Q.sub.N   7 L      5 L         3 L            4 C               9 C   14 C                        24 C                            36 C                                54 C                                    71 C__________________________________________________________________________             Q.sub.NLmax = 7 MVAr             Q.sub.NCmax = 118 MVAr__________________________________________________________________________1,05  Q.sub.V   180,5      177,5         174,5            166,5               160,5 154,5                        144,5                            131,5                                113,5                                    B   (b)1,025 Q.sub.V   198,6      195,6         193,6            186,6               180,6 174,6                        165,6                            152,6                                135,6                                    117,61,0   Q.sub.V   217,0      215,0         213,0            206,0               201,0 196,0                        186,0                            174,0                                156,0                                    139,01,05  Q.sub.V   26,2      23,2         20,2            12,2               6,2   0,2                        -9,8                            -22,8                                -40,8                                    B   (c)1,025 Q.sub.V   51,5      48,5         46,5            39,5               33,5  27,5                        18,5                            5,5 -11,5                                    -29,51,0   Q.sub.V   77,0      75,0         73,0            66,0               61,0  56,0                        46,0                            34,0                                16,0                                    -1,0__________________________________________________________________________ 
    
     
                                           TABLE III__________________________________________________________________________Network N2aRANGE A Powerflow direction N1 to N2a                    Powerflow direction N2a to N1Voltage P (MW)           P (MW)in p.u. 200      150         100            50 20 0  20 50 100                              150                                 200__________________________________________________________________________1,05  Q.sub.N   9 C      12 C         17 C            24 C               28 C  33 C                        39 C                           50 C                              B  B  (a)1,025 Q.sub.N   36 L      33 L         28 L            22 L               18 L  12 L                        6 L                           5 C                              17 C                                 30 C1,0   Q.sub.N   80 L      77 L         73 L            68 L               B     B  51 L                           40 L                              28 L                                 15 L__________________________________________________________________________             Q.sub.NLmax = 80 MVAr             Q.sub.NCmax = 50 MVAr__________________________________________________________________________1,05  Q.sub.V   222,5      219,5         214,5            207,5               203,5 198,5                        192,5                           181,5                              B  B  (b)1,025 Q.sub.V   256,6      253,6         248,6            242,6               238,6 232,6                        226,6                           215,6                              203,6                                 190,61,0   Q.sub.V   290,0      287,0         283,0            278,0               B     B  261,0                           250,0                              238,0                                 225,01,05  Q.sub.V   68,2      65,2         60,2            53,2               49,2  44,2                        38,2                           27,2                              B  B  (c)1,025 Q.sub.V   109,5      106,5         101,5            95,5               91,5  85,5                        79,5                           68,5                              56,5                                 43,51,0   Q.sub.V   150,0      147,0         143,0            138,0               B     B  121,0                           110,0                              98,0                                 85,0__________________________________________________________________________ 
    
     
                                           TABLE IV__________________________________________________________________________Network N2bRANGE A Powerflow direction N1 to N2b                    Powerflow direction N2b to N1Voltage P (MW)           P (MW)in p.u. 200      150         100            50 20 0  20 50  100 150 200__________________________________________________________________________1,05  Q.sub.N   60 C      63 C         68 C            74 C               80 C  84 C                        92 C                            104 C                                B   B   (a)1,025 Q.sub.N   12 C      15 C         19 C            25 C               30 C  35 C                        42 C                            54 C                                67 C                                    81 C1,0   Q.sub.N   36 L      33 L         30 L            25 L               B     B  8 L 3 C 15 C                                    29 C__________________________________________________________________________ Q.sub.NLmax =  Q.sub.NLmean =  9,5 MVAr                          for all Networks N2,Q.sub.NCmax = 104 MVAr          Q.sub.NCmean = 60,5 MVAr                         N2a, N2b together__________________________________________________________________________1,05  Q.sub.V   171,5      168,5         163,5            157,5               151,5 147,5                        139,5                            127,5                                B   B   (b)1,025 Q.sub.V   208,6      205,6         201,6            195,6               190,6 185,6                        178,6                            166,6                                153,6                                    139,61,0   Q.sub.V   246,0      243,0         240,0            235,0               B     B  218,0                            207,0                                195,0                                    181,01,05  Q.sub.V   17,2      14,2         9,2            3,2               -2,8  -6,8                        -14,8                            -26,8                                B   B   (c)1,025 Q.sub.V   61,5      58,5         54,5            48,5               43,5  38,5                        31,5                            19,5                                6,5 -7,51,0   Q.sub.V   106,0      103,0         100,0            95,0               B     B  78,0                            67,0                                55,0                                    41,0__________________________________________________________________________ 
    
     
                       TABLE V______________________________________Powerflow direction N1 to N2            Common region under                  unfavorable                             favorable                  Network    Network    N1    N2      conditions conditions______________________________________200 MW Q.sub.Vmax          208,0   217,0 103,0-146,1                                 35,8-208,0P.sub.n  Q.sub.Vover          146,1   180,5  Q.sub.Vunder          103,0    77,0  Q.sub.Vmin           35,8    26,2150 MW Q.sub.Vmax          195,8   215,0 88,7-95,4                                 23,2-195,8P.sub.Ref  Q.sub.Vover           95,4   177,5  Q.sub.Vunder           88,7    75,0  Q.sub.Vmin          -20,4    23,2100 MW Q.sub.Vmax          195,5   213,0 87,1-105,4                                 20,2-195,5P.sub.Ref  Q.sub.Vover          105,4   174,5  Q.sub.Vunder           87,1    73,0  Q.sub.min          -10,4    20,2 50 MW Q.sub.Vmax          175,1   206,0 66,0-112,4                                 12,2-175,1P.sub.Ref  Q.sub.Vover          112,4   166,5  Q.sub.Vunder           64,8    66,0  Q.sub.Vmin          -3,4     12,2 20 MW Q.sub.Vmax          178,6   201,0 68,3-117,4                                  6,2-178,6P.sub.Ref  Q.sub.Vover          117,4   160,5  Q.sub.Vunder           68,3    61,0  Q.sub.Vmin           1,6     6,2______________________________________ 
    
     
                       TABLE VI______________________________________Powerflow direction N2 to N1            Common region under                  unfavorable                             favorable                  Network    Network    N1    N2      conditions conditions______________________________________200 MW Q.sub.Vmax          221,8   139,0 114,7-117,6                                 75,8-139,0P.sub.n  Q.sub.Vover          186,1   117,6  Q.sub.Vunder          114,7   -1,0  Q.sub.Vmin           75,8   -29,5150 MW Q.sub.Vmax          221,8   156,0 113,5-114,7                                 9,6-156,0P.sub.Ref  Q.sub.Vover          125,4   113,5  Q.sub.Vunder          114,7    16,0  Q.sub.Vmin           9,6    -40,8100 MW Q.sub.Vmax          210,5   174,0 102,1-123,4                                 7,6-174,0P.sub.Ref  Q.sub.Vover          123,4   131,5  Q.sub.Vunder          102,1    34,0  Q.sub.min           7,6    -22,8 50 MW Q.sub.Vmax          183,1   186,0  72,8-121,4                                 5,6-183,1P.sub.Ref  Q.sub.Vover          121,4   144,5  Q.sub.Vunder           72,8    46,0  Q.sub.Vmin           5,6    -9,8 20 MW Q.sub.Vmax          182,1   196,0  71,8-120,4                                 4,6-182,1P.sub.Ref  Q.sub.Vover          120,4   154,5  Q.sub.Vunder           71,8    56,0  Q.sub.Vmin           4,6     0,2______________________________________ 
    
     
                       TABLE VII______________________________________Powerflow direction N1 to N2a            Common region under                  unfavorable                             favorable                  Network    Network    N1    N2a     conditions conditions______________________________________200 MW Q.sub.Vmax          208,0   290,0 146,1-150,0                                 68,2-208,0P.sub.n  Q.sub.Vover          146,1   222,5  Q.sub.Vunder          103,0   150,0  Q.sub.Vmin           35,8    68,2150 MW Q.sub.Vmax          195,8   287,0  95,4-147,0                                 65,2-195,8P.sub.Ref  Q.sub.Vover           95,4   219,5  Q.sub.Vunder           88,7   147,0  Q.sub.Vmin          -20,4    65,2100 MW Q.sub.Vmax          195,5   283,0 105,4-143,0                                 60,2-195,5P.sub.Ref  Q.sub.Vover          105,4   214,7  Q.sub.Vunder           87,1   143,0  Q.sub.min          -10,4    60,2 50 MW Q.sub.Vmax          175,1   278,0 112,4-138,0                                 53,2-175,1P.sub.Ref  Q.sub.Vover          112,4   207,5  Q.sub.Vunder           64,8   138,0  Q.sub.Vmin          -3,4     53,2 20 MW Q.sub.Vmax          178,6   238,6  91,5-117,4                                 49,2-178,6P.sub.Ref  Q.sub.Vover          117,4   203,5  Q.sub.Vunder           68,3    91,5  Q.sub.Vmin           1,6     49,2______________________________________ 
    
     
                       TABLE VIII______________________________________Powerflow direction N2a to N1            Common Region under                  unfavorable                             favorable                  Network    Network    N1    N2a     conditions conditions______________________________________200 MW Q.sub.Vmax          221,8   225,0 114,7-186,1                                 75,8-221,8P.sub.n  Q.sub.Vover          186,1   190,6  Q.sub.Vunder          114,7    85,0  Q.sub.Vmin           75,8    43,5150 MW Q.sub.Vmax          221,8   238,0 114,7-125,4                                 56,5-221,8P.sub.Ref  Q.sub.Vover          125,4   203,6  Q.sub.Vunder          114,7    98,0  Q.sub.Vmin           9,6     56,5100 MW Q.sub.Vmax          210,5   250,0 110,0-123,4                                 27,2-210,5P.sub.Ref  Q.sub.Vover          123,4   181,5  Q.sub.Vunder          102,1   110,0  Q.sub.min           7,6     27,2 50 MW Q.sub.Vmax          183,1   261,0 121,0-121,4                                 38,2-183,1P.sub.Ref  Q.sub.Vover          121,4   192,5  Q.sub.Vunder           72,8   121,0  Q.sub.Vmin           5,6     38,2 20 MW Q.sub.Vmax          182,1   232,6  85,5-120,4                                 44,2-182,1P.sub.Ref  Q.sub.Vover          120,4   198,5  Q.sub.Vunder           71,8    85,5  Q.sub.Vmin           4,6     44,2______________________________________ 
    
     
                       TABLE IX______________________________________Powerflow direction N1 to N2b            Common Region under                  unfavorable                             favorable                  Network    Network    N1    N2b     conditions conditions______________________________________200 MW Q.sub.Vmax          208,0   246,0 106,0-146,1                                 35,8-208,0P.sub.n  Q.sub.Vover          146,1   171,5  Q.sub.Vunder          103,0   106,0  Q.sub.Vmin           35,8    17,2150 MW Q.sub.Vmax          195,8   243,0  95,4-103,0                                 14,2-195,8P.sub.Ref  Q.sub.Vover           95,4   168,5  Q.sub.Vunder           88,7   103,0  Q.sub.Vmin          -20,4    14,2100 MW Q.sub.Vmax          195,5   240,0 100,0-105,4                                 9,2-195,5P.sub.Ref  Q.sub.Vover          105,4   163,5  Q.sub.Vunder           87,1   100,0  Q.sub.min          -10,4    9,2 50 MW Q.sub.Vmax          175,1   235,0  95,0-112,4                                 3,2-175,1P.sub.Ref  Q.sub.Vover          112,4   157,5  Q.sub.Vunder           64,8    95,0  Q.sub.Vmin          -3,4     3,2 20 MW Q.sub.Vmax          178,6   190,6  68,3-117,4                                 1,6-178,6P.sub.Ref  Q.sub.Vover          117,4   151,5  Q.sub.Vunder           68,3    43,5  Q.sub.Vmin           1,6    -2,8______________________________________ 
    
