Читать книгу Power Flow Control Solutions for a Modern Grid Using SMART Power Flow Controllers - Kalyan K. Sen - Страница 4

1 Chapter 1Figure 1‐1 Part of a large interconnected transmission system supplying elec...Figure 1‐2 Power flow along a controlled path.Figure 1‐3 Representation of a transmission line between sending and receivi...Figure 1‐4 Power flow along a transmission line between sending and receivin...Figure 1‐5 (a) Electric grid: power flow along a lossless transmission line ...Figure 1‐6 Power flow in a lossless line with a series‐compensating reactanc...Figure 1‐7 Power flow in a lossless line with a series‐compensating voltage ...Figure 1‐8 (a) Power transmission system with a series‐compensating voltage ...Figure 1‐9 Transmission line Voltage Regulators: (a) Two‐winding Transformer...Figure 1‐10 Transmission line voltage Phase Angle Regulators: (a) asymmetric...Figure 1‐11 Transmission line Reactance Regulators: (a) Thyristor‐Controlled...Figure 1‐12 Ranges of voltages (V_s and V_s′) at the primary and seconda...Figure 1‐13 Response time of the first commercial STATCON for 100 Mvar capac...Figure 1‐14 Voltage profile without an SVC (left) and with an SVC (right) (f...Figure 1‐15 A single‐line diagram of an electric arc furnace.Figure 1‐16 Instantaneous plant input bus voltage (v_plant) and plant current...Figure 1‐17 Use of a STATCOM, combined with a Fixed Capacitor (FC) as an arc...Figure 1‐18 Instantaneous reactive power drawn by the furnace and supplied b...Figure 1‐19 Instantaneous active power drawn by a compensator (STATCOM and F...Figure 1‐20 Flicker measurements without and with a compensation provided by...Figure 1‐21 Single‐line diagram of a Dynamic Voltage Restorer (DVR).Figure 1‐22 Sag correction by a DVR (field performance) (Sen 2015).Figure 1‐23 (a) Basic circuits for Active Power Line Conditioner and (b) Uni...Figure 1‐24 Cost versus features in various solutions (Case 1: “do nothing;”...Figure 1‐25 World’s first UPFC at the AEP Inez substation (left) versus a co...Figure 1‐26 Point‐to‐point transfer of power with local reactive power compe...Figure 1‐27 Ranges of Q_r versus P_r at the receiving end of the transmission ...Figure 1‐28 Independent active and reactive power flow controller with local...Figure 1‐29 Independent power flow control by impedance regulation (field pe...Figure 1‐30 Simultaneous power flow control by reactance regulation (field p...Figure 1‐31 Sen Transformer (ST).Figure 1‐32 Autotransformer/PAR (asym).Figure 1‐33 Multiline power flow concepts.Figure 1‐34 Choices for transmission line control equipment.Figure 1‐35 Today’s grid with traditional generation and integrated renewabl...Figure 1‐36 Interconnected transmission system, integrated with a SMART Powe...Figure 1‐37 Voltage regulation with an SVC and independent power flow regula...

2 Chapter 2Figure 2‐1 Representation of a power system network, consisting of a voltage...Figure 2‐2 (a) One‐generator/one‐line power system network and (b) phasor di...Figure 2‐3 Variations of load voltage (V_r) and load power (P_r) as a function...Figure 2‐4 (a) One‐generator/one‐line power system network with a shunt volt...Figure 2‐5 Variations of load voltage (V_r) and load power (P_r) as a function...Figure 2‐6 Two‐generator/one‐line power system network and the related phaso...Figure 2‐7 Two‐generator/one‐line power system network and the related phaso...Figure 2‐8 Voltage profile along a line when pu and .Figure 2‐9 Two methods of controlling the line voltage: (a) with a shunt‐com...Figure 2‐10 Scheme for implementing a shunt‐compensating voltage.Figure 2‐11 Scheme for implementing a series‐compensating voltage.Figure 2‐12 Electrical machine‐based solutions for power flow controllers us...Figure 2‐13 VSC‐based solutions for power flow controllers using (a) Shunt–S...Figure 2‐14 Transformer/LTCs‐based solutions for power flow controllers usin...Figure 2‐15 Q_s′ and V_s′ versus P_s′ at the modified sending...Figure 2‐16 Q_r versus P_r at the receiving end of the line for voltage regula...Figure 2‐17 (a) Power transmission system with a