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Transient stability analysis of commercial scale open cycle disk MHD generator connected to power system
Energy Conversion and Management 44 (2003) 731–741 www.elsevier.com/locate/enconman
Transient stability analysis of commercial scale open cycle disk MHD generator connected to power system I. Inoue
a,*
, Y. Inui b, N. Hayanose c, M. Ishikawa
d
a
d
Department of Information and Electronics Engineering, Yatsushiro National College of Technology, 2627 Hirayama-Shinmachi, Yatsushiro 866–8501, Japan b Department of Electrical and Electronic Engineering, Toyohashi University of Technology, Tempaku-cho, Toyohashi 441–8580, Japan c Department of Electrical Engineering, Kobe City College of Technology, Gakuen-Higashimachi, Nishi-ku, Kobe 651–2194, Japan Institute of Engineering Mechanics and Systems, University of Tsukuba, Tennodai, Tsukuba 305–8573, Japan Received 15 October 2001; accepted 20 March 2002
Abstract The authors propose and investigate a new improvement method for transient stability under power line fault condition for commercial scale open cycle subsonic inflow disk MHD generators connected to the power system through time dependent numerical simulations of a total interconnecting system. The method needs no additional device and only manipulates the inverter control angle. Its control sequence has two steps. At first, the control angle is reduced toward 90° just after the fault occurs. Next, it is linearly increased to its rated value with constant increasing rate per cycle after the removed line is reconnected. It is made clear through the simulations that the proposed control scheme of the inverter is very effective to improve the transient stability of the generator. The reduction of the inverter control angle is effective to stabilize the MHD generator, but the operating state shifts from its normal operation. The increase of the control angle can stably return the generator to its normal operation. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Commercial scale open cycle MHD; Subsonic disk generator; Power system; Transient stability; Inverter control
*
Corresponding author. Tel.: +81-965-53-1313; fax: +81-965-53-1319. E-mail address: [email protected] (I. Inoue).
0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 0 8 4 - 5
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1. Introduction To connect MHD generators to commercial power systems, stable operation of the total interconnecting system between the generator and the power system through the power inversion system should be guaranteed. Since large scale open cycle MHD generators for commercial use tend to be unstable, many stability analyses of the generators were performed [1–3], and it has already been made clear that subsonic inflow disk generators with multi-loads are stable under usual operating conditions [4–6]. Power line faults, however, give remarkable fluctuations to the generator and make it fairly unstable. After the fault is removed, therefore, the generator usually does not return to its stable steady state before the fault [5,6]. This means that some control scheme that can maintain transient stability of the generator against power line faults is required. The authors, therefore, have been investigating improvement methods for transient stability under power line fault condition for commercial scale subsonic open cycle disk MHD generators connected to the power system and have proposed to utilize the system dumping resistor (SDR) circuit in a previous paper [7]. This circuit is very effective to maintain transient stability because it suppresses the fluctuations induced by power line faults and assists the generator in returning to the stable steady state after the fault is removed. This method, however, needs to install an additional device of the SDR circuit. Considering the above background, in this paper, the authors propose and investigate a new improvement method for transient stability under power line fault condition for the generator through time dependent numerical simulations of a total interconnecting system between the generator and the power system. The method needs no additional device and uses only the inverter control angle as a manipulated variable to return the generator to the stable steady state.
2. Simulation model Fig. 1 illustrates the schematic diagram of the simulation model of an interconnecting system between the MHD generator and the power system used in the present study. The model is assumed to consist of a commercial scale open cycle subsonic inflow disk MHD generator with two pairs of power output terminals, d.c. reactors, cascade connected two line commutated inverter bridges, two inverter transformers, filters, a shunt capacitor connected through a transformer,
Fig. 1. Schematic diagram of simulated interconnecting system between open cycle inflow disk MHD generator and power system.
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double circuit transmission lines and an infinite bus. In this section, a brief explanation is given for each component of the model system. 2.1. MHD generator The authors have already succeeded in designing a commercial scale open cycle MHD generator that is stable under the usual operating conditions [5,6], and therefore, the designed generator is adopted in the present simulation model. It is a subsonic inflow disk type with two pairs of power output terminals using coal fired combustion gas with 1% (weight) potassium seed. Fig. 2 shows the schematic diagram of the generator. The radii of the anode, first and second cathode are 6, 4 and 2 m, respectively, and the two loads of the upstream and downstream sides are connected between the anode and the first cathode and between the first and the second cathode, respectively. The thermal input is 1300 MW, and the electrical output is 234.7 MW, which is the sum total of 150.7 MW on the upstream side and 84.0 MW on the downstream side. Table 1 lists the size and the rated conditions of the generator. 2.2. Power system To simplify the simulation system and make the calculations easy, the power system is treated as double circuit transmission lines and a three phase infinite bus connected at the end of the lines. The rated effective line voltage and frequency are 275 kV and 60 Hz, respectively, and the total inductance of the transmission lines is selected as 0:01 þ j0:1 pu on the basic capacity of 250 MVA. 2.3. Inversion system Some inversion system is needed to supply the d.c. output of the MHD generator to the a.c. power system, and therefore, cascade connected two line commutated inverter bridges are equipped in the system. Each bridge is connected to the generator through a d.c. reactor and inverts the output current of the generator of the upstream or downstream side. The inductance of the upstream side reactor is selected as 250 mH and that of the downstream side 100 mH. An inverter transformer is connected with each bridge. The rated capacity, voltage and impedance of
Fig. 2. Schematic diagram of inflow disk MHD generator with two pairs of power output terminals.
