Non-linear Control Theory and Applications in Power and Energy

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Non-linear Control Theory and Applications in Power and Energy Systems

George Konstantopoulos Control and Power Systems Dept. of Automatic Control and Systems Engineering The University of Sheeld United Kingdom

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Power systems

Increase of renewable energy systems penetration Connection to the electrical grid through power electronic devices (converters) Power grid stability becomes fragile! Conventional control techniques and linear systems analysis enough

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems



not

Control in Power Systems Power electronics have high eciency. Power Grid stability lies on the system operation and control. Control

Bridge

Power

The need to create a bridge between 'automatic control' and 'power systems' communities Accurate modelling of power systems Advanced control design Passivity - Stability Non-linear control theory is essential!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Control in Power Systems Power electronics have high eciency. Power Grid stability lies on the system operation and control. Control

Bridge

Power

The need to create a bridge between 'automatic control' and 'power systems' communities Accurate modelling of power systems Advanced control design Passivity - Stability Non-linear control theory is essential!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Control in Power Systems Power electronics have high eciency. Power Grid stability lies on the system operation and control. Control

Bridge

Power

The need to create a bridge between 'automatic control' and 'power systems' communities Accurate modelling of power systems Advanced control design Passivity - Stability Non-linear control theory is essential!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

DC/DC boost converter u=0

L i

u=1

E

+ v

C

R

-

PV, energy storage, wind, etc. Regulation of the dc output voltage v to a higher level than the dc input voltage E by controlling the switch u .

Applications: Aim:

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Average model Average modelling (u → µ ) high switching frequency duty-ratio µ = tTon

Li˙ = −(1 − µ)v + E 1

C v˙ = (1 − µ)i − v R where the control input µ is continuous-time and µ ∈ [0, 1].

The system is non-linear! George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

AC/DC or DC/AC converter S1

D1

d1

Ua

L

R

Ia

Ub

L

R

Ib

Uc

L

R

Ic

S3

D3

d3

S5

D5

d5

Va Vb

C

Vdc

Vc S2

D2

d2

S4 d4

D4

S6

D6

d6

Average modelling + Park transformation (a − b − c → George Konstantopoulos (The University of Sheeld)

d

− q)

Non-linear Control in Power Systems

RL

Average model Li˙d = −Rid + ωs Lid − md Li˙q = −Riq − ωs Liq − mq C V˙ dc =

Vdc 2

Vdc 2

+ Ud + Uq

3 V (md id + mq iq ) − dc 4 RL

external uncontrolled inputs: grid voltages Ud , Uq (constants, usually Uq = 0) control inputs: duty-ratio signals md = 2VVdcd , mq = 2VVdcq

In order to operate the converter in the 'linear modulation' (PWM): md2 + mq2 ≤ 1 George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Existing control techniques Traditional techniques: linearisation (small-signal model) linear control (PI) Advanced techniques: non-linear techniques (passivity-based, feedback linearisation) → asymptotic stability dependence from system parameters Can we nd a non-linear parameter-free controller with guaranteed closed-loop stability? George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Generalised dissipative Hamiltonian form DC/DC boost converter:

Li˙ = −(1 − µ)v + E 1

C v˙ = (1 − µ)i − v R → M x˙ = (J (µ) − R ) x + Gu  v T , µ is the control input and

where x = i u = E is the external uncontrolled input.     L 0 0 −(1 − µ) M = 0 C , J (µ) = (1 − µ) ,R = 0    T 0 0 ,G = 1 0 1 0 R 

The system is passive w.r.t. the external uncontrolled input u independently from µ→Dynamic controller?

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Non-linear control design M x˙ = (J (x , µ) − R ) x + G (x )u Passive w.r.t. the constant uncontrolled input u Control input µ is a saturated signal in the interval (−1, 1) or [0, 1) Control task: Regulate state xi at xiref . Oscillator-based non-linear controller: 

µ = z1 + c    z˙1 = 0 −k (xi − xiref ) z˙2 k (xi − xiref ) 0

z1 z2

 ,

k >0

µ ∈ (−1, 1): c = 0, z1 (0) = −1 + γ , z2 (0) = 0 1−γ µ ∈ [0, 1): c = 1−γ 2 , z1 (0) = − 2 , z2 (0) = 0 George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Non-linear control design M x˙ = (J (x , µ) − R ) x + G (x )u Passive w.r.t. the constant uncontrolled input u Control input µ is a saturated signal in the interval (−1, 1) or [0, 1) Control task: Regulate state xi at xiref . Oscillator-based non-linear controller: 

