Power Electronics, HVDC, FACTS and Power Quality

Introduction

Modern electrical power systems are increasingly shaped by semiconductor converters. High-voltage direct current (HVDC) transmission enables economical long-distance and submarine transmission; flexible AC transmission systems (FACTS) control voltage, impedance and power flow; and power-electronic interfaces connect renewable generation, storage and controllable loads. At the same time, converter switching, nonlinear loads and network disturbances create power-quality challenges.

This chapter develops the operating principles, mathematical models, control structures, design calculations and practical limitations of these technologies. Particular attention is given to converter harmonics, HVDC control and protection, voltage-source converter (VSC) dynamics, modular multilevel converters (MMCs), FACTS devices, power-quality indices and disturbance mitigation.

Learning objectives

After studying this chapter, the reader should be able to:

1. Explain line-commutated and self-commutated converter operation.
2. Derive the principal DC-voltage and overlap relationships for a six-pulse bridge.
3. Compare two-level, three-level and modular multilevel VSCs.
4. Model an HVDC link, including its DC line, controls, commutation effects and stability constraints.
5. Evaluate the operating principles of SVC, STATCOM, TCSC, UPFC and related FACTS devices.
6. Quantify voltage sag, unbalance, harmonics, flicker and power factor using appropriate definitions.
7. Select practical mitigation, filtering, protection and control strategies.

1. Power-electronic conversion in power systems

A power converter changes voltage, current, frequency or power-flow direction using controlled semiconductor switches. The main classes are:

• Uncontrolled rectifiers, normally using diodes.
• Line-commutated converters (LCCs), using thyristors and the AC system voltage for commutation.
• Voltage-source converters (VSCs), using self-commutated IGBTs, IGCTs or MOSFET-based switches.
• Current-source converters (CSCs), whose DC link has a relatively large inductance.
• Modular multilevel converters (MMCs), formed from many controllable submodules in each arm.

A converter is never an ideal independent source. Its behavior is coupled to network short-circuit strength, transformer leakage, control bandwidth, switching frequency, filters, grounding and protection. Consequently, converter design must combine circuit theory, electromagnetic transients, control systems and insulation coordination.

Six-pulse bridge and commutation overlap

For a three-phase, six-pulse bridge supplied through a transformer with commutation reactance, the ideal average DC voltage for a fully controlled bridge is

where V_LL is the RMS line-to-line AC voltage and V_d0 is the ideal no-load average DC voltage. With firing angle α and continuous, sufficiently smooth DC current, a commonly used idealized expression is

where ω is angular frequency, L_c is the effective commutating inductance per phase referred to the converter AC side, and I_d is DC current. The second term represents the average voltage consumed by commutation overlap.

The overlap angle μ is not independent of current. For a symmetrical converter with constant AC line-to-line voltage approximation, one useful relation is

The exact coefficient depends on the definition of commutation reactance and voltage base; therefore the circuit convention must be stated before applying the equation. Increasing current or commutation inductance increases μ and reduces the available DC voltage.

During overlap, two valves conduct simultaneously. The outgoing valve current transfers to the incoming valve through leakage inductance rather than changing instantaneously. This limits di/dt, but causes reactive-power consumption, voltage notches and additional commutation stress.

Twelve-pulse conversion and harmonics

A twelve-pulse converter combines two six-pulse bridges supplied by transformer secondary voltages displaced by 30°. In an ideal balanced system, characteristic AC harmonics of each six-pulse bridge are of order

Thus the 5th and 7th harmonics from the two bridges have opposite phase relationships and are cancelled or substantially reduced at the common AC bus. The 11th and 13th are similarly treated. The characteristic AC harmonics that remain for the twelve-pulse arrangement are

namely 11th, 13th, 23rd, 25th and so on. On the DC side, the principal ripple orders are multiples of 12 rather than multiples of 6. Cancellation is imperfect when transformer ratio errors, firing-angle mismatch, unequal leakage reactance, unbalance or control errors exist.

A six-pulse converter also produces reactive power. For an ideal bridge with negligible overlap and continuous current, the fundamental displacement factor is approximately cos α; distortion and overlap reduce the true power factor further. LCC stations therefore require AC filters, reactive compensation and harmonic-performance guarantees.

Figure: Two six-pulse bridges supplied by 30°-displaced transformer windings form a twelve-pulse converter.

2. VSCs and dynamic models

A two-level VSC produces three phase-leg pole voltages by switching upper and lower valves. Pulse-width modulation synthesizes a fundamental voltage whose magnitude and phase are controlled. A simplified averaged model is more useful than individual switching waveforms for electromechanical studies.

For a VSC connected to an AC bus through reactor R + jX, the complex power transferred from the converter to the bus is governed by the voltage magnitude and angle difference. In a lossless approximation,

where V_c is converter internal fundamental voltage, V_g is grid voltage, δ is the converter-to-grid angle and X is coupling reactance. Positive or negative reactive power depends on the adopted sign convention. The equations show why converter voltage angle primarily controls active power while voltage magnitude primarily influences reactive power, although the variables are coupled in a weak or strong grid.

In a synchronously rotating dq frame aligned with the grid voltage, a commonly used averaged current model is

For a d-axis aligned with the grid, v_gq = 0 and instantaneous powers are

Thus, under alignment, active power is mainly controlled by i_d and reactive power by i_q. Decoupling feed-forward terms ωLi_q and ωLi_d improve current-loop response. The model assumes balanced sinusoidal quantities, a valid phase-locked loop, averaged switching and a sufficiently small current ripple.

DC-link dynamics

For a VSC DC capacitor,

A perturbation around an operating point gives approximately

before including controller and network dynamics. This relationship explains why active-power imbalance changes DC voltage and why the outer DC-voltage controller must be slower than the inner current controller.

MMC operation and arm-energy dynamics

An MMC arm contains N submodules, arm inductance L_arm, resistance R_arm and a series connection of capacitor voltages. The upper and lower arm insertion indices are n_u and n_l, normally between 0 and 1. The total stored capacitor energy is

where C_SM is the submodule capacitance and the summation covers all submodule capacitors. Its dynamic balance is

At arm level, neglecting switching ripple,

The signs depend on current reference directions. The arm currents can be decomposed into AC output current and circulating current. A second-harmonic circulating component often appears because the instantaneous power exchanged between AC and DC sides contains a double-frequency component. A circulating-current controller reduces capacitor-voltage ripple and semiconductor losses.

MMC control normally includes: outer active/reactive-power or DC-voltage loops; inner dq current loops; circulating-current suppression; capacitor-voltage balancing; nearest-level modulation or phase-shifted carrier modulation; and submodule bypass/blocking logic. Capacitor balancing is essential because unequal voltages can overstress individual devices even when the average arm voltage is correct.

3. HVDC transmission

An HVDC scheme consists of converter transformers, converters, smoothing reactors, DC conductors or cables, filters, reactive compensation, controls, measurement systems and protection. Configurations include:

• Monopolar: one high-voltage conductor with ground, metallic return or dedicated return conductor.
• Bipolar: positive and negative poles, normally with a neutral or metallic return for unbalanced operation.
• Back-to-back: two converters at one site, with no long DC line, used to interconnect asynchronous AC systems.
• Multi-terminal: three or more converter stations sharing a DC network.

DC-line model

For a quasi-steady bipolar link, the DC circuit can be represented by

where V_dR and V_dI are rectifier and inverter terminal DC voltages with a consistent current direction, and R_d includes conductor, return-path and smoothing-reactor resistance as appropriate. For a distributed line, the frequency-dependent telegrapher equations are

where R, L, G and C are per-unit-length parameters. A lumped model is adequate for slow control studies; traveling-wave models are required for fast protection and cable transients.

LCC control and commutation failure

An LCC rectifier commonly controls current through firing angle α, while the inverter controls extinction angle γ or current margin. The extinction angle must remain sufficiently large for thyristor recovery. AC voltage depression, phase shifts and increased current reduce the available commutation margin and can cause commutation failure.

The short-circuit ratio is commonly defined as

where S_sc is AC short-circuit power at the converter bus and P_dN is rated DC power. A low SCR indicates a weak AC system. For multi-infeed systems, effective short-circuit ratio (ESCR) accounts approximately for the influence of nearby converters and shunt compensation:

where Q_c is the relevant shunt compensation rating. Exact multi-infeed interaction requires a network model; ESCR is an index, not a complete stability criterion.

VSC-HVDC is particularly valuable for weak grids, offshore wind, black-start-capable networks and underground or submarine cables. It independently controls active and reactive power, supports voltage control and avoids the LCC requirement for a strong commutating AC voltage. Its limitations include converter losses, semiconductor fault-current constraints and the need for sophisticated DC protection.

HVDC control hierarchy and stability

A practical hierarchy contains:

1. Fast inner current control.
2. Outer active-power, DC-voltage, AC-voltage or reactive-power control.
3. Supervisory power-order, ramp-rate and power-modulation control.
4. Protection and emergency control.

The control bandwidths must be separated. If outer-loop bandwidth approaches inner-loop bandwidth, neglected dynamics and delays can produce poor damping or instability. Digital delay, PLL dynamics, weak-grid impedance variation and converter control interaction should be assessed using eigenvalue analysis, impedance-based stability, electromagnetic-transient simulation or frequency scans.

For a linearized state-space model,

stability of the local linear model requires all eigenvalues of A to have negative real parts. This condition is local and does not guarantee large-disturbance stability or stable operation under controller saturation.

HVDC protection

Protection must distinguish internal DC faults from external AC faults and withstand high di/dt. LCC links use DC reactors, line differential or traveling-wave methods, valve protection, AC breaker backup and controlled deblocking/blocking. VSC links require fast fault detection because DC capacitors discharge rapidly and ordinary AC current zero interruption is unavailable on the DC side. Options include hybrid DC circuit breakers, resonant breakers, fast disconnectors, fault-blocking submodules and DC-grid protection zones.

4. FACTS technology

FACTS devices use power electronics to increase controllability and transfer capability without necessarily constructing new lines.

Figure: Shunt, series and combined FACTS connection concepts.

For a lossless two-bus line,

A series compensator changes effective X, while a shunt device regulates voltage magnitude and reactive balance. A STATCOM behaves approximately as a controllable AC voltage behind reactance. Unlike a thyristor-switched capacitor, it can maintain reactive-current capability during depressed voltage more effectively, although its reactive power in a voltage-source implementation is approximately proportional to voltage for a fixed current limit.

A UPFC combines a shunt converter that exchanges real power with the AC system and a series converter that injects a controllable voltage. The common DC link permits real-power exchange between the two converters. Practical constraints include converter current, DC-link voltage, thermal limits, transformer insulation, control interaction and protection during faults.

5. Power quality

Power quality describes the deviation of voltage, current and frequency from ideal sinusoidal, balanced and regulated conditions. Applicable limits depend on the jurisdiction and connection point. IEEE 1159 provides terminology and monitoring guidance; IEEE 519 addresses harmonic control at points of common coupling; IEC 61000 series contains compatibility and emission standards. Standards should be applied using their current editions and local regulatory requirements.

Voltage variation and sag

A voltage sag is a reduction in RMS voltage, typically lasting from about half a cycle to one minute, while the exact classification depends on the adopted standard. Retained voltage is commonly expressed as

For a fault at bus f, use the Thevenin network, not an ordinary load-divider formula. Let V_m0 be the prefault voltage at observation bus m, Z_ff the Thevenin driving-point impedance at the fault bus, Z_mf the transfer impedance between observation and fault buses, and Z_f the fault impedance. Then, for a balanced positive-sequence fault,

This formulation includes fault location and network coupling. For a fault directly at the observed bus, m = f and, if V_m0 = V_f0 = E,

where Z_th = Z_ff. The special case of a bolted fault, Z_f = 0, gives zero voltage at the faulted bus, but not necessarily zero voltage elsewhere. Unbalanced faults require symmetrical components and separate sequence networks.

Worked sag calculation

Suppose a 11-kV positive-sequence network has prefault bus voltage 1.00 pu. A remote three-phase fault has Z_ff = j0.20 pu, fault impedance Z_f = j0.05 pu, and transfer impedance from the monitored bus is Z_mf = j0.10 pu. The prefault voltage at the fault location is 1.00 pu. Therefore

I_f = 1/(j0.20+j0.05) = −j4.00 pu.

The monitored-bus voltage is

V_m = 1 − (j0.10)(−j4.00) = 1 − 0.40 = 0.60 pu.

Thus the retained voltage is 60%, and the sag magnitude is 40%. This result is location-dependent; it is not obtained from Z_load.

Voltage unbalance and sequence components

For phase phasors V_a, V_b and V_c, let a = e^j120°. Symmetrical components are

where V₁ is positive sequence, V₂ negative sequence and V₀ zero sequence. A commonly used voltage-unbalance factor is

The exact compliance method may instead use phase magnitude deviation or a standard-specific algorithm. Negative-sequence voltage produces double-frequency currents and heating in rotating machines; zero sequence is associated with neutral displacement and ground-current paths.

Harmonics and distortion

For RMS harmonic components V_h, total harmonic distortion is

and similarly for current. The denominator is the fundamental RMS component, not total RMS. Total RMS voltage is

A nonlinear load draws nonsinusoidal current. Network impedance converts harmonic current into voltage distortion according to V_h = Z_h I_h, where Z_h is the system impedance at harmonic order h. Parallel resonance between capacitor banks and system inductance can amplify a particular harmonic; filter design must therefore use frequency-dependent network studies rather than only fundamental-frequency calculations.

Worked harmonic calculation

A bus voltage has fundamental component 230 V RMS, a 5th harmonic of 11.5 V and a 7th harmonic of 6.9 V. Then

THD_V = 100√(11.5²+6.9²)/230.

The numerator is √(132.25+47.61)=13.41 V, so THD_V = 5.83%. If the 5th-harmonic system impedance is 0.40 Ω and a load injects 18 A RMS at the 5th harmonic, the approximate 5th-harmonic voltage is V₅ = 0.40×18 = 7.2 V. A tuned or active filter must reduce the injected current and avoid creating a new parallel-resonance problem.

Other disturbances

• Interruption: RMS voltage falls below the specified threshold.
• Transient: A short-duration impulsive or oscillatory event caused by switching, lightning or faults.
• Voltage swell: RMS voltage rises above its nominal range.
• Flicker: Perceptible light modulation caused by low-frequency voltage-amplitude variation.
• Rapid voltage change: A change that may not meet sag duration criteria but can disturb equipment.
• Frequency deviation: Departure from nominal frequency caused by active-power imbalance.
• Poor power factor: The combination of displacement and waveform distortion effects.

6. Mitigation, monitoring and protection

Mitigation should begin with measurement at the point of common coupling. Record RMS trends, event waveforms, harmonic spectra, phase angles, sequence components, frequency and triggering information. Instruments must have adequate sampling rate, anti-aliasing, synchronization and uncertainty documentation.

Typical measures include:

• point-on-wave controlled switching and pre-insertion resistors;
• transformer phase-shifting and twelve-pulse or higher-pulse conversion;
• passive tuned and high-pass filters;
• active power filters and hybrid filters;
• dynamic voltage restorers and UPS systems;
• STATCOMs and synchronous condensers for voltage support;
• load balancing, phase relocation and neutral conductors;
• converter modulation, harmonic compensation and selective harmonic elimination;
• protection coordination with converter blocking, bypass and thermal limits.

Passive filters are economical but sensitive to detuning, component tolerances and resonance. Active filters are flexible but have finite current rating, switching losses and control delay. A mitigation device should be evaluated for normal operation, contingencies, background distortion, resonance, temporary overvoltage and failure modes.

7. Applied case: offshore wind export using VSC-HVDC

Background

Consider an illustrative 600-MW offshore wind plant located 120 km from an onshore grid. The export cable is long, the offshore network has no naturally strong synchronous source, and the onshore receiving grid occasionally operates at low short-circuit strength.

Problem

An LCC link would require substantial offshore reactive compensation and a sufficiently strong commutating voltage. Cable charging and commutation dependence would complicate operation during weak-grid conditions. The project also requires controlled energization and support during onshore voltage disturbances.

Analysis and approach

A bipolar VSC-HVDC link is selected. Offshore converters regulate AC voltage and frequency for the wind-plant collector system while controlling exported active power. The onshore converter controls DC voltage and reactive power, with current limits prioritizing either active-power transfer or voltage support according to grid-code requirements.