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Table 1 Size and rated conditions of commercial scale open cycle subsonic inflow disk MHD generator used in the study Thermal input Magnetic flux density Inlet radius Inlet stagnation temperature Inlet static temperature Inlet stagnation pressure Inlet static pressure Inlet Mach number Inlet swirl ratio Upstream side current Upstream side voltage Upstream side output Total electrical output Channel length Radial gas velocity Outlet radius Outlet stagnation temperature Outlet static temperature Outlet stagnation pressure Outlet static pressure Outlet Mach number Outlet swirl ratio Downstream side current Downstream side voltage Downstream side output Enthalpy extraction ratio
1300 MW 8.5 T 6m 2700 K 2550 K 4.2 atm 2.64 atm 0.9 2.0 9850 A 15.3 kV 150.7 MW 234.7 MW 4m 370 m/s 2m 2259 K 2195 K 1.0 atm 0.84 atm 0.54 0.77 7000 A 12.0 kV 84.0 MW 18.1%
the upstream side transformer are 200 MVA, 13/275 kV and 0:007 þ j0:15 pu, respectively, and those of the downstream side are 110 MVA, 10/275 kV and 0:007 þ j0:15 pu, respectively. The rated control angle of the inverter is selected as 140.3° because the total output power of the generator coincides with its designed value under this condition. The line commutated inverter generates a large amount of harmonics and reactive power. The fifth, seventh, ninth and eleventh single tuned type (the quality factor Q ¼ 50) and the high pass type (24th, Q ¼ 3) filters are equipped to absorb the harmonics, and each capacity is selected as 15, 10, 15, 10 and 25 MVA, respectively. The shunt capacitor with capacity of 50 MVA is also equipped to absorb the reactive power, and it is connected through a transformer with rated capacity, voltage and impedance of 50 MVA, 275/77 kV and 0:007 þ j0:1 pu, respectively.
3. Simulation method 3.1. MHD generator The calculations of the time dependent behavior of the disk MHD generator are performed by using the quasi-one dimensional time dependent simulation code developed by the authors.
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In the code, the cylindrical coordinate system shown in Fig. 2 is used, and the following quasione dimensional time dependent equations of the plasma flow in the generator are taken into account as the basic equations of the calculations: Mass conservation: o o ðqAÞ þ ðqur AÞ ¼ 0 ot or
ð1Þ
Momentum conservation: q
our our u2 op ¼ Jh B fr þ qur q hþ or ot or r
ð2Þ
q
ouh ouh ur uh þ qur þq ¼ Jr B fh ot or r
ð3Þ
Energy conservation: o u2r þ u2h o u2r þ u2h op hþ hþ þ qur ¼ þ Jr Er q q ot or ot 2 2
ð4Þ
Generalized Ohm’s law: Jr ¼
r ðEr þ uh B þ bur BÞ 1 þ b2
ð5Þ
Jh ¼
r ðbEr þ buh B ur BÞ 1 þ b2
ð6Þ
Current continuity: o ðJr AÞ ¼ 0 or
ð7Þ
In the above equations, q is the mass density, ur and uh the r and h components of the gas velocity, respectively, p the gas pressure, T the gas temperature, h the enthalpy, Jr and Jh the r and h components of the current density, respectively, Er the r component of the electric field, B the magnetic flux density, fr and fh the r and h components of the friction loss, respectively, q the heat loss, r the conductivity, b the Hall parameter and A the generator cross-section. The basic equations of the gas dynamical quantities, Eqs. (1)–(4), are solved by adopting the MacCormack predictor–corrector method [8]. In addition to the generation channel, the nozzle and the diffuser (0.2 and 0.4 m long, respectively) are also included in the analytical region, and the gas dynamical and electrical quantity distributions in the extent between the nozzle inlet (the radius r ¼ 6:2 m) and the diffuser outlet (r ¼ 1:6 m) are calculated. Concerning the boundary conditions of the gas dynamical quantities, the stagnation temperature and stagnation pressure are fixed at the inlet boundary, and the stagnation pressure is fixed at the outlet boundary because the gas flow is subsonic and these conditions are most suitable for this flow.
