Nonlinear 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)
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear 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 Nonlinear control theory is essential!
George Konstantopoulos (The University of Sheeld)
Nonlinear 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 Nonlinear control theory is essential!
George Konstantopoulos (The University of Sheeld)
Nonlinear 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 Nonlinear control theory is essential!
George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear Control in Power Systems
Average model Average modelling (u → µ ) high switching frequency dutyratio µ = tTon
Li˙ = −(1 − µ)v + E 1
C v˙ = (1 − µ)i − v R where the control input µ is continuoustime and µ ∈ [0, 1].
The system is nonlinear! George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear 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: dutyratio 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)
Nonlinear Control in Power Systems
Existing control techniques Traditional techniques: linearisation (smallsignal model) linear control (PI) Advanced techniques: nonlinear techniques (passivitybased, feedback linearisation) → asymptotic stability dependence from system parameters Can we nd a nonlinear parameterfree controller with guaranteed closedloop stability? George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear Control in Power Systems
Nonlinear 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 . Oscillatorbased nonlinear 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)
Nonlinear Control in Power Systems
Nonlinear 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 . Oscillatorbased nonlinear 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)
Nonlinear Control in Power Systems
Controller operation z2
B 1− γ Α − ,0 2
1− γ Aμ=Ο
μj* μ*
aγ
z1, μ
W(0) Wμ(0)
→ the control input µ is bounded in [0, 1 − γ]! George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Closedloop system stability Closedloop 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)
Nonlinear Control in Power Systems
Closedloop system stability The converter system and the controller system can be handled independently: converter system with µ ∈ [0, 1 − γ] is bounded inputbounded output (BIBO) stable. controller system is BIBO with zero gain! (bounded output independently from the input) →closedloop system is BIBO w.r.t. the external input u . Since u is constant, then the closedloop 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)
Nonlinear 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
Nonlinear Control in Power Systems
Stability of ISpS plant systems Consider a nonlinear 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: Inputtostate practical stability The BIC introduces a zerogain property.
George Konstantopoulos (The University of Sheeld)
Nonlinear 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 Nonlinear Control in Power Systems
]
Integrator replacement Bounded integral control (BIC): output
g(x)
upper bound
⇒ bounded output
g(x)
steadystate
lower bound
BIC
normal conditions: slow down the integration near the bounds abnormal conditions: failsafe operation George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear 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 oscillatorbased nonlