HVDC Modeling and Analysis in Transient Stability March 18, 2014 Kate Davis, Ph.D. 2001 South First Street Champaign, Illinois 61820 +1 (217) 384.6330 [email protected] http://www.powerworld.com HVDC Lines and Stability HVDC lines can rapidly control the transmitted power so they need to be modeled in transient stability

Rectifier Terminals Inverter Terminals Transient Stability 2014 PowerWorld Corporation 2 HVDC Overview HVDC lines are often configured using a bipolar link Made up of groups of transformers, converters, and filters Modeled as a single record, two-terminal DC line, in transient stability software Each terminal has two converters of equal rating The junction between the converters is grounded Normally, there is no ground current

Transient Stability 2014 PowerWorld Corporation 3 Transient Stability Modeling Reference The most comprehensive book on this type of analysis is by Prabha Kundar and is called Power System Stability and Control published in 1994. Book is too detailed for a classroom textbook, but it is a really great as a reference book once youre working Covers HVDC Transmission Transient Stability 2014 PowerWorld Corporation 4

Three-Phase Bridge Converter Converters control the power flow through the HVDC link Refer to any power electronics book For this analysis, assume Ideal AC voltage source in series with inductance Direct current Id is ripple free due to smoothing reactor Ideal switches Transient Stability 2014 PowerWorld Corporation 5 Simplified Bridge Model Switches are labeled in firing order Transient Stability 2014 PowerWorld Corporation

6 Bridge Firing Order + + +

Va is more positive than Vb and Vc, 1 conducts Vb is more negative than Vc and Va, 6 conducts Output voltage is eab Transient Stability 2014 PowerWorld Corporation 7 Bridge Firing Order

60<<0 + + + Va is more positive than Vb

and Vc, 1 conducts Vc is more negative than Va and Vb, 2 conducts Output voltage is eac Transient Stability 2014 PowerWorld Corporation 8 Bridge Firing Order 0<<60

Transient Stability + + + 2014 PowerWorld Corporation 9

Bridge Firing Order 60<<120 + + + Transient Stability 2014 PowerWorld Corporation

10 Bridge Firing Order 120<<180 Transient Stability + +

+ 2014 PowerWorld Corporation 11 Bridge Firing Order 180<<240 +

+ + Transient Stability 2014 PowerWorld Corporation 12 Bridge Firing with a Delay Angle Firing can be delayed by , causing current to lag the voltage

0 Rectified DC voltage, no firing angle = = 0 0 < <180 Transient Stability Rectified DC voltage, firing angle 2014 PowerWorld Corporation 13 Power Factor Angle

As varies, the phase displacement between the voltage and current also varies 0<<90, P decreases and Q increases Transient Stability 2014 PowerWorld Corporation 14 Power Factor Angle The converter always draws reactive power from the AC system =90, P is zero, Q is max 90<<180, P increases, Q decreases Transient Stability 2014 PowerWorld Corporation 15

Commutation Overlap The analysis so far assumes there are always exactly two switches on at one time In reality, there is some overlap Denoted by commutation angle 0<<60, typically is 15 to 25 Commutation begins after and ends after extinction angle =+ Source inductances matter here Transient Stability 2014 PowerWorld Corporation 16 Commutation Overlap Voltage Waveforms = 0

Effect of commutation overlap is a voltage drop Transient Stability 2014 PowerWorld Corporation 17 Rectifier vs. Inverter With no commutation overlap, the converter = 0 becomes an inverter at ==90 With commutation overlap, this occurs later, 0 + =0 ( + ) = 2 t =

2 The converter transitions at The direct current rectifier to inverter from the = + Resistances are small, so small changes in voltage can have a large impact on current! Transient Stability 2014 PowerWorld Corporation

18 Summary of Angles Rectifier Inverter Ignition delay angle Overlap angle Extinction angle = Ignition advance angle Extinction advance angle Overlap angle Control of voltage and current (or power) is achieved by controlling the internal voltages and Control firing angles (fast) or tap changers (slow) Typically the rectifier maintains the DC current and the inverter maintains the DC voltage Transient Stability