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3.2. Power circuit Time dependent analyses of power circuits are usually performed on their simplified equivalent single phase circuits because the transient response time of the usual power machinery and apparatus is much longer than one cycle of a.c. The transient response time of the MHD generator is, however, very short, and the influence of the detailed switching operation of the inverter upon the time dependent behavior of the generator cannot be neglected. Considering the above facts, the simulations are performed on the detailed three phase circuit of the model, not on the simplified equivalent single phase circuit. In the calculations, the thyristors in the inverter bridges are simulated as ideal switches, which are turned on by the gate signal under the forward voltage condition and are turned off at the timing of zero current, and the circuit breakers, equipped at both ends of the transmission lines, are also simulated as ideal switches, which break the circuit at the timing of the first zero current condition after their operation of current interruption. The node voltages are selected as unknown quantities and the Bergeron method, which is the same scheme as EMTP [9], is adopted as the analytical scheme.
4. Investigation on improvement method of transient stability 4.1. Stabilization of MHD generator To propose and investigate a new improvement method for transient stability, which only manipulates the inverter control angle, under power line fault condition for the MHD generator, numerical simulations are performed in the case of a three phase ground fault occurring at the center of one of the double circuit transmission lines. The stable steady state of the nominal operating condition, using the rated inverter control angle of 140.3°, is selected as the initial condition. In the calculation sequence, the fault is assumed to continue for 5 cycles (83.3 ms) and be cleared by disconnecting the faulted line through operation of the circuit breakers located at both ends of the line. After that, the operation continues using only another transmission line for 18 cycles (300 ms), and finally the removed line is reconnected and the circuit returns to the normal connection because the usual grounding spontaneously disappears by this time. At first, a numerical simulation is performed in the case when the inverter control angle is fixed constant at its rated angle of 140.3° regardless of the occurrence of the fault. Fig. 3 depicts the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the a.c. voltages at the sending end of the transmission line, the output currents of the upstream side inverter and the active power provided to the power system. It is made clear from this figure that the commutation failure, induced by the drop of the a.c. voltage during the ground fault period, does not disappear even after the fault is cleared and the a.c. voltage recovers to its normal level. This continuous commutation failure makes the MHD generator unstable, resulting in continuous acceleration of the gas velocity in the generator and irregular fluctuation of the active power. This result tells that in the case of no control, power line faults make the generator unstable owing to the continuous commutation failure, indicating that some control aimed at restoration
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Fig. 3. Time variations of (a) radial gas velocity distribution in MHD generator, (b) a.c. voltages at sending end of transmission line, (c) output currents of upstream side inverter and (d) active power provided to power system when inverter control angle is fixed constant at its rated angle of 140.3°.
of normal commutation is needed to stabilize the generator. The power line fault increases the output currents of the generator, leading to lack of the margin angle of the inverter, and this is supposed to be the cause of this continuous commutation failure. The authors, therefore, propose to reduce the inverter control angle toward 90° after the fault occurs because it is most effective to guarantee the margin angle. To confirm the validity of this proposed control method, numerical simulations are performed in the cases when the control angle is reduced to various levels between 90° and 140.3° just after the fault occurs, and it is successfully found that stable operation of the generator is realized after the fault is cleared in the cases when the control angle is reduced to 110° and under. As an example, Fig. 4 shows the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the output currents of the upstream side inverter, the output currents of the MHD generator and the active power provided to the power system in the case when the angle is reduced to 110°. It can be confirmed from this figure that the commutation failure induced by the fault disappears after the fault is cleared, and the generator converges to a new stable steady state corresponding to the new control angle of 110°. The above results indicate that the proposed method, reduction of the inverter control angle just after the fault occurs, is effective in stabilization of the MHD generator, though there exists a possibility that the highest critical control angle changes, depending on the sort and duration time of the fault.
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Fig. 4. Time variations of (a) radial gas velocity distribution in MHD generator, (b) output currents of upstream side inverter, (c) output currents of MHD generator and (d) active power provided to power system in case where inverter control angle is reduced to 110° just after power line fault occurs.
4.2. Restoration to normal operation The MHD generator is successfully stabilized, whereas its operating state is quite different from its normal operation because it converges to a different steady state corresponding to the new control angle. The gas flow in the generator is in an accelerated state and the output current of the generator upstream side reaches about twice as large as its normal value. The inverter control angle, therefore, should be returned to its rated value. Considering this fact, the authors propose the following new control scheme to improve the transient stability under power line fault condition for the generator: (Step 1) Reduce the control angle toward 90° just after the fault occurs. (Step 2) Increase the control angle linearly to its rated value with constant increasing rate per cycle after the removed line is reconnected. To reveal the behavior of the generator when the above proposed control scheme is implemented, numerical simulations are performed in the cases where the control angle is at first reduced to 110° just after the fault occurs, and next, it is returned to its rated value of 140.3° at various increasing rates of the control angle after the faulted line is reconnected. It is made clear from the simulations that this control is very effective and can stably return the generator to its normal operation, though too high an increasing rate makes the generator unstable again, and there exists an upper critical rate. The critical rate is about 1.1°/cycle in this case. As examples, Figs. 5 and 6 illustrate the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the inverter margin angles, the output currents of the
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Fig. 5. Time variations of (a) radial gas velocity distribution in MHD generator, (b) inverter margin angles, (c) output currents of MHD generator and (d) active power provided to power system in case where reduction level of inverter control angle is 110° and its increasing rate is 1.0°/cycle.
MHD generator and the active power provided to the power system in the cases when the increasing rate is 1.0° and 1.2°/cycle, respectively. It can be confirmed from Fig. 5 that the generator, which at once fluctuates by the fault, stably converges to its normal operation within 2 s when the increasing rate is 1.0°/cycle. As shown in Fig. 6, the margin angle of the upstream side inverter reaches zero, and the generator becomes unstable again when the increasing rate exceeds its upper critical value. The increase of the control angle requires a decrease of the d.c. current to maintain normal commutation, and this response is realized by the drooping voltage–current characteristic of the generator. This response, however, needs a transient time lag, which is determined by the time dependent characteristics of the generator and the d.c. reactors, and commutation failure occurs in the case where the required decreasing rate of the d.c. current exceeds its highest possible rate. This is considered to be the cause of the unsuccessful result when the increasing rate of the control angle is too high. Next, to make clear the influence of the control parameters, the reduction level and the increasing rate of the inverter control angle, numerical simulations are performed for various sets of them. Fig. 7 summarizes the numerically obtained time variations of the output current of the generator upstream side. This figure tells that selecting a high increasing rate makes the control speedy. The upper limit of the increasing rate is always about 1.1°/cycle, regardless of the reduction level. This result is quite reasonable because the characteristic time of the transient response is fixed by the system design and is independent of the inverter control angle. The upper
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Fig. 6. Time variations of (a) radial gas velocity distribution in MHD generator, (b) inverter margin angles, (c) output currents of MHD generator and (d) active power provided to power system in case where reduction level of inverter control angle is 110° and its increasing rate is 1.2°/cycle.
Fig. 7. Time variations of upstream side current of MHD generator in cases where reduction level of inverter control angle is (a) 110°, (b) 105°, (c) 100°, (d) 95° and its increasing rate is 1.2°, 1.0°, 0.75°, 0.5°/cycle.
limit of the increasing rate is, therefore, considered to be peculiar to the system. It is also made clear that using a lower reduction level of near 90°, to increase the margin angle of the inverter
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and to guarantee the stabilization of the generator, is quite all right because the influence of the reduction level on the transient response time is relatively small. These results indicate that the proposed control scheme of the inverter can stably return the generator to its normal operation and, therefore, is very effective to improve the transient stability of the generator.
5. Conclusions In this paper, the authors propose and investigate a new improvement method for transient stability under power line fault condition for commercial scale open cycle subsonic inflow disk MHD generators connected to power system through time dependent numerical simulations of a total interconnecting system. The simulation model is assumed to consist of the generator with two pairs of power output terminals, d.c. reactors, cascade connected two line commutated inverter bridges, two inverter transformers, filters, a shunt capacitor connected through a transformer, double circuit transmission lines and an infinite bus. The method needs no additional device and only manipulates the inverter control angle. Its control sequence has two steps. At first, the control angle is reduced toward 90° just after the fault occurs. Next, it is linearly increased to its rated value with constant increasing rate per unit cycle after the removed line is reconnected. It is made clear through the simulations under power line fault condition that the proposed control scheme of the inverter is very effective to improve the transient stability of the generator. The reduction of the inverter control angle is effective to stabilize the MHD generator, but the operating state shifts from its normal operation. The increase of the control angle can stably return the generator to its normal operation when its increasing rate is lower than its upper critical value, and the critical value is peculiar to the system.
References [1] Marty SM, Simpson SW, Messerle HK. Proceedings of the 28th Symposium on Engineering Aspects of MHD, 1990. p. IX.5-1. [2] Ishikawa M, Kyogoku A, Umoto J. Trans IEE Jpn 1995;115-B:68 (in Japanese). [3] Matsuo T, Ishikawa M, Umoto J. Trans IEE Jpn 1995;115-B:847 (in Japanese). [4] Matsuo T, Ishikawa M, Umoto J. Trans IEE Jpn 1996;116-B:360 (in Japanese). [5] Hayanose N, Tan A, Matsuo T, Inui Y, Ishikawa M, Umoto J. Proceedings of the 12th International Conference on MHD Electrical Power Generation, vol. 2, 1996. p. 843. [6] Inoue I, Hayanose N, Inui Y, Ishikawa M. Procedings of the 1998 International Symposium on Advanced Energy Technology, 1998. p. 521. [7] Hayanose N, Inui Y, Ishikawa M. Energy Convers Mgmt 2001;42:1191. [8] MacCormack RW. AIAA Paper, AIAA–69–354, 1969. [9] Dommel HW. IEEE Trans Pwr App Syst 1969;PAS-88:388.
Transient stability analysis of commercial scale open cycle disk MHD generator connected to power system I. Inoue
a,*
, Y. Inui b, N. Hayanose c, M. Ishikawa
d
a
d
Department of Information and Electronics Engineering, Yatsushiro National College of Technology, 2627 Hirayama-Shinmachi, Yatsushiro 866–8501, Japan b Department of Electrical and Electronic Engineering, Toyohashi University of Technology, Tempaku-cho, Toyohashi 441–8580, Japan c Department of Electrical Engineering, Kobe City College of Technology, Gakuen-Higashimachi, Nishi-ku, Kobe 651–2194, Japan Institute of Engineering Mechanics and Systems, University of Tsukuba, Tennodai, Tsukuba 305–8573, Japan Received 15 October 2001; accepted 20 March 2002
Abstract The authors propose and investigate a new improvement method for transient stability under power line fault condition for commercial scale open cycle subsonic inflow disk MHD generators connected to the power system through time dependent numerical simulations of a total interconnecting system. The method needs no additional device and only manipulates the inverter control angle. Its control sequence has two steps. At first, the control angle is reduced toward 90° just after the fault occurs. Next, it is linearly increased to its rated value with constant increasing rate per cycle after the removed line is reconnected. It is made clear through the simulations that the proposed control scheme of the inverter is very effective to improve the transient stability of the generator. The reduction of the inverter control angle is effective to stabilize the MHD generator, but the operating state shifts from its normal operation. The increase of the control angle can stably return the generator to its normal operation. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Commercial scale open cycle MHD; Subsonic disk generator; Power system; Transient stability; Inverter control
*
Corresponding author. Tel.: +81-965-53-1313; fax: +81-965-53-1319. E-mail address: [email protected] (I. Inoue).
0196-8904/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 2 ) 0 0 0 8 4 - 5
732
I. Inoue et al. / Energy Conversion and Management 44 (2003) 731–741
1. Introduction To connect MHD generators to commercial power systems, stable operation of the total interconnecting system between the generator and the power system through the power inversion system should be guaranteed. Since large scale open cycle MHD generators for commercial use tend to be unstable, many stability analyses of the generators were performed [1–3], and it has already been made clear that subsonic inflow disk generators with multi-loads are stable under usual operating conditions [4–6]. Power line faults, however, give remarkable fluctuations to the generator and make it fairly unstable. After the fault is removed, therefore, the generator usually does not return to its stable steady state before the fault [5,6]. This means that some control scheme that can maintain transient stability of the generator against power line faults is required. The authors, therefore, have been investigating improvement methods for transient stability under power line fault condition for commercial scale subsonic open cycle disk MHD generators connected to the power system and have proposed to utilize the system dumping resistor (SDR) circuit in a previous paper [7]. This circuit is very effective to maintain transient stability because it suppresses the fluctuations induced by power line faults and assists the generator in returning to the stable steady state after the fault is removed. This method, however, needs to install an additional device of the SDR circuit. Considering the above background, in this paper, the authors propose and investigate a new improvement method for transient stability under power line fault condition for the generator through time dependent numerical simulations of a total interconnecting system between the generator and the power system. The method needs no additional device and uses only the inverter control angle as a manipulated variable to return the generator to the stable steady state.
2. Simulation model Fig. 1 illustrates the schematic diagram of the simulation model of an interconnecting system between the MHD generator and the power system used in the present study. The model is assumed to consist of a commercial scale open cycle subsonic inflow disk MHD generator with two pairs of power output terminals, d.c. reactors, cascade connected two line commutated inverter bridges, two inverter transformers, filters, a shunt capacitor connected through a transformer,
Fig. 1. Schematic diagram of simulated interconnecting system between open cycle inflow disk MHD generator and power system.
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double circuit transmission lines and an infinite bus. In this section, a brief explanation is given for each component of the model system. 2.1. MHD generator The authors have already succeeded in designing a commercial scale open cycle MHD generator that is stable under the usual operating conditions [5,6], and therefore, the designed generator is adopted in the present simulation model. It is a subsonic inflow disk type with two pairs of power output terminals using coal fired combustion gas with 1% (weight) potassium seed. Fig. 2 shows the schematic diagram of the generator. The radii of the anode, first and second cathode are 6, 4 and 2 m, respectively, and the two loads of the upstream and downstream sides are connected between the anode and the first cathode and between the first and the second cathode, respectively. The thermal input is 1300 MW, and the electrical output is 234.7 MW, which is the sum total of 150.7 MW on the upstream side and 84.0 MW on the downstream side. Table 1 lists the size and the rated conditions of the generator. 2.2. Power system To simplify the simulation system and make the calculations easy, the power system is treated as double circuit transmission lines and a three phase infinite bus connected at the end of the lines. The rated effective line voltage and frequency are 275 kV and 60 Hz, respectively, and the total inductance of the transmission lines is selected as 0:01 þ j0:1 pu on the basic capacity of 250 MVA. 2.3. Inversion system Some inversion system is needed to supply the d.c. output of the MHD generator to the a.c. power system, and therefore, cascade connected two line commutated inverter bridges are equipped in the system. Each bridge is connected to the generator through a d.c. reactor and inverts the output current of the generator of the upstream or downstream side. The inductance of the upstream side reactor is selected as 250 mH and that of the downstream side 100 mH. An inverter transformer is connected with each bridge. The rated capacity, voltage and impedance of
Fig. 2. Schematic diagram of inflow disk MHD generator with two pairs of power output terminals.
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Table 1 Size and rated conditions of commercial scale open cycle subsonic inflow disk MHD generator used in the study Thermal input Magnetic flux density Inlet radius Inlet stagnation temperature Inlet static temperature Inlet stagnation pressure Inlet static pressure Inlet Mach number Inlet swirl ratio Upstream side current Upstream side voltage Upstream side output Total electrical output Channel length Radial gas velocity Outlet radius Outlet stagnation temperature Outlet static temperature Outlet stagnation pressure Outlet static pressure Outlet Mach number Outlet swirl ratio Downstream side current Downstream side voltage Downstream side output Enthalpy extraction ratio
1300 MW 8.5 T 6m 2700 K 2550 K 4.2 atm 2.64 atm 0.9 2.0 9850 A 15.3 kV 150.7 MW 234.7 MW 4m 370 m/s 2m 2259 K 2195 K 1.0 atm 0.84 atm 0.54 0.77 7000 A 12.0 kV 84.0 MW 18.1%
the upstream side transformer are 200 MVA, 13/275 kV and 0:007 þ j0:15 pu, respectively, and those of the downstream side are 110 MVA, 10/275 kV and 0:007 þ j0:15 pu, respectively. The rated control angle of the inverter is selected as 140.3° because the total output power of the generator coincides with its designed value under this condition. The line commutated inverter generates a large amount of harmonics and reactive power. The fifth, seventh, ninth and eleventh single tuned type (the quality factor Q ¼ 50) and the high pass type (24th, Q ¼ 3) filters are equipped to absorb the harmonics, and each capacity is selected as 15, 10, 15, 10 and 25 MVA, respectively. The shunt capacitor with capacity of 50 MVA is also equipped to absorb the reactive power, and it is connected through a transformer with rated capacity, voltage and impedance of 50 MVA, 275/77 kV and 0:007 þ j0:1 pu, respectively.
3. Simulation method 3.1. MHD generator The calculations of the time dependent behavior of the disk MHD generator are performed by using the quasi-one dimensional time dependent simulation code developed by the authors.
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In the code, the cylindrical coordinate system shown in Fig. 2 is used, and the following quasione dimensional time dependent equations of the plasma flow in the generator are taken into account as the basic equations of the calculations: Mass conservation: o o ðqAÞ þ ðqur AÞ ¼ 0 ot or
ð1Þ
Momentum conservation: q
our our u2 op ¼ Jh B fr þ qur q hþ or ot or r
ð2Þ
q
ouh ouh ur uh þ qur þq ¼ Jr B fh ot or r
ð3Þ
Energy conservation: o u2r þ u2h o u2r þ u2h op hþ hþ þ qur ¼ þ Jr Er q q ot or ot 2 2
ð4Þ
Generalized Ohm’s law: Jr ¼
r ðEr þ uh B þ bur BÞ 1 þ b2
ð5Þ
Jh ¼
r ðbEr þ buh B ur BÞ 1 þ b2
ð6Þ
Current continuity: o ðJr AÞ ¼ 0 or
ð7Þ
In the above equations, q is the mass density, ur and uh the r and h components of the gas velocity, respectively, p the gas pressure, T the gas temperature, h the enthalpy, Jr and Jh the r and h components of the current density, respectively, Er the r component of the electric field, B the magnetic flux density, fr and fh the r and h components of the friction loss, respectively, q the heat loss, r the conductivity, b the Hall parameter and A the generator cross-section. The basic equations of the gas dynamical quantities, Eqs. (1)–(4), are solved by adopting the MacCormack predictor–corrector method [8]. In addition to the generation channel, the nozzle and the diffuser (0.2 and 0.4 m long, respectively) are also included in the analytical region, and the gas dynamical and electrical quantity distributions in the extent between the nozzle inlet (the radius r ¼ 6:2 m) and the diffuser outlet (r ¼ 1:6 m) are calculated. Concerning the boundary conditions of the gas dynamical quantities, the stagnation temperature and stagnation pressure are fixed at the inlet boundary, and the stagnation pressure is fixed at the outlet boundary because the gas flow is subsonic and these conditions are most suitable for this flow.