µ = z1 + c    z˙1 = 0 −k (xi − xiref ) z˙2 k (xi − xiref ) 0

z1 z2

 ,

k >0

µ ∈ (−1, 1): c = 0, z1 (0) = −1 + γ , z2 (0) = 0 1−γ µ ∈ [0, 1): c = 1−γ 2 , z1 (0) = − 2 , z2 (0) = 0 George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Controller operation z2

B  1− γ  Α − ,0 2  

1− γ Aμ=Ο

μj* μ*

a-γ

z1, μ

W(0) Wμ(0)

→ the control input µ is bounded in [0, 1 − γ]! George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Closed-loop system stability Closed-loop system:   ˜ ˜ ˜ ˜u ˙ Mx ˜ = J (˜ x) − R x ˜+G

with x˜ =

 T x 

M˜ = 

z1

M

0

0

0

1

0

0

0

1



J

J˜(˜x ) =  0

z2

T



.



R00

˜ =0 , R



0 0,

G˜ =



1

0

0

T

,

0 0 0 0 0

ref ) 0 k (xi − xi

0



−k (xi − xiref )  0

We keep the same passive structure! George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Closed-loop system stability The converter system and the controller system can be handled independently: converter system with µ ∈ [0, 1 − γ] is bounded input-bounded output (BIBO) stable. controller system is BIBO with zero gain! (bounded output independently from the input) →closed-loop system is BIBO w.r.t. the external input u . Since u is constant, then the closed-loop system solution is bounded in an area where the desired equilibrium exists! Unique equilibrium, no limit cycles → convergence to the equilibrium!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Bounded integral controller (BIC) Traditional Integral controller for regulating a scalar function g (x ) to zero: Z

u (t ) =

t

0

g (x (τ)) d τ

which introduces a dynamic controller that can be written as

u w˙

= =

w g (x )

Bounded Integral Controller (BIC): u=w

    2 w 2 + (wq −b) −   − k g ( x ) c 2 2   umax ε w˙   w = 2   2 2 w˙ q wq (wq −b) − u 2ε g (x )c −k uw + − 2 ε2



1



max

b ≥ 0, k , umax , ε, c > 0.

George Konstantopoulos (The University of Sheeld)

max

1

Non-linear Control in Power Systems

Stability of ISpS plant systems Consider a non-linear system: x˙ = f (t , x , u , u1 ) u1

plant

w

BIC

x

g(x)

Proposition The feedback interconnection of plant system with the proposed BIC is ISpS, when the plant system is ISpS. ISpS: Input-to-state practical stability The BIC introduces a zero-gain property.

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

BIC with given bound u ∈ [−u

max

,u

max

2

umax Normal conditions: b = 0, ε = 1, k > 0, c = u2wq −( 2 max u ∗ ) ∗ where u is the nominal value of u . Abnormal conditions: b = c = 1, k > 0 (suciently large), ε > 0 (suciently small)

1+ε 1 1−ε

θɺ

Ο

u*

C

wq

wq 1 wq*

umax w

W0

0

−umax

0

umax

w

Normal conditions George Konstantopoulos (The University of Sheeld)

Abnormal conditions Non-linear Control in Power Systems

]

Integrator replacement Bounded integral control (BIC): output

g(x)

upper bound

⇒ bounded output

g(x)

steady-state

lower bound

BIC

normal conditions: slow down the integration near the bounds abnormal conditions: fail-safe operation George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

DC/DC boost converter control L i E

u=0 u=1

+ v

C

R

-

Control task: Regulate the output voltage v at vref . Using the oscillator-based non-linear controller: 1−γ µ = z1 + 2      z˙1 = 0 −k (v − vref ) z1 z˙2 k (v − vref ) 0 z2 γ small positive constant k >0 initial conditions z1 (0) = − 1−γ 2 , z2 (0) = 0 George Konstantopoulos (The University of Sheeld) Non-linear Control in Power Systems

Results simulation experiment

24 22

output voltage (V)