series‐compensating voltage...Figure 2‐18 Variations of the magnitude (V_s′) and phase‐shift angle (ψ...Figure 2‐19 (a) Two‐generator/one‐line power system network and its series‐c...Figure 2‐20 (a) Two‐generator/one‐line power system network and its series‐c...Figure 2‐21 (a) Variation of the sending‐end active and reactive power flows...Figure 2‐22 (a) Variation of the receiving‐end active and reactive power flo...Figure 2‐23 (a) Variation of the modified sending‐end active and reactive po...Figure 2‐24 (a) Power transmission system with a series‐compensating voltage...Figure 2‐25 (a) Variation of the exchanged active and reactive powers (P_se a...Figure 2‐26 (a) Two‐generator/one‐line power system network and its series‐c...Figure 2‐27 Phasor diagram of a PAR (sym): (a) active power flow, pu; (b) ...Figure 2‐28 RR operating with a reactance control method: (a) two‐generator/...Figure 2‐29 Series‐reactance regulation as combined effects voltage regulati...Figure 2‐30 RR operating with a reactance control method: (a) two‐generator/...Figure 2‐31 Effect of a compensating reactance on the power flow and the eff...Figure 2‐32 RR operating with a reactance control method: (a) two‐generator/...Figure 2‐33 RR operating with a reactance control method: (a) two‐generator/...Figure 2‐34 Effect of a compensating reactance on the power flow and the eff...Figure 2‐35 RR operating with a voltage control method with a compensating v...Figure 2‐36 RR operating with a voltage control method when a compensating v...Figure 2‐37 Effect of compensating voltage on power flow, effective line rea...Figure 2‐38 Power flow characteristics at the modified sending end and recei...Figure 2‐39 Variations of series‐compensating resistance and reactance of an...Figure 2‐40 Variations of series‐compensating impedance magnitude and its ph...Figure 2‐41 Variations of series‐compensating resistance and reactance of a ...Figure 2‐42 Variations of series‐compensating impedance magnitude and its ph...Figure 2‐43 Variations of series‐compensating resistance and reactance of a ...Figure 2‐44 Variations of series‐compensating impedance magnitude and its ph...Figure 2‐45 Variations of series‐compensating resistance and reactance of a ...Figure 2‐46 Variations of series‐compensating impedance magnitude and its ph...Figure 2‐47 Variations of series‐compensating resistance and reactance of a ...Figure 2‐48 Variations of series‐compensating impedance magnitude and its ph...Figure 2‐49 (a) Autotransformer. (b) Two‐winding transformer. (c) Ratio of t...Figure 2‐50 Q_r vs P_r at the receiving end of the line for the range of modif...Figure 2‐51 (a) Voltage‐Regulating Transformer (Shunt–Series configuration);...Figure 2‐52 Thyristor‐Controlled Load Tap Changer.Figure 2‐53 Static Var Compensator (SVC).Figure 2‐54 Concept of a STATCOM.Figure 2‐55 Static synchronous compensator (STATCOM).Figure 2‐56 Two‐generator/one‐line power system network with (a) no compensa...Figure 2‐57 Two‐generator/one‐line power system network with (a) no compensa...Figure 2‐58 (a) Phase Angle Regulator (asym), (b) phasor diagram for decreas...Figure 2‐59 (a) Phase Angle Regulator (sym), (b) phasor diagram for decreasi...Figure 2‐60 Thyristor‐Controlled Series Capacitor (TCSC) (k is the number of...Figure 2‐61 Static Synchronous Series Compensator (SSSC).Figure 2‐62 The UPFC (with a common DC link) and its building blocks: STATCO...Figure 2‐63 Two‐generator/one‐line power system network, integrated with a P...Figure 2‐64 Unified Power Flow Controller (UPFC).Figure 2‐65 Basic UPFC model, integrated in a two‐generator/one‐line power s...Figure 2‐66 Sen Transformer (ST).Figure 2‐67 Basic ST model, integrated in a two‐generator/one‐line power sys...Figure 2‐68 Basic BTB‐SSSC, integrated in a three‐generator/two‐line power s...Figure 2‐69 Implementation of a basic BTB‐SSSC.Figure 2‐70 BTB‐SSSC scheme as an asynchronous tie, integrated in a four‐gen...Figure 2‐71 Basic MST, integrated in a three‐generator/two‐line power system...Figure 2‐72 Multiline Sen transformer (MST).Figure 2‐73 Back‐To‐Back STATCOM (BTB‐STATCOM), integrated