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3.2. Power circuit Time dependent analyses of power circuits are usually performed on their simplified equivalent single phase circuits because the transient response time of the usual power machinery and apparatus is much longer than one cycle of a.c. The transient response time of the MHD generator is, however, very short, and the influence of the detailed switching operation of the inverter upon the time dependent behavior of the generator cannot be neglected. Considering the above facts, the simulations are performed on the detailed three phase circuit of the model, not on the simplified equivalent single phase circuit. In the calculations, the thyristors in the inverter bridges are simulated as ideal switches, which are turned on by the gate signal under the forward voltage condition and are turned off at the timing of zero current, and the circuit breakers, equipped at both ends of the transmission lines, are also simulated as ideal switches, which break the circuit at the timing of the first zero current condition after their operation of current interruption. The node voltages are selected as unknown quantities and the Bergeron method, which is the same scheme as EMTP [9], is adopted as the analytical scheme.
4. Investigation on improvement method of transient stability 4.1. Stabilization of MHD generator To propose and investigate a new improvement method for transient stability, which only manipulates the inverter control angle, under power line fault condition for the MHD generator, numerical simulations are performed in the case of a three phase ground fault occurring at the center of one of the double circuit transmission lines. The stable steady state of the nominal operating condition, using the rated inverter control angle of 140.3°, is selected as the initial condition. In the calculation sequence, the fault is assumed to continue for 5 cycles (83.3 ms) and be cleared by disconnecting the faulted line through operation of the circuit breakers located at both ends of the line. After that, the operation continues using only another transmission line for 18 cycles (300 ms), and finally the removed line is reconnected and the circuit returns to the normal connection because the usual grounding spontaneously disappears by this time. At first, a numerical simulation is performed in the case when the inverter control angle is fixed constant at its rated angle of 140.3° regardless of the occurrence of the fault. Fig. 3 depicts the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the a.c. voltages at the sending end of the transmission line, the output currents of the upstream side inverter and the active power provided to the power system. It is made clear from this figure that the commutation failure, induced by the drop of the a.c. voltage during the ground fault period, does not disappear even after the fault is cleared and the a.c. voltage recovers to its normal level. This continuous commutation failure makes the MHD generator unstable, resulting in continuous acceleration of the gas velocity in the generator and irregular fluctuation of the active power. This result tells that in the case of no control, power line faults make the generator unstable owing to the continuous commutation failure, indicating that some control aimed at restoration
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Fig. 3. Time variations of (a) radial gas velocity distribution in MHD generator, (b) a.c. voltages at sending end of transmission line, (c) output currents of upstream side inverter and (d) active power provided to power system when inverter control angle is fixed constant at its rated angle of 140.3°.
of normal commutation is needed to stabilize the generator. The power line fault increases the output currents of the generator, leading to lack of the margin angle of the inverter, and this is supposed to be the cause of this continuous commutation failure. The authors, therefore, propose to reduce the inverter control angle toward 90° after the fault occurs because it is most effective to guarantee the margin angle. To confirm the validity of this proposed control method, numerical simulations are performed in the cases when the control angle is reduced to various levels between 90° and 140.3° just after the fault occurs, and it is successfully found that stable operation of the generator is realized after the fault is cleared in the cases when the control angle is reduced to 110° and under. As an example, Fig. 4 shows the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the output currents of the upstream side inverter, the output currents of the MHD generator and the active power provided to the power system in the case when the angle is reduced to 110°. It can be confirmed from this figure that the commutation failure induced by the fault disappears after the fault is cleared, and the generator converges to a new stable steady state corresponding to the new control angle of 110°. The above results indicate that the proposed method, reduction of the inverter control angle just after the fault occurs, is effective in stabilization of the MHD generator, though there exists a possibility that the highest critical control angle changes, depending on the sort and duration time of the fault.
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Fig. 4. Time variations of (a) radial gas velocity distribution in MHD generator, (b) output currents of upstream side inverter, (c) output currents of MHD generator and (d) active power provided to power system in case where inverter control angle is reduced to 110° just after power line fault occurs.