The design process includes:

1. Selecting DC voltage from insulation, current, cable-loss and equipment constraints.
2. Determining converter MVA rating from active power and reactive-current requirements.
3. Designing MMC submodule capacitance from allowable voltage ripple and energy variation.
4. Selecting arm inductance to limit fault current and circulating current.
5. Performing harmonic, resonance and cable-frequency studies.
6. Testing PLL, current control and outer-loop interaction over a range of grid strengths.
7. Defining DC fault zones, fast detection and breaker or fault-blocking requirements.
8. Validating controls in electromagnetic-transient simulation and hardware-in-the-loop tests.

Outcome

The VSC arrangement provides independent active and reactive control, facilitates offshore voltage formation and avoids dependence on a strong offshore AC system. However, it increases control and protection complexity. The final design must demonstrate thermal margins, ride-through behavior, harmonic compliance, stable operation under weak-grid conditions and safe isolation of DC faults.

Learning points

The technically superior converter is not selected solely from nominal efficiency. Network strength, fault behavior, cable characteristics, control interaction, standards, maintainability and protection architecture determine system suitability.

8. Contemporary developments and governance

Research and industry development are moving toward hybrid AC/DC grids, multi-terminal HVDC, grid-forming VSCs, silicon-carbide devices, higher-voltage MMCs, adaptive harmonic control, wide-area measurements and data-driven condition monitoring. Grid-forming converters regulate an internal voltage waveform and can contribute to frequency and voltage establishment, unlike grid-following converters that depend strongly on a PLL and an external voltage waveform.

AI can assist anomaly detection, converter-health estimation and adaptive tuning, but it should not replace deterministic protection without validation. Training-data bias, cyberattack surfaces, explainability, fail-safe operation and responsibility for automated decisions are governance concerns. Digital substations and converter controls require secure time synchronization, access control, patch management and independent protection paths.

Engineering practice should document assumptions, parameter uncertainty, model validation, software versions, test coverage and compliance evidence. Harmonic and stability claims should be reproducible from clearly defined network models and measurement locations.

Chapter Summary

Power electronics provides controllable interfaces for modern power systems. Six-pulse LCC bridges have an average voltage approximately proportional to 1.35V_LL cos α, reduced by commutation overlap and dependent on AC-system strength. Twelve-pulse converters cancel selected six-pulse harmonics and leave characteristic AC harmonics of order 12k ± 1. VSCs provide independent active and reactive control through dq current regulation, while MMCs add scalable voltage capability but require capacitor balancing, circulating-current suppression and detailed energy control.

HVDC links may be monopolar, bipolar, back-to-back or multiterminal. LCC-HVDC is economical at high power but consumes reactive power and depends on commutation voltage; VSC-HVDC offers black-start, weak-grid and independent P–Q control advantages. SCR and ESCR indicate system strength but do not replace dynamic stability studies. DC lines require lumped or distributed models according to the study time scale, and protection must address rapid DC fault development.

FACTS devices regulate voltage, line impedance, phase angle and power flow through shunt, series or combined connections. Power quality must be quantified using correct network models and standard-based definitions. Voltage sag at a monitored bus is obtained from the fault-network transfer relationship, not a generic load-divider equation. Sequence components quantify unbalance, THD quantifies waveform distortion, and filters or compensators must be designed with frequency-dependent network resonance in mind.

Key Terms

Power converter: Semiconductor-based apparatus that changes electrical power form or control variables.

LCC: Line-commutated converter using the AC system for valve commutation.

VSC: Voltage-source converter with a stiff DC-voltage characteristic.

MMC: Modular multilevel converter using series-connected submodules.

Commutation overlap: Interval during which outgoing and incoming converter valves conduct simultaneously.

Characteristic harmonic: Harmonic order predicted by converter pulse number and balanced operation.

HVDC: High-voltage direct-current transmission.

SCR: Short-circuit ratio measuring AC-system strength relative to converter rating.

ESCR: Effective short-circuit ratio including relevant compensation effects.

FACTS: Flexible AC transmission systems using power-electronic control.

STATCOM: Shunt VSC providing dynamic reactive-current control.

TCSC: Thyristor-controlled series compensator.

UPFC: Unified power-flow controller combining series and shunt VSCs.

Voltage sag: Temporary reduction in RMS voltage magnitude.

Symmetrical components: Positive-, negative- and zero-sequence representation of unbalanced phasors.

VUF: Ratio of negative-sequence to positive-sequence voltage magnitude.

THD: RMS magnitude of harmonic components relative to the fundamental.