in a two‐generato...Figure 2‐74 Back‐to‐Back STATCOM.Figure 2‐75 Generalized power flow controller with a DC link.Figure 2‐76 Generalized power flow controller with an AC link.Figure 2‐77 Phasor diagram of VR for (a) no power flow, (b) 100% power flow ...Figure 2‐78 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐79 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐80 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐81 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐82 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐83 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐84 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐85 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐86 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐87 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐88 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐89 (a) Phasor diagram of a two‐generator/one‐line power system netw...Figure 2‐90 Phasor diagram of a PAR (sym) for (a) no power flow, (b) 100% po...Figure 2‐91 Phasor diagram of PAR (asym) for (a) no power flow, (b) 100% pow...Figure 2‐92 Phasor diagram of a RR for (a) no power flow, (b) 100% power flo...Figure 2‐93 Phasor diagram of an IR for (a) no power flow, (b) 100% power fl...Figure 2‐94 Voltage magnitude at the modified sending end for the range of a...Figure 2‐95 Power flow characteristics at the modified sending end for the r...Figure 2‐96 Power flow characteristics at the receiving end for the range of...Figure 2‐97 Reactive Power Index (RPI) versus active power (P_r) flow at the ...Figure 2‐98 Loss Index (LI) versus active power (P_r) flow at the receiving e...Figure 2‐99 Apparent Power Rating (APR) versus active power (P_r) flow at the...Figure 2‐100 Two‐generator/one‐line power system network without any compens...Figure 2‐101 Two‐generator/one‐line power system network with shunt compensa...Figure 2‐102 Two‐generator/one‐line power system network with two shunt comp...Figure 2‐103 Two‐generator/one‐line power system network with three shunt co...Figure 2‐104 Active power at the sending and receiving ends versus number of...Figure 2‐105 RPI versus number of shunt compensators.Figure 2‐106 LI versus number of compensators.Figure 2‐107 Total APR of shunt compensation versus number of shunt compensa...Figure 2‐108 Series‐compensating reactance versus equivalent shunt compensat...Figure 2‐109 RPI of series compensation versus equivalent shunt compensation...Figure 2‐110 LI of series compensation versus equivalent shunt compensation ...Figure 2‐111 APR of series compensation versus equivalent shunt compensation...Figure 2‐112 Two‐generator/one‐line power system network with a lossy line a...Figure 2‐113 Two‐generator/one‐line power system network with a lossy line a...Figure 2‐114 Reactive Power Index (RPI) versus active power flow (P_r) at the...Figure 2‐115 Loss Index (LI) versus active power flow (P_r) at the receiving ...Figure 2‐116 Apparent Power Rating (APR) versus active power flow (P_r) at th...Figure 2‐117 Sen Index (SI) versus active power flow (P_r) at the receiving e...

3 Chapter 3Figure 3‐1 EMTP modeling structure.Figure 3‐2 Single‐line diagram of a three‐phase voltage source with a source...Figure 3‐3 The A‐phase voltage (v_1Apu).Figure 3‐4 Single‐line diagram of a two‐generator/single‐line power system n...Figure 3‐5 Single‐line diagram of a two‐generator/single‐line power system w...Figure 3‐6 A‐phase line voltage (v_1A) at the BUS01 node, its peak value (V₁)...Figure 3‐7 Block diagram of a Vector PLL.Figure 3‐8 A‐phase line voltage (v_1A) at BUS01 node, its peak value (V₁), an...Figure 3‐9 Block diagram of a steady‐state resistance calculator.Figure 3‐10 Mathematical models of the (a) ST and (b) UPFC. Note the differe...Figure 3‐11 Variations of exchanged powers between the transmission line and...Figure 3‐12 Variations of node voltage with the operation of the ST and UPFC...Figure 3‐13 Variations of the receiving‐end power flows with the operations ...Figure 3‐14 Comparison of the active and reactive power flows at the receivi...