4.2. Restoration to normal operation The MHD generator is successfully stabilized, whereas its operating state is quite different from its normal operation because it converges to a different steady state corresponding to the new control angle. The gas flow in the generator is in an accelerated state and the output current of the generator upstream side reaches about twice as large as its normal value. The inverter control angle, therefore, should be returned to its rated value. Considering this fact, the authors propose the following new control scheme to improve the transient stability under power line fault condition for the generator: (Step 1) Reduce the control angle toward 90° just after the fault occurs. (Step 2) Increase the control angle linearly to its rated value with constant increasing rate per cycle after the removed line is reconnected. To reveal the behavior of the generator when the above proposed control scheme is implemented, numerical simulations are performed in the cases where the control angle is at first reduced to 110° just after the fault occurs, and next, it is returned to its rated value of 140.3° at various increasing rates of the control angle after the faulted line is reconnected. It is made clear from the simulations that this control is very effective and can stably return the generator to its normal operation, though too high an increasing rate makes the generator unstable again, and there exists an upper critical rate. The critical rate is about 1.1°/cycle in this case. As examples, Figs. 5 and 6 illustrate the numerically obtained time variations of the radial gas velocity distribution in the MHD generator, the inverter margin angles, the output currents of the
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Fig. 5. Time variations of (a) radial gas velocity distribution in MHD generator, (b) inverter margin angles, (c) output currents of MHD generator and (d) active power provided to power system in case where reduction level of inverter control angle is 110° and its increasing rate is 1.0°/cycle.
MHD generator and the active power provided to the power system in the cases when the increasing rate is 1.0° and 1.2°/cycle, respectively. It can be confirmed from Fig. 5 that the generator, which at once fluctuates by the fault, stably converges to its normal operation within 2 s when the increasing rate is 1.0°/cycle. As shown in Fig. 6, the margin angle of the upstream side inverter reaches zero, and the generator becomes unstable again when the increasing rate exceeds its upper critical value. The increase of the control angle requires a decrease of the d.c. current to maintain normal commutation, and this response is realized by the drooping voltage–current characteristic of the generator. This response, however, needs a transient time lag, which is determined by the time dependent characteristics of the generator and the d.c. reactors, and commutation failure occurs in the case where the required decreasing rate of the d.c. current exceeds its highest possible rate. This is considered to be the cause of the unsuccessful result when the increasing rate of the control angle is too high. Next, to make clear the influence of the control parameters, the reduction level and the increasing rate of the inverter control angle, numerical simulations are performed for various sets of them. Fig. 7 summarizes the numerically obtained time variations of the output current of the generator upstream side. This figure tells that selecting a high increasing rate makes the control speedy. The upper limit of the increasing rate is always about 1.1°/cycle, regardless of the reduction level. This result is quite reasonable because the characteristic time of the transient response is fixed by the system design and is independent of the inverter control angle. The upper
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Fig. 6. Time variations of (a) radial gas velocity distribution in MHD generator, (b) inverter margin angles, (c) output currents of MHD generator and (d) active power provided to power system in case where reduction level of inverter control angle is 110° and its increasing rate is 1.2°/cycle.
Fig. 7. Time variations of upstream side current of MHD generator in cases where reduction level of inverter control angle is (a) 110°, (b) 105°, (c) 100°, (d) 95° and its increasing rate is 1.2°, 1.0°, 0.75°, 0.5°/cycle.
limit of the increasing rate is, therefore, considered to be peculiar to the system. It is also made clear that using a lower reduction level of near 90°, to increase the margin angle of the inverter
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and to guarantee the stabilization of the generator, is quite all right because the influence of the reduction level on the transient response time is relatively small. These results indicate that the proposed control scheme of the inverter can stably return the generator to its normal operation and, therefore, is very effective to improve the transient stability of the generator.
5. Conclusions In this paper, the authors propose and investigate a new improvement method for transient stability under power line fault condition for commercial scale open cycle subsonic inflow disk MHD generators connected to power system through time dependent numerical simulations of a total interconnecting system. The simulation model is assumed to consist of the generator with two pairs of power output terminals, d.c. reactors, cascade connected two line commutated inverter bridges, two inverter transformers, filters, a shunt capacitor connected through a transformer, double circuit transmission lines and an infinite bus. The method needs no additional device and only manipulates the inverter control angle. Its control sequence has two steps. At first, the control angle is reduced toward 90° just after the fault occurs. Next, it is linearly increased to its rated value with constant increasing rate per unit cycle after the removed line is reconnected. It is made clear through the simulations under power line fault condition that the proposed control scheme of the inverter is very effective to improve the transient stability of the generator. The reduction of the inverter control angle is effective to stabilize the MHD generator, but the operating state shifts from its normal operation. The increase of the control angle can stably return the generator to its normal operation when its increasing rate is lower than its upper critical value, and the critical value is peculiar to the system.
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