Grid-forming converter: Converter that establishes or regulates an internal voltage waveform.

Point of common coupling: Network location at which a customer installation connects to other users.

Review Questions

Short Answer Questions

1. Why does commutation overlap reduce the average DC voltage of an LCC?
2. State the characteristic AC harmonic orders of an ideal six-pulse and twelve-pulse converter.
3. What is the physical meaning of SCR in an HVDC system?
4. Distinguish the principal functions of a STATCOM and a TCSC.
5. Define voltage-unbalance factor using symmetrical components.

Long Answer Questions

1. Derive the ideal average DC voltage of a three-phase fully controlled bridge and discuss the effects of firing angle and overlap.
2. Explain the dq model of a grid-connected VSC and show how active and reactive powers are controlled.
3. Describe MMC arm-energy dynamics, capacitor-voltage balancing and circulating-current suppression.
4. Compare LCC-HVDC and VSC-HVDC with respect to reactive power, weak-grid operation, faults, harmonics and control.
5. Explain the operating principles and applications of SVC, STATCOM, TCSC, SSSC and UPFC.

Analytical / Application Questions

1. A monitored bus has V_m0=1.0 pu, Z_ff=j0.25 pu, Z_f=j0.05 pu and Z_mf=j0.12 pu for a remote balanced fault. Calculate fault current and retained monitored-bus voltage using the Thevenin transfer-impedance method.
2. A bus contains a 230-V fundamental, 9.2-V fifth harmonic and 4.6-V seventh harmonic. Calculate voltage THD and explain whether a passive filter can be specified without a resonance study.
3. Design a control and protection concept for a 500-MW VSC-HVDC connection to a weak grid. State the control hierarchy, measurements, limiting functions, stability studies and DC-fault protection requirements.

Multiple Choice Questions with Answers

Question 1

For an ideal three-phase six-pulse bridge, the average no-load DC voltage is approximately:

A. 0.707 V_LL

B. 1.35 V_LL

C. 2.00 V_LL

D. 3.00 V_LL

Correct Answer: B. 1.35 V_LL

Question 2

The characteristic AC harmonics of an ideal twelve-pulse converter are:

A. 6k

B. 6k ± 1 only

C. 12k ± 1

D. 12k

Correct Answer: C. 12k ± 1

Question 3

In a grid-voltage-aligned dq reference frame, active power is primarily controlled by:

A. The d-axis current

B. The q-axis voltage only

C. The zero-sequence current

D. The DC-line resistance

Correct Answer: A. The d-axis current

Question 4

For a monitored bus m and remote fault f, the appropriate fault-voltage relation is:

A. V_m = Z_load I_load

B. V_m = V_m0 − Z_mf I_f

C. V_m = V_m0 + Z_load I_load for every fault

D. V_m = 0 for every remote fault

Correct Answer: B. V_m = V_m0 − Z_mf I_f

Question 5

If the positive-sequence voltage is 1.0 pu and the negative-sequence voltage is 0.02 pu, the voltage-unbalance factor is:

A. 0.02%

B. 0.2%

C. 2%

D. 20%

Correct Answer: C. 2%

Question 6

A six-pulse converter has a principal characteristic AC harmonic of which order?

A. 3rd

B. 5th

C. 6th

D. 12th

Correct Answer: B. 5th

Question 7

The principal purpose of an HVDC smoothing reactor is to:

A. Increase AC frequency

B. Reduce DC-current ripple and limit current changes

C. Eliminate all converter losses

D. Provide mechanical inertia to the turbine

Correct Answer: B. Reduce DC-current ripple and limit current changes

Question 8

A STATCOM is connected primarily in:

A. Shunt

B. Series only

C. The DC conductor only

D. The transformer neutral only

Correct Answer: A. Shunt

Question 9

If the fundamental voltage is 200 V RMS and the only harmonic is 10 V RMS, the voltage THD is:

A. 2%

B. 5%

C. 10%

D. 20%

Correct Answer: B. 5%

Question 10

A low SCR generally indicates that the converter is connected to:

A. A strong AC system with very low impedance

B. A weak AC system with lower short-circuit strength

C. An isolated DC load with no AC connection

D. A system with zero harmonic current

Correct Answer: B. A weak AC system with lower short-circuit strength