20

LabVIEW

18 16 14

NI hardware 12

dc power supply

dc/dc boost converter

R-L load

10 0

0.5

1

1.5

2 time (sec)

2.5

1 0.8 X: 0.4

0.6

B Y: 0.4858

0.4

z2

0.2 0

Aμ=Ο

-0.2 -0.4 Wμ(0)

-0.6 -0.8 -1 -1

-0.5

0

0.5

1

z1

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

3

3.5

4

Wind Power Application ZLQG WXUELQH

JHQHUDWRUVLGHFRQYHUWHU

S JHDUER[

D

S

D

D

Vas Vbs

Vdc Vcs

S

Sa



6&,*

D

S

D

S

JULGVLGHFRQYHUWHU

i

S

D

Sb

D

Sc

Vag Vbg

C Rdc

Vcg

 Sa

D

D

Sb

D

Sc

D

XWLOLW\JULG

ia

Lg

Rg

Va

ib

Lg

Rg

Vb

ic

Lg

Rg

Vc

D

Generator-side converter control: Maximum Power Point Tracking (MPPT) Field-oriented control (FOC) of SCIG Grid-side converter control: dc-link bus voltage regulation unity power factor (

Qg = 0)

Traditional control techniques: PI and cascaded PI control (complete system stability has not been investigated yet)

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Wind Power Application ZLQG WXUELQH

JHQHUDWRUVLGHFRQYHUWHU

S JHDUER[

D

S

D

D

Vas Vbs

Vdc Vcs

S

Sa



6&,*

D

S

D

S

JULGVLGHFRQYHUWHU

i

S

D

Sb

D

Sc

Vag Vbg

C Rdc

Vcg

 Sa

D

D

Sb

D

Sc

D

XWLOLW\JULG

ia

Lg

Rg

Va

ib

Lg

Rg

Vb

ic

Lg

Rg

Vc

D

Generator-side converter control: Maximum Power Point Tracking (MPPT) Field-oriented control (FOC) of SCIG Grid-side converter control: dc-link bus voltage regulation unity power factor (

Qg = 0)

Traditional control techniques: PI and cascaded PI control (complete system stability has not been investigated yet)

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Wind power system modelling Complete wind power system modelling: SCIG dynamics + ac/dc/ac converter dynamics

M x˙ = J x , mds , mqs , mdg , mqg



−R



x +u

where x = ids iqs λdr λqr ωr id iq Vdc T is the T  is the state vector, u = 0 Vm 0 0 − 23 Tm 0 0 0 uncontrolled external input vector. 



Generator-side converter control inputs: mds , mqs Grid-side converter control inputs: mdg , mqg

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Wind power system modelling M = diag

 σ,σ,

1 1 2

2

Lr , Lr , 3 Jm , Lg , Lg , 3 C

Rr L2m + R s L2r 

0



R

       =        

0

− Rr L2m

Lr

0 0 0 0 0 0

  −σ ωe     J =  Lm  − Lr p λqr     1m −2 ds

0 0 0 0

Rr L2m L2r

0

+ Rs

σ ωe

Lm p λ Lr dr

0

− Rr L2m

0 0 0 0 0

Rr L2r

Lr

0

Rr L2r

0 0 0 0

− 2 mqs

0

Lr

Lr

0 0 1

,

− Rr L2m

− Rr L2m

0 0 0



0 0 0

0 0

−p ωr − ωe L

0 0 0 0

0 0 0 0

ωe −p ωr

r

George Konstantopoulos (The University of Sheeld)

Lr

0 0 0 0 0

0 0 0 0

2b 3

0 0 0

0 0 0 0 0 Rg 0 0

Lm p λqr Lr − LLm p λdr r

0 0 0 0 0 0

0 0 0 0 0 0 Rg 0 0 0 0 0 0 0

−ωs Lg

1 2 mdg

0 0 0 0 0 0 03



2Rdc

        ,       

0 0 0 0 0

ωs L g

0 1

2 mqg

Non-linear Control in Power Systems

1m 2 ds 1m 2 qs

0 0 0

1m −2 dg 1m −2 qg

0

              

Wind power system control How can we control the complete wind power system to achieve the desired operation and guarantee stability? → use the same oscillator-based non-linear controller for the duty-ratio inputs as in the case of the dc/dc boost converter The limits of the duty-ratio signals are not independent: m

2 2 ds + mqs ≤ 1,

George Konstantopoulos (The University of Sheeld)

m

2 2 dg + mqg ≤ 1

Non-linear Control in Power Systems

Wind power system control How can we control the complete wind power system to achieve the desired operation and guarantee stability? → use the same oscillator-based non-linear controller for the duty-ratio inputs as in the case of the dc/dc boost converter The limits of the duty-ratio signals are not independent: m

2 2 ds + mqs ≤ 1,

George Konstantopoulos (The University of Sheeld)

m

2 2 dg + mqg ≤ 1

Non-linear Control in Power Systems

Wind power system control How can we control the complete wind power system to achieve the desired operation and guarantee stability? → use the same oscillator-based non-linear controller for the duty-ratio inputs as in the case of the dc/dc boost converter The limits of the duty-ratio signals are not independent: m

2 2 ds + mqs ≤ 1,

George Konstantopoulos (The University of Sheeld)

m

2 2 dg + mqg ≤ 1

Non-linear Control in Power Systems

Wind power system control Generator-side converter control: MPPT: regulate rotor speed ωr to ωrref ref FOC: regulate d −axis stator current ids to ids

ds = mqs =

m

 

z˙1 z˙2 z˙3





 =

 k1

0

k1

0

 ids − idsref k2

z1 z2

 ref  −k1 ids − ids  0 −k2 ωr − ωrref     ref ωr − ωr −c1 z12 + z22 + z32 − 1 0

, k2 are non-negative constants

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

z1 z2 z3

 

Wind power system control The 3 controller states are attracted and exclusively move on  the surface of the sphere Cr 1 = z1 , z2 , z3 : z12 + z22 + z32 = 1 .

0.5

A

Cr 1

1 z3

disk D 0

-0.5 0.5

0.

z2

0

0 -0.5

z1

-0.5

The duty-ratio inputs mds = z1 and mqs = z2 take values 2 + m 2 ≤ 1. inside the disk D , i.e. mds qs

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Real-time simulation results ωr: [0.2pu/div]

iqs: [0.5pu/div]

iq: [0.5pu/div] ωrref: [0.2pu/div]

ids: [0.5pu/div] id: [0.5pu/div]

Vdc: [0.25pu/div] vga: [1pu/div] Pg: [0.5pu/div]

Pwind: [0.5pu/div] Qg: [0.5pu/div]

George Konstantopoulos (The University of Sheeld)

iga: [1pu/div]

Non-linear Control in Power Systems

Power network - microgrid A power network has a large number of buses:

The inverters can be assumed as voltage sources to simplify the analysis.

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Power network - microgrid

Is non-linear control theory important in this case? Basic control structure: Droop control (non-linear structure of P and Q ) Accurate load modelling (linear, non-linear) → Linearisation and local stability is not enough

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Power network - microgrid

Is non-linear control theory important in this case? Basic control structure: Droop control (non-linear structure of P and Q ) Accurate load modelling (linear, non-linear) → Linearisation and local stability is not enough

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Parallel operation of inverters i1

R1

L1

L2

vo

R2

i2

vr1

rC1

C1

load

iL C2

rC 2

vr 2

Proportional load sharing: Conventional droop control (i ∈ {1, 2}):

Ei = E ∗ − ni Qi θ˙i = ω ∗ − mi Pi =⇒



vri = 2Ei sin (θi )

cannot achieve accurate load sharing when inverters introduce dierent output impedances. George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Robust Droop Controller i1

R1

L1

L2

vo

R2

i2

vr1

rC1

C1

load

iL C2

rC 2

vr 2

Robust Droop Controller (RDC):

E˙i = Ke (E ∗ − Vo ) − ni Qi θ˙ i = ω ∗ − mi Pi accurate load sharing output voltage regulation near E ∗ Linear/non-linear plant + non-linear controller =⇒Is the closed-loop system stable? George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Robust Droop Controller i1

R1

L1

L2

vo

R2

i2

vr1

rC1

C1

load

iL C2

rC 2

vr 2

Robust Droop Controller (RDC):

E˙i = Ke (E ∗ − Vo ) − ni Qi θ˙ i = ω ∗ − mi Pi accurate load sharing output voltage regulation near E ∗ Linear/non-linear plant + non-linear controller =⇒Is the closed-loop system stable? George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Bounded droop controller (BDC) Bounded droop controller (BDC): RMS dynamics: ˙i = (Ke (E ∗ − Vo ) − ni Qi ) cE qi

E

˙ qi = − (Ke (E ∗ − Vo ) − ni Qi ) cEi

E

where c is a positive constant. frequency dynamics: ˙i = (ω ∗ − mi Pi ) z qi , z ˙qi = − (ω ∗ − mi Pi ) zi . √ =⇒ vri = 2Ei zi z

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

BDC operation RMS voltage dynamics Eqi

frequency dynamics zqi

starting point

Eqi 0 desired steady-state equilibrium point

φi Vi ȝj*

ȅ

starting point

1

θi

a-Ȗ E * i

Ei

ȝj*

ȅ

rotating with angular velocity equal to θɺi

a-Ȗ

zi

Wi 0

φ˙i = (Ke (E ∗ − Vo ) − ni Qi ) c

θ˙i = ω ∗ − mi Pi

i , Eqi ∈ [−Vi , V√ i] √ ∈ [−  √ zi , zqi  1, 1] Therefore vri = 2Ei zi ∈ − 2Vi , 2Vi ⇒ BDC

E

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Real-time simulations (non-linear load) BDC

RDC

P2: [400W/div]

P2: [400W/div]

P1: [400W/div]

P1: [400W/div]

Q1: [400Var/div]

Q1: [400Var/div]

Q2: [400Var/div]

Q2: [400Var/div]

vo: [200V/div]

vo: [200V/div]

i1: [10A/div]

i1: [10A/div]

i2: [10A/div]

i2: [10A/div]

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Real-time simulations (non-linear load) BDC Eq1: [200V/div] E1: [200V/div]

zq1: [1/div]

z1: [1/div]

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Grid-friendly inverters (Synchronverters) (θ = 0 )

Rotor field axis

Rs , L Rotation

M

M

N

Field voltage Rs , L

Rs , L

M

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Synchronverter parts

Power Part

+ Rs, Ls VDC

ea eb ec

Rg, Lg

Circuit Breaker

ia

va

ib

vb

vgb

ic

vc

vgc

vga

C -

Control Part

R a

v

b

v

c

v

a DC b

ωr

c

-

θg

Reset Pset

p θɺ

Tm

n

ω

1 Js

-

R

Te a DC

1 Ks

Qs M f if

e i

Vn

b PWM generation c

c

-

Dq

George Konstantopoulos (The University of Sheeld)

θc a

Formulas of Te, Q, e b

Qset

-

θ

1 s

Vg

From\to the power part

Dp

Non-linear Control in Power Systems

Synchronverter control and stability Stability is still not proven! Can we guarantee system stability while maintaining the synchronverter original operation? Solution: Use BIC for the integrators of the frequency ω and the eld-excitation current if loops. Guarantee specic bounds for the synchronverter voltage and frequency according the its technical limits: E

∈ [Emin , Emax ] and ω ∈ [ωn − ∆ωmax , ωn + ∆ωmax ]

Guarantee convergence to the desired unique equilibrium

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Synchronverter control and stability Stability is still not proven! Can we guarantee system stability while maintaining the synchronverter original operation? Solution: Use BIC for the integrators of the frequency ω and the eld-excitation current if loops. Guarantee specic bounds for the synchronverter voltage and frequency according the its technical limits: E

∈ [Emin , Emax ] and ω ∈ [ωn − ∆ωmax , ωn + ∆ωmax ]

Guarantee convergence to the desired unique equilibrium

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Real-time simulation results Ps: [50W/div]

Qs: [500Var/div]

fg-fn: [0.1Hz/div]

f-fn: [0.1Hz/div]

Time: [5s/div]

Time: [5s/div]

va: [10V/div] E: [0.5pu/div]

V-Vg: [0.5V/div]

iga: [10A/div]

Time: [5s/div]

George Konstantopoulos (The University of Sheeld)

Time: [20ms/div]

Non-linear Control in Power Systems

Completely Autonomous Power Systems (CAPS)

Big step for guaranteeing stability of CAPS! George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Closed-loop system stability A fundamental problem in control systems is that closed-loop system stability is not guaranteed even if the plant is BIBO. Bounded?

reference +

error Controller

controller output

BIBO Plant

plant output

-

Sensor

The BIC can guarantee a bounded input for the plant independently from the error signal (zero-gain property). However, a given bound for a state of the plant (e.g. current) is not guaranteed. Can we design such a controller?

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Closed-loop system stability A fundamental problem in control systems is that closed-loop system stability is not guaranteed even if the plant is BIBO. Bounded?

reference +

error Controller

controller output

BIBO Plant

plant output

-

Sensor

The BIC can guarantee a bounded input for the plant independently from the error signal (zero-gain property). However, a given bound for a state of the plant (e.g. current) is not guaranteed. Can we design such a controller?

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Closed-loop system stability A fundamental problem in control systems is that closed-loop system stability is not guaranteed even if the plant is BIBO. Bounded?

reference +

error Controller

controller output

BIBO Plant

plant output

-

Sensor

The BIC can guarantee a bounded input for the plant independently from the error signal (zero-gain property). However, a given bound for a state of the plant (e.g. current) is not guaranteed. Can we design such a controller?

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Single-phase rectier with limited current i

vs

r

L u

u

u

u

+ v -

C

di dt dV CVdc dc dt L

RL

-

= −ri − v + vs =

vi −

Vdc2 RL

Control tasks: ref dc output voltage regulation at Vdc unity power factor operation (ac side) Control input: converter voltage v

George Konstantopoulos (The University of Sheeld)

iL + Vdc

Non-linear Control in Power Systems

Traditional control of rectiers i

r

L u

u

u

u

+ v

vs

-

iL + vdc

C

RL

-

u

u

Hysteresis current control -

×

sin(ωt) PLL

I*

vdcref

PI

Outer-loop PI controller for voltage regulation Inner-loop Hysteresis controller for current control → stability and a given limit for the input current are not guaranteed, requires a PLL

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Current limiting non-linear controller Current limiting non-linear controller (CLNC): 

v (t )  w˙ w˙ q

=

w (t )i (t ) "

=

cw

− ∆w 2q

max

0

ref 

Vdc − Vdc

Vdc − Vdcref  2 −k (w∆−ww2m ) + wq2 − 1 max cwq 



#

w − wm wq

with c , k , wm , ∆wmax > 0. ref , then If the output voltage Vdc is regulated at Vdc ∗ and w = w ∗ v (t ) = w i (t ) which guarantees unity power factor at the input of the rectier. George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems



Experimental results normal operation:

Vdcref

= 90V → 115V Vdc

(ripple)

Vdc

vs

vs i

i

Time: [100ms/div]

Time: [10ms/div]

current limit with

Imax = 4A:

Vdcref = 90V → 140V Vdc

(ripple)

Vdc

vs

vs i

i Time: [10ms/div]

George Konstantopoulos (The University of Sheeld)

Time: [100ms/div]

Non-linear Control in Power Systems

Outline Introduction Model of some typical Power & Energy Systems Non-linear control theory Dissipative Hamiltonian form Oscillator-based non-linear controller Bounded integral controller (BIC)

Applications

DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Grid-friendly inverters (Synchronverters): CAPS Rectiers with limited current

Summary George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Summary Stability and advanced operation of power systems require: accurate system modelling (average analysis → dissipative Hamiltonian structure) non-linear control design (parameter-free) Oscillator-based non-linear controller Bounded integral controller

Non-linear system analysis Passivity analysis zero-gain property of the controller convergence to the desired equilibrium George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Control applications Power converters Renewable energy systems (wind, solar, etc.) Parallel operation of inverters Grid-friendly inverters (Synchronverters) many more...

Non-linear control theory can change the way power systems have been treated so far!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Control applications Power converters Renewable energy systems (wind, solar, etc.) Parallel operation of inverters Grid-friendly inverters (Synchronverters) many more...

Non-linear control theory can change the way power systems have been treated so far!

George Konstantopoulos (The University of Sheeld)

Non-linear Control in Power Systems

Loading...

Non-linear Control Theory and Applications in Power and Energy

Non-linear Control Theory and Applications in Power and Energy Systems George Konstantopoulos Control and Power Systems Dept. of Automatic Control an...

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