high-voltage direct-current transmission systems from theory … · 2015-10-12 · high-voltage...

High-Voltage Direct-Current Transmission Systems

� From theory to practice and back

Daniele Zonettia, Romeo Ortegaa, Abdelkrim Benchaibb

aLaboratoire de Signaux et Systèmes - Centrale SupélecbAlstom Grid

Journées de l'Automatique GDR MACS 2015Grenoble, 5-6 October 2015

Outline

1. Introduction: system topology & control architecture

2. Energy�based modeling

3. Power �ow analysis

4. PI controllers:

I Design: a GAS PI passivity�based controller

I Analysis: stability and performance limitations

5. Overcoming performance limitations

6. Conclusions & future research

Multiterminal HVDC Transmission Systems

Multiterminal HVDC transmission system

De�nition. Direct Current (DC) electrical system operating at highvoltage (HV), with no loads. It is constituted by n terminals that areconnected via ` DC transmission lines.

I AC sources connected to the terminals are associated to MVACgrids or Renewable Energy Sources (RES)

I and interfaced through power converters, that:I perform the AC/DC conversion

I regulate the power �ow and/or the voltages

Background and motivation

Popularity of HVDC transmission is due to the following features

I interconnection of non�synchronous AC grids (B2B connection)

I transmission over long distances (reduced losses)I island systems connectionI integration of remotely located RES

but introduces new control problems...

I intermittent nature of RES

I power converters are highly nonlinear elements

I no time�scale separation between transmission and generation

Control architecture: nominal conditions

...and still employs the control structure of traditional AC systems

CENTRALIZEDREFERENCESCALCULATOR

POWERCONVERTER

INNERCONTROL

DESIREDBEHAVIOR

POWERFLOW

minutes

10 seconds-3

I Refs calculator: takes as inputs the desired behavior and thepower �ow and derive the setpoints

I Feasible setpoints (nominal conditions) are stabilized by the in-ner control, under perfect knowledge of the power �ow

Control architecture: perturbed conditions

...and still employs the control structure of traditional AC systems

PRIMARY CONTROL


POWERCONVERTER

INNERCONTROL

DESIREDBEHAVIOR

POWERFLOW

minutes

10 seconds-3

Power �ow uncertainty or contingencies (perturbed conditions) ⇒not feasible setpoints. Primary control adapts refs to

I be feasible and so stabilizable by the inner control

I preserve suitable properties (power sharing,voltage stability)

Outline




4. PI controllers:





Energy�based modeling

Power system as interconnection of single components

I Power converters (C ), that are nonlinear elements

I DC transmission lines (L), as linear π-models

I Interconnection laws described by standard KVL, KCL

Assumptions

A1. Balanced operation of the line voltages

A2. Ideal four�quadrant operation of the converter

A3. Switches can be operated su�ciently fast⇒ Average model of the converter

A4. Synchronized operation of the converters⇒ AC source≡ constant amplitude/frequency voltage source⇒ dq�frame representation≡ constant steady�states


Components modeled as port�Hamiltonian (pH) systems

x = (J (x)−R(x))∇xH(x) + G (x)u + E

y = G>(x)∇xH(x),

with: x the state in physical variables (�uxes,charges), J = −J >(x),R(x) ≥ 0, H(x) the interconnection, damping and energy functions.

Interconnection laws captured by a graph G(V , E ,M)

For a graph G with set of vertices V, edges E and incidence matrixM, KCL and KVL are given respectively by

MIE = 0, M>VV = VE ,

with IE , VE current and voltages of the edges and VV voltages at thevertices (nodes).


n = 3 CONVERTERS: xC ∈ R3n, u ∈ R2n

xC = (JC −RC )∇HC + gC (xC )u

+ EC − GC iC

vC = G>C ∇HC ,

with HC := 1

2x>C QC xC .

` = 2 LINES: xL ∈ R`

xL = −RL∇HL + vL

iL = ∇HL,

with HL := 1

2x>L QLxL.

WF1

WF2

GRID


n = 3 CONVERTERS: xC ∈ R3n, u ∈ R2n

xC = (JC −RC )∇HC + gC (xC )u

+ EC − GC iC

vC = G>C ∇HC ,

with HC := 1

2x>C QC xC .

` = 2 LINES: xL ∈ R`

xL = −RL∇HL + vL

iL = ∇HL,

with HL := 1

2x>L QLxL.