4 Chapter 4Figure 4‐1 (a) Autotransformer. (b) two‐winding transformer.Figure 4‐2 Ranges of voltage magnitudes (V_s and V_s′) at the sending an...Figure 4‐3 Ranges of voltage magnitudes (V_s and V_s′) at the sending an...Figure 4‐4 (a) Voltage‐regulating transformer (shunt‐series configuration); ...Figure 4‐5 Ranges of voltage magnitudes (V_s and V_s′) at the sending an...Figure 4‐6 (a) Voltage‐regulating transformer (two‐winding transformer); (b)...Figure 4‐7 Ranges of voltage magnitudes (V_s and V_s′) at the sending an...Figure 4‐8 Ranges of voltage magnitudes (V_s and V_s′) at the sending an...Figure 4‐9 (a) PAR (asym) configuration in a bypass‐mode of operation; (b) p...Figure 4‐10 (a) PAR (asym) configuration for decreasing power flow; (b) phas...Figure 4‐11 (a) PAR (asym) configuration for increasing power flow; (b) phas...Figure 4‐12 Ranges of voltage magnitudes (V_s and V_s′) at the sending a...Figure 4‐13 (a) PAR (sym) configuration in a bypass‐mode of operation; (b) p...Figure 4‐14 (a) PAR (sym) configuration for decreasing power flow; (b) phaso...Figure 4‐15 (a) PAR (sym) configuration for increasing power flow; (b) phaso...Figure 4‐16 Ranges of voltage magnitudes (V_s and V_s′) at the sending a...

5 Chapter 5Figure 5‐1 Shunt‐compensating, mechanically‐switched capacitor, connected to...Figure 5‐2 Exchanged reactive power (Q_sh), the resulting voltages (v_1A and VFigure 5‐3 Shunt‐compensating, mechanically‐switched capacitor with a series...Figure 5‐4 Exchanged reactive power (Q_sh), the resulting voltages (v_1A and VFigure 5‐5 Shunt‐compensating, mechanically‐switched reactor, connected to a...Figure 5‐6 Exchanged reactive power (Q_sh), the resulting voltages (v_1A and VFigure 5‐7 Series‐compensating, mechanically‐switched reactors, connected in...Figure 5‐8 Exchanged reactive power (Q_se), the resulting voltage (v_12A), cur...Figure 5‐9 Instantaneous filtered magnitude of the voltage across the series...Figure 5‐10 Instantaneous filtered phase angle of the voltage across the ser...Figure 5‐11 Instantaneous magnitude (i_dq) of the current through the series‐...Figure 5‐12 Instantaneous phase angle (θ_idq) of the current through the...Figure 5‐13 Instantaneous magnitude (z_se) of the impedance of the series‐com...Figure 5‐14 Instantaneous phase angle (θ_zse) of the voltage across the ...Figure 5‐15 Instantaneous relative phase angle (β) of the series‐compen...Figure 5‐16 Two superimposed voltages: (1) A phase of the calculated compens...Figure 5‐17 Series‐compensating, mechanically‐switched capacitor with a reac...Figure 5‐18 Exchanged reactive power (Q_se), the resulting voltage (v_12A), cu...Figure 5‐19 Series‐compensating voltage to emulate a reactor in series with ...Figure 5‐20 Validation of a series‐compensating voltage as an emulated react...

6 Chapter 6Figure 6‐1 (a) Two‐generator/one‐line uncompensated power system network; (b...Figure 6‐2 (a) Two‐generator/one‐line power system network with a series‐com...Figure 6‐3 Transformer/LTCs‐based solutions for power flow controllers using...Figure 6‐4 (a) Voltage‐Regulating Transformer (Shunt‐Series configuration); ...Figure 6‐5 (a) Voltage‐Regulating Transformer (Shunt‐Shunt configuration); (...Figure 6‐6 (a) Phase Angle Regulator (asymmetric); (b) phasor diagram.Figure 6‐7 (a, b) Effect of a series‐compensating voltage on power flow in a...Figure 6‐8 (a) Voltage‐Regulating Transformer for increasing line voltage; (...Figure 6‐9 (a) Voltage‐Regulating Transformer for decreasing line voltage; (...Figure 6‐10 (a) ST for voltage regulation; (b) phasor diagram.Figure 6‐11 (a) ST for voltage compensation with and β = 106.1°; (b) ...Figure 6‐12 Compensating points with the use of the ST within the entire ran...Figure 6‐13 Series impedance emulation control block diagram of the ST.Figure 6‐14 Series resistance emulation control block diagram of the ST.Figure 6‐15 Series reactance emulation control block diagram of the ST.Figure 6‐16 Closed‐loop automatic power flow control block diagram of the ST...Figure 6‐17 Open‐loop Compensating‐Voltage Unit control block diagram of the...Figure 6‐18 Selection of