2014 PowerWorld Corporation 19 Generic PowerWorld Converter Model ACBase DCBase Eac 1 :TR Pac X c , Rc commutatin g reactance and resistance , Firing Angle Vdc dc system

tap 1 : 1 FixedTap DCBase ACBase * TR Qac Eac V perunitac ACBase * TR tap FixedTap 1 I dc V perunitac DCBase tap FixedTap 1

Fixed parameters and setpoints of the converter Nr , Ni Rc , X c ACBase, DCBase TR FixedTap step Setpoint Transient Stability number of bridges at rectifier/inverter commutating resistance and reactance in Ohms per phase base AC voltage and DC voltage in kV ratio of DCBase to ACBase fixed AC tap step size for the variable AC tap Each converter has a single setpoint value. This value depends on the mode of the converter. The mode is either Current a desired current in Amps

Power a desired power in MW Voltage a desired DC voltage in kV Note that there can be only one voltage controlling converter 2014 PowerWorld Corporation 20 Generic Converter Model Limiting parameters of the converter tap min ,tap max min , max , min , max I max PF Margin Minimum/maximum values for the variable AC tap Minimum/maximum firing angle limits at converter Maximum current allowed through the converter Converter participation factor Rectifier margin (only used at rectifier converters).

Variables calculated in AC power balance equations Pac Qac V perunitac AC real power injection at the converter AC reactive power injection at the converter Voltage in the AC system in per units. Variables calculated in DC system Vdc , I dc , tap Transient Stability DC system voltage and current in kV and kAmps firing angle at rectifier/inverter

Variable AC tap 2014 PowerWorld Corporation 21 Bridge Equivalent Circuits Transient Stability 2014 PowerWorld Corporation 22 Summary of Equations Transient Stability 2014 PowerWorld Corporation 23

Generic DC Network A completely general multi-terminal DC network can be modeled Challenge is fault clearing Transient Stability 2014 PowerWorld Corporation 24 Generic DC Network Each converter is given a setpoint: voltage, current, or power Only one bus can have a voltage set point ndc V V f k V Pdcsetk Vk I dcsetk Vk m k 0 m 1

Solve using Newtons method ndc V 2Vk f k V I dcsetk m Vk Rmk m1 0 Rmk f k V Vk Vm Rmk

Assumes unconstrained operation Then account for limits and update AC equations Transient Stability 2014 PowerWorld Corporation 25 MULTI-TERMINAL DC LINE DYNAMIC MODEL FOR THE PACIFIC DC INTERTIE Transient Stability 2014 PowerWorld Corporation 26 PDCI Model Total leaving Rectifier

3104 MW Total leaving Inverters -2734 MW Total leaving BIG EDDY(40111) 1110 MW Each line 555 MW BIG EDDY 40111 BIG EDDY 1,3 41341, 41343 Each line 997 MW Total leaving SYLMAR S(24147) -978 MW CELILO1 41311

CELILO2 41312 CELILO3 41313 SYLMAR1 26097 CELILO3P 3 (DC) SYLMAR3P 4 (DC) DC Line Setpoint 1552 MW each CELILO4P 5 (DC) SYLMAR4P 6 (DC)

CELILO4 41314 Total leaving BIG EDDY(41341 and 41343) 1994 MW LEGEND Transient Stability SYLMAR2 26099 SYLMAR S 24147 SYLMAR3 26098 SYLMARLA 26094 SYLMAR4

26100 Each line -878 MW Total leaving SYLMARLA(26094) -1756 MW (Note: 41341 are tied by zero-impedance branch, so basically they are the same node) 230 kV Bus Each line -489 MW 500 kV Bus Now only one converter here AC/DC converter

symbol 2014 PowerWorld Corporation 27 PDCI Model Documentation Existing text files contained the actual user-defined model implementation of the PDCI model (pdci_ns3.p and pdci_sn3.p) Dmitry Kosterev provided partial documentation describing the pdci_sn3.ps file. Most helpful for the newer CELILO 500 kV converters which had recently been upgraded PowerWorld took the actual 1,300 lines of code in the pdci_ns3.p file and determined the block diagram being modeled for this important device (also looked through pdci_sn3.p) The pdci_ns3.p code encompasses a model for controlling the firing angle on two converters at Celilo and one converter at Sylmar Transient Stability

2014 PowerWorld Corporation 28 PowerWorld Implementation and User Experience Code Implementation Simulators internal code has been written to make the interaction of the dynamic multi-terminal DC line model and converters with the network boundary equations generic This will make adding new DC line models much easier Also will permit the creation of an interface to a user-defined multi-terminal DC line model For immediate use, the user need only check a box asking that this model be used Simulator will look for the PDCI in the case and automatically include the dynamic model if appropriate All parameters of the model are hard-coded then Version 17 allows the user to explicitly add the dynamic models and also

see the internal states of these models if desired All parameters of the model will remain hard-coded for model May change this eventually if desired Transient Stability 2014 PowerWorld Corporation 29 Implementation Overview in Simulator Assign dynamic model MTDC_PDCI to the multi-terminal DC Line record Assign appropriate dynamic converter models to the various DC converters: CONV_CELILO_E, CONV_CELILO_N, CONV_SYLMAR Transient Stability 2014 PowerWorld Corporation 30

MTDC_PDCI The model is assigned to one Multi-Terminal DC record For the PDCI, two separate MTDC records are modeled, one for each pole of the PDCI Inputs From DC Converters: States of the sensed DC Current, DC Voltage, and AC Voltage at each converter AC Network: Direct access to network boundary equation AC voltages is also used Other MTDC_PDCI: There is some feedback between the two poles in the DC voltage measurement Outputs Feeds a reference current to each DC Converter model: id_ref_CN, id_ref_CE, and id_ref_S Also feeds a flag for VacLow as needed to the CONV_CELILO_N Transient Stability 2014 PowerWorld Corporation

31 MTDC_PDCI: Low Voltage Detection Average DC voltage across both poles of the MTDC Transient Stability 2014 PowerWorld Corporation 32 MTDC_PDCI: Current Order Allocation Transient Stability

Outputs 2014 PowerWorld Corporation 33 MTDC_PDCI: Parameters and Initialization Parameters and initialization depends on flow direction on the PDCI (North to South) or (South to North) Transient Stability 2014 PowerWorld Corporation 34 CONV_CELILO_E Model assigned to the converter at the 230 kV bus at Celilo the Existing Control Inputs

Reference current id_ref_CE from MTDC_PDCI Network boundary equation converter values: Idc, Vdc, Vac Output Cosine of the control angle (Alpha or Beta as appropriate) Transient Stability 2014 PowerWorld Corporation 35 CONV_CELILO_E Transient Stability 2014 PowerWorld Corporation 36

CONV_CELILO_E Parameters and Initialization Parameters and initialization depends on flow direction on the PDCI (North to South) or (South to North) Transient Stability 2014 PowerWorld Corporation 37 CONV_CELILO_N Model assigned to the converter at the 500 kV bus at Celilo the New Control Inputs Reference current id_ref_CN from MTDC_PDCI VacLow from MTDC_PDCI Network boundary equation converter values: Idc, Vdc, Vac

Output Cosine of the control angle (Alpha or Beta as appropriate) Transient Stability 2014 PowerWorld Corporation 38 CONV_CELILO_N Transient Stability 2014 PowerWorld Corporation 39 CONV_CELILO_N Parameters and Initialization Parameters and initialization depends on flow direction on the PDCI (North to South)

or (South to North) Transient Stability 2014 PowerWorld Corporation 40 CONV_SYLMAR Model assigned to the converter at the Sylmar Inputs Reference current id_ref_S from MTDC_PDCI Network boundary equation converter values: Idc, Vdc, Vac Output Cosine of the control angle (Alpha or Beta as appropriate) Transient Stability

2014 PowerWorld Corporation 41 CONV_SYLMAR Transient Stability 2014 PowerWorld Corporation 42 CONV_SYLMAR Parameters and Initialization Parameters and initialization depend on flow direction on the PDCI (North to South) or (South to North) Transient Stability 2014 PowerWorld Corporation

43 Handling of the Interaction with Network Boundary Equations DC Converter equations Written in terms of Alpha at rectifiers Written in terms of Beta at inverters Beta is different than the Gamma traditionally used when writing the static power flow equations Equations are as follows Also force currents to be positive Transient Stability 2014 PowerWorld Corporation 44 DC Network Model

Transient Stability 2014 PowerWorld Corporation 45 What is Different Than a Power Flow Solution? In static power flow solutions, DC converter control is instantaneous Power Flow Firing Angles (Alpha and Gamma) are assumed to move instantaneously PDCI model does not make this assumption, it models the dynamics of the firing angle control Power flow solutions also ignore the inductance of the DC transmission line In power flow, DC currents change instantaneously PDCI models inductance, so DC currents become state variables Note: can also be capacitance in the DC transmission lines We are NOT modeling in the PDCI presently For cable DC lines (underwater for instance), the capacitance may become large

enough that modeling will be important Transient Stability 2014 PowerWorld Corporation 46 Traditional Explicit Numerical Integration Routines Always use the initial algebraic variables to back-solve and obtain the initial values of all dynamic states PowerWorlds transient stability tool then uses explicit integration (2nd order Runga-Kutta) 1. Use numerical integration (with a time-step) to update dynamic state 2. Update algebraic variables such as the AC system voltage and angle (by solving network boundary equations) 3. Go back to 2 and repeat until simulation finished Multi-terminal DC simulation will be inserted between steps 1 and 2

Transient Stability 2014 PowerWorld Corporation 47 Implementation in Numerical Solution Engine of MTDC 1. Numerical integration integrate the MTDC_PDCI, CONV_CELILO_E, CONV_CELILO_N, and CONV_SYLMAR model states Updated variables are cos() and cos() 2. Take the cos() and cos() terms and use them to model a step change in the DC voltages seen by the DC network equations. Use numerical integration to solve for new DC voltages and DC currents Updated variables are DC voltage and DC Currents

3. Solving normal AC network boundary equations except modify DC line equations to assume that cos() and cos() , and DC currents are a constant. When network boundary equations are solved update the DC voltages Updated variables are DC voltages 4. Back to Step 1 and repeat Transient Stability 2014 PowerWorld Corporation 48 Step 2: Numerical Integration of DC Network Equations DC Converter Model Constant angle so model as a constant voltage source

DC Transmission Line Model Model as RL circuit using trapezoidal rule DC Bus Equation Just use Kirchoffs Current Law Transient Stability 2014 PowerWorld Corporation 49 Step 2: Numerical Integration of DC Network Equations Solve the previous set of equations using subinterval integration Assume at beginning that no converters are stuck at the zero current limit At each sub-interval, if the calculation yields a converter current with the wrong sign then redo the sub-interval replacing the DC converter equation with the equation I=0

Assume that once a current goes to zero it remains zero during the remaining sub-interval time-steps Transient Stability 2014 PowerWorld Corporation 50 Step 2: Numerical Integration of DC Network Equations: Matrices The following is an example matrix setup for the PDCI If Celilo1 current becomes negative, then replace equation with the following Transient Stability 2014 PowerWorld Corporation 51

Step 3: Solution of algebraic change in DC voltages DC Converter Model Constant angle and constant current DC Transmission Line Model Model as RL circuit, but current is constant dI/dt is the unknown variable DC Bus Equation Just use Kirchoffs Current Law, but for the derivatives Transient Stability 2014 PowerWorld Corporation 52

Step 3: Solution of algebraic change in DC voltages: Matrices The following is a sample of the matrix setup for the PDCI Transient Stability 2014 PowerWorld Corporation 53 Summary Basics of HVDC operation and modeling PowerWorld models in steady state and in transient stability Brief overview of PDCI example Resources available to find more information Transient Stability 2014 PowerWorld Corporation

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