INTERCONNECTION LAWS

M =

[In M−1>n 0>`

]

(KCL) iC +MiL = 0n, −1>n iC = 0

(KVL) v = vC , M>v = vL


Port�Hamiltonian modeling

A multiterminal HVDC transmission system can be modeled as

x =˙[xCxL

]=

[(JR −RC) −GCMM>G>C −RL

]︸︷︷︸

J−R

∇H+

[gC (xC )

0

]︸︷︷︸

g(x)

u +

[EC

0

]︸︷︷︸

E

(1)

I quadratic energy function H := 1

2x>Qx , Q = blockdiag{QC ,QL}

I structured (linear) input matrix g(x) := [JdQx JqQx ]

I topology matrix M issued by the graph G

why dq�frame?

I Constant steady�states ⇒ pure regulation problem

I Control active/reactive power ⇒ control of dq currents

Outline




4. PI controllers:





Passivity as a tool for power �ow analysis

Admissible equilibria

Proposition 1. The admissible equilibria of (1) are the solution of

0 = HC ,i (x?C ,i , x

?L) (2)

x?L = Cx?C . (3)

with C := (RLQL)−1M>G>C QC , i ∈ [1, n].

why is this interesting?

I Lines equilibria linearly depend on converters equilibria

I Eqs.(2) are the power �ow steady�states equations (PFSSE)

I De�nes all steady states achievable by ANY stabilizing controller

Passivity as a tool for power �ow analysis


POWERCONVERTER

INNERCONTROL

DESIREDBEHAVIOR

POWERFLOW

minutes

10 seconds-3

PFSSE can be used to formulate the problem of refs calculation

Outline




4. PI controllers:





Passivity as a tool for inner�loop control design

Control design:

I de�ne the incremental forms

I �nd a passive output for the incremental model

I design the controller over the passive output

Incremental formulation

Let x? an admissible equilibrium point and u? the equilibrium control,i.e.

(x?, u?) : (J −R)Qx? + g(x?)u? + E = 0.

We de�ne then

x := x − x?, u := u − u?, H(x) := 1

2x>Qx ,

i.e. the errors and the correspondent (incremental) energy function.


A passive output for the incremental model

Proposition 2. Consider the incremental model of (1). The map-ping u → y is passive with storage function H(x), with

y := g>(x?)Qx .

why is important to �nd a passive output?

I the feedback interconnection of two passive systems is passive

I the system can be controlled with PIs


Decentralized PI Regulation

Proposition 3. Let an assignable reference x?, verifying the PFSSE.Then the PI controller

z = −y , u = −KPy + KI z ,

globally asymptotically regulates the system (1) to (x?C , Cx?C ) for anypositive gains KI , KP .

I PI control: simple, easily implementable, intrinsically robust

I Blockdiag matrices for the gains guarantee decentralization

Outline




4. PI controllers:





Other controllers reported in literature

Power converter states: dq�currents (id , iq), DC voltage vC

1. PI passivity�based controller based on the output

yd := v?C id − i?dvC yq := v?C iq − i?qvC

2. PI dq-currents controller based on the output

y Id := id − i?d , y Iq := iq − i?q

3. PI voltage controller based on the output

yVd := vC − v?C , yVq := iq − i?q ,

Simulations: a three�terminal benchmark

WF1

WF2

GRID

CONTROL OBJECTIVES

I All terminals are required to keep reactive power near to zero

I Terminals WF1, WF2 regulate the injected (absorbed) active power

I The remaining terminal regulates the DC voltage (slack bus, SB)

Simulations: a three�terminal benchmark

WF1

WF2

GRID

SIMULATED SCENARIO: From 0 to 5T s, with changes every T s

SB WF 1 WF 2 SB WF 1 WF 2

0 −1260 900 1000 100 142.595 158.951T −1588 900 1800 100 153.650 179.6912T −266 500 −200 100 109.004 104.0043T 905 −400 −200 100 69.419 60.8774T −849 1300 −200 100 128.708 124.532

highly stressed scenario � many power reversals

Simulations: standard PI controllers

Setpoint changes every T = 4 s.

0 5 10 15 20−5000

0

5000

I d (A

)

SLACK BUS (SB)

0 5 10 15 20500

1000

1500WIND FARM 1 (WF1)

0 5 10 15 20500

1000

1500WIND FARM (WF2)

0 5 10 15 20−2

0

2

4x 10

6

i q (A

)

0 5 10 15 20−10

−5

0

5

10

0 5 10 15 20−10

−5

0

5

10

0 5 10 15 20−1

−0.5

0

0.5

1

x 105

time (s)

v C (

V)

0 5 10 15 20

1

2

3

4

5x 10

5

time (s)0 5 10 15 20

0

1

2

3

4

5x 10

5

time (s)

Good performances but instability arises under full power �ow reversal

Performance limitations analysis: standard PI controllers

Zero dynamics analysis � Current and voltage outputs y I , yV

Proposition 4. The zero dynamics of the single converter wrt tothe output y I and output yV are given by the scalar system

ζ = −βζ + α

ζ− γ, β, γ > 0.

I the system admits two equilibria, but only one is physicallymeaningful

I If α > 0, i.e. the converter is injecting power the equilibrium isstable

I If α < 0, i.e. the converter is absorbing power the equilibriumis unstable

⇒ zero dynamics become unstable when the power �ow is reversed

Simulations: passivity�based controllers


0 2000 4000 6000 8000 10000−2000

−1000

0

1000

2000

i d (A

)

SLACK BUS (SB)

0 2000 4000 6000 8000 10000−1000

0

1000


0 2000 4000 6000 8000 10000−1000

0

1000


0 2000 4000 6000 8000 10000−0.02

0

0.02

0.04

i q (A

)

0 2000 4000 6000 8000 10000−0.02

−0.01

0

0.01

0.02

0 2000 4000 6000 8000 10000−5

0

5

10

15x 10

−3

0 2000 4000 6000 8000 100000.5

1

1.5

x 105

time (s)

v C (

V)

0 2000 4000 6000 8000 100000.5

1

1.5

2x 10

5

time (s)0 2000 4000 6000 8000 10000

0.5

1

1.5

2x 10

5

time (s)

Stable for any scenario (as expected) but unacceptably slow transients

Performance limitations analysis: passivity�based controllers

Zero dynamics analysis � Passive output y

Proposition 5. The zero dynamics of the single converter wrt to theoutput y is linear and exponentially stable and converge at a rate

λ :=1

2

P? − P?dc

H(x?C ).

I λ ≈ 0.04 for HV power converters

I analogous to have a slow zero in linear systems

I increasing the gains, one pole of the closed�loop system is at-tracted by the slow zero

Outline




4. PI controllers:





Adding a linear feedback to the passivity�based inner loop

Decentralized PI Regulation plus linear feedback

Proposition 6. Let an assignable reference x?, verifying the PFSSE.Then the controller

z = −y , u = −KPy + KI z − KLQ(x − x?),

globally asymptotically regulates the system (1) to (x?C , Cx?C ) for anypositive gains KI , KP , KL verifying

R+ g(x?)KPg>(x?) +

1

2

[g(x?)KL + K>L g>(x?)

]> 0.

I the additional feedback a�ects only the P�part of the controller

I d.o.f. added to render the Lyapunov derivative negative de�nite

I the condition can be intended as a constraint for tuning of the gains

Adding a linear feedback to the passivity�based inner loop

Inspired by the ubiquitous voltage droop control, we choose KL

such that[uduq

]=

[−kPdyd + kIdzd − kD(vC − v?C )

−kPqyq + kIqzq.

],

˙[zdzq

]=

[−yd−yq

],

i.e. we introduced an additional linear feedback in the voltage error

⇒ ∃ gains kP , kI , kL that ensure

I GAS of the equilibrium point

I decentralization of the controller

I to speed up convergence to ms

I drawback: in contrast with conventional droop, stability is lostin perturbed conditions

Simulations: passivity�based control plus linear feedback


0 2 4 6 8 10−1000

0

1000

2000WIND FARM (WF1)

0 2 4 6 8 10−1000

0

1000


0 2 4 6 8 10−0.02

0

0.02

0.04

i q (A

)

0 2 4 6 8 10−0.02

−0.01

0

0.01

0.02

0 2 4 6 8 10−0.01

0

0.01

0.02

0 2 4 6 8 100.5

1

1.5

2x 10

5

time (s)0 2 4 6 8 10

0.5

1

1.5

2x 10

5

time (s)

0 2 4 6 8 10−2000

−1000

0

1000

2000

i d (A

)

SLACK BUS (SB)

0 2 4 6 8 100.5

1

1.5

2x 10

5

time (s)

v C (

V)

Stable for any scenario with good performances

Outline




4. PI controllers:





Conclusions

I uni�ed framework for modeling using a pH representation

I power �ow analysis (assignable behavior)

I PI passivity�based control (PI�PBC) design

I performance and stability analysis of standard PI controllers

I modi�ed PI�PBC to improve performances

[Zonetti, D., Ortega, R., Benchaib, A. Modeling and control of HVDC transmissionsystems � From theory to practice and back. Control Engineering Practice (2015)]

Future research

I extend to lines modeled by PDEs

I include AC dynamics to explore AC/DC interactions

I design a new GAS primary controller

I behind the PI�PBC: adaptive control?

I as a whole with the inner loop⇒ stabilization to an unknown operating point

I extend the theory to DC distribution networks interfaced to ACgrids ⇒ including loads

in partnership with FREEDM Center at

North Carolina State University �

http://www.freedm.ncsu.edu/

Merci.

high-voltage direct-current transmission systems from theory … · 2015-10-12 · high-voltage...

Documents