tap positions (Faruque and Dinavahi‐2007).Figure 6‐19 Compensating voltage (V_s′s) in pu during the entire range ...Figure 6‐20 Modified sending‐end voltage (V_s′) in pu during the entire...Figure 6‐21 Phase‐shift angle (ψ) in degrees during the entire range of...Figure 6‐22 (a) Active power (P_r), (b) reactive power (Q_r), and (c) apparent...Figure 6‐23 Active power (P_r) versus reactive power (Q_r) in pu at the receiv...Figure 6‐24 (a) Exchanged active power (P_se), (b) reactive power (Q_se), and ...Figure 6‐25 Exchanged active power (P se ) versus reactive power (Q_se) in pu...Figure 6‐26 Line current (I) in pu during the entire range of the relative p...Figure 6‐27 (a) Sen Transformer (ST) and (b) phasor diagram.Figure 6‐28 Magnitude of the simulated sending‐end voltage (V_s) in pu during...Figure 6‐29 Magnitude of the simulated compensating voltage (V_s′s) in ...Figure 6‐30 Magnitude of the simulated modified sending‐end voltage (V_s′...Figure 6‐31 (a) Active power (P_r) in MW, (b) reactive power (Q_r) in Mvar, an...Figure 6‐32 Active power (P_r) in MW versus reactive power (Q_r) in Mvar at th...Figure 6‐33 (a) Exchanged active power (P_se) in MW, (b) reactive power (Q_se)...Figure 6‐34 Exchanged active power (P_se) in MW versus reactive power (Q_se) i...Figure 6‐35 Line current (I) in A_rms during the entire range of the relative...Figure 6‐36 Three‐generator/four‐line power system network, integrated with ...Figure 6‐37 Magnitude of the simulated sending‐end voltage (V_s) in pu during...Figure 6‐38 Magnitude of the simulated compensating voltage (V_s′s) in ...Figure 6‐39 Magnitude of the simulated modified sending‐end voltage (V_s′...Figure 6‐40 (a) Active power (P_r) in MW, (b) reactive power (Q_r) in Mvar, an...Figure 6‐41 Active power (P_r) in MW versus reactive power (Q_r) in Mvar at th...Figure 6‐42 (a) Exchanged active power (P_se) in MW, (b) reactive power (Q_se)...Figure 6‐43 Exchanged active power (P_se) in MW versus reactive power (Q_se) i...Figure 6‐44 Line current (I) in A_rms during the entire range of the relative...Figure 6‐45 Factory test circuit of an ST.Figure 6‐46 The instantaneous A‐phase, sending‐end current (i_sA), line curre...Figure 6‐47 Magnitude of the simulated sending‐end voltage (V_s) in pu during...Figure 6‐48 Magnitude of the simulated compensating voltage (V_s′s) in ...Figure 6‐49 Magnitude of the simulated modified sending‐end voltage (V_s′...Figure 6‐50 (a) Active power (P_r) in MW, (b) reactive power (Q_r) in Mvar, an...Figure 6‐51 Active power (P_r) in MW versus reactive power (Q_r) in Mvar flows...Figure 6‐52 (a) Exchanged active power (P_se) in MW, (b) reactive power (Q_se)...Figure 6‐53 Exchanged active power (P_se) in MW versus reactive power (Q_se) i...Figure 6‐54 Line current ( I ) in A_rms during the entire range of the relati...Figure 6‐55 Modified factory test circuit of an ST.Figure 6‐56 The instantaneous A‐phase, sending‐end current (i_sA), line curre...Figure 6‐57 Magnitude of the simulated sending‐end voltage (V_s) during the e...Figure 6‐58 Magnitude of the simulated compensating voltage (V_s′s) dur...Figure 6‐59 Magnitude of the simulated modified sending‐end voltage (V_s′...Figure 6‐60 (a) Active power (P_r), (b) reactive power (Q_r), and (c) apparent...Figure 6‐61 Active power (P_r) versus reactive power (Q_r) flows at the receiv...Figure 6‐62 (a) Exchanged active power (P_se), (b) reactive power (Q_se), and ...Figure 6‐63 Exchanged active power (P_se) versus reactive power (Q_se) by the ...Figure 6‐64 Line current ( I ) in A_rms during the entire range of the relati...Figure 6‐65 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐66 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐67 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐68 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐69 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐70 (a) ST operating with a compensating voltage in the range of 0 ≤...Figure 6‐71 (a) ST configuration using LTCs with lower current rating; (b) p...Figure 6‐72 ST configuration using taps with lower voltage and current ratin...Figure 6‐73 ST configuration with six secondary windings in a two‐core desig...Figure 6‐74 The voltage compensating points for the A phase in Option 1 conf...Figure 6‐75 ST configuration with three secondary windings in a two‐core des...Figure 6‐76 Operating points of an ST in Option 1: (a) modified sending‐end ...Figure 6‐77 The voltage compensating points for the A phase in Option 2 conf...Figure 6‐78 ST configuration with three secondary windings in a two‐core des...Figure 6‐79 Operating points of an ST in Option 2: (a) modified sending‐end ...Figure 6‐80 The voltage compensating points for the A phase in Option 3 conf...Figure 6‐81 An ST configuration with three secondary windings in a two‐core ...Figure 6‐82 Operating points of an ST in Option 3: (a) modified sending‐end ...Figure 6‐83 (a) UPFC test results for automatic power flow control mode of o...Figure 6‐84 Practical voltage compensation range with the use of the VRT, PA...Figure 6‐85 Multiline Sen Transformer (MST).Figure 6‐86 ST’s Compensating‐Voltage Unit is connected to the stepped‐down ...Figure 6‐87 ST’s Compensating‐Voltage Unit is connected to the stepped‐up vo...Figure 6‐88 Two methods of modifying a transmission line voltage: (a) with a...Figure 6‐89 (a) Shunt‐Shunt configuration of ST for the A phase; (b) phasor ...Figure 6‐90 Shunt‐Shunt configuration of ST for the A phase, B phase, and C ...Figure 6‐91 (a) ST with shunt‐connected Compensating‐Voltage Unit operating ...Figure 6‐92 (a) ST with shunt‐connected Compensating‐Voltage Unit operating ...Figure 6‐93 (a) ST with shunt‐connected Compensating‐Voltage Unit operating ...Figure 6‐94 MST with shunt‐connected Compensating‐Voltage Units.Figure 6‐95 Generalized Sen Transformer (GST).

7 Appendix AFigure A‐1 Three‐phase voltage source supplying currents to a balanced three...Figure A‐2 Balanced three‐phase voltages in a timing diagram and the corresp...Figure A‐3 Balanced three‐phase currents in a timing diagram and the corresp...Figure A‐4 Unbalanced three‐phase variables in a timing diagram and the corr...Figure A‐5 Balanced three‐phase, positive‐sequence components in a timing di...Figure A‐6 Balanced three‐phase, negative‐sequence components in a timing di...Figure A‐7 Zero‐sequence component in a timing diagram and the corresponding...Figure A‐8 A‐phase, positive‐, negative‐, and zero‐sequence components in a ...Figure A‐9 B‐phase positive‐, negative‐, and zero‐sequence components in a t...Figure A‐10 C‐phase positive‐, negative‐, and zero‐sequence components in a ...Figure A‐11 Triple frequency, zero‐sequence component in a timing diagram an...Figure A‐12 Unbalanced three‐phase variables in a timing diagram and the cor...Figure A‐13 A‐phase positive‐, negative‐, and zero‐sequence components in a ...Figure A‐14 B‐phase positive‐, negative‐, and zero‐sequence components in a ...Figure A‐15 C‐phase positive‐, negative‐, and zero‐sequence components in a ...Figure A‐16 Three‐phase voltage source supplying three‐phase unbalanced load...Figure A‐17 Representation of three‐phase, positive‐sequence variables in (a...Figure A‐18 Representation of three‐phase, negative‐sequence variables in (a...Figure A‐19 Conversion of A, B, and C stationary variables into (a) d‐q stat...

8 Appendix BFigure B‐1 Two‐generator/one‐line power system network with a lossy line and...Figure B‐2 Natural current (I_n) as a function of power angle from 0^° ...Figure B‐3 (a) Natural active power (P_sn), (b) natural reactive power (Q_sn),...Figure B‐4 (a) Natural active power (P_rn), (b) natural reactive power (Q_rn),...Figure B‐5 (a) Q_sn vs P_sn at the sending end and (b) Q_rn vs P_rn at the recei...Figure B‐6 Modified sending‐end voltage (V_s′) with a shunt‐compensatin...Figure B‐7 Modified sending‐end voltage (V_s′) with a series‐compensati...

9 Appendix CFigure C‐1 Sen transformer representation.Figure C‐2 Phasor diagram of the Sen transformer.Figure C‐3 Proposed Sen transformer representation in PSS^®E load flow....Figure C‐4 Chilean power system – study area.Figure C‐5 Chilean system post Sen transformer.Figure C‐6 Operating limits of the two‐transformer model and the Sen transfo...