     
                       TABLE X______________________________________Powerflow direction N2b to N1            Common Region under                  unfavorable                             favorable                  Network    Network    N1    N2b     conditions conditions______________________________________200 MW Q.sub.Vmax          221,8   181,0 114,7-139,6                                 75,8-181,0P.sub.n  Q.sub.Vover          186,1   139,6  Q.sub.Vunder          114,7    41,0  Q.sub.Vmin           75,8   -7,5150 MW Q.sub.Vmax          221,8   195,0 114,7-125,4                                 9,6-195,0P.sub.Ref  Q.sub.Vover          125,4   153,6  Q.sub.Vunder          114,7    55,0  Q.sub.Vmin           9,6     6,5100 MW Q.sub.Vmax          210,5   207,0 102,1-123,4                                 7,6-207,0P.sub.Ref  Q.sub.Vover          123,4   127,5  Q.sub.Vunder          102,1    67,0  Q.sub.Vmin           7,6    -26,8 50 MW Q.sub.Vmax          183,1   218,0  78,0-121,4                                 5,6-218,0P.sub.Ref  Q.sub.Vover          121,4   139,5  Q.sub.Vunder           72,8    78,0  Q.sub.Vmin           5,6    -14,8 20 MW Q.sub.Vmax          182,1   185,6  71,8-120,4                                 4,6-182,1P.sub.Ref  Q.sub.Vover          120,4   147,5  Q.sub.Vunder           71,8    38,5  Q.sub.Vmin           4,6    -6,8______________________________________ 
    
     
                       TABLE XI______________________________________Largest common operating region of the reactiveconverter power of all a-c-networks   under favorable                under unfavorable   Network conditions                Network conditions______________________________________P =  200 MW   75,8 to 139,0 MVAr                          103,0 to                                 139,0 MVAr150 MW   65,2    156,0 MVAr                           88,7  147,0 MVAr100 MW   60,2    174,0 MVAr                           87,1  143,0 MVAr 50 MW   53,2    175,1 MVAr                           66,0  138,0 MVAr 20 MW   49,2    178,6 MVAr                           68,3  120,4 MVAr______________________________________ 
    
     
                       TABLE XII______________________________________Data of the converter transformers of the d-c-tie                  Short         max. per-                  circuitVoltage limits for the         missible voltage transformationbus bar voltage         tap      U.sub.Kn                          ratio______________________________________smaller to   209,29 kV 35       0,1530                            0,110449                                   u.sub.max209,3   213,89    34       0,1550                            0,108696213,9   216,19    33       0,1560                            0,106997216,2   220,79    32       0,1580                            0,105351220,8   223,09    31       0,1590                            0,103755223,1   227,69    30       0,1600                            0,102206227,7   229,99    29       0,1610                            0,100703230,0   234,59    28       0,1620                            0,099244234,6   236,89    27       0,1630                            0,097826236,9   241,49    26       0,1650                            0,096448241,5   243,79    25       0,1660                            0,095109243,8   248,39    24       0,1670                            0,093806248,4   250,69    23       0,1690                            0,092538250,7   greater   22       0,1700                            0,091304                                   nominal value             21       0,1720                            0,090103             20       0,1730                            0,088933             19       0,1740                            0,087793             18       0,1720                            0,086681             17       0,1730                            0,085598             16       0,1740                            0,08541             15       0,1760                            0,083510             14       0,1770                            0,082504                                   u.sub.min             13       0,1780                            0,081522             12       0,1790                            0,080563             11       0,1800                            0,796626             10       0,1810                            0,078711             9        0,1830                            0,077816             8        0,1840                            0,076942             7        0,1850                            0,076087             6        0,1860                            0,075251             5        0,1870                            0,074433             4        0,1880                            0,073633             3        0,1900                            0,072849             2        0,1910                            0,072082             1        0,1920                            0,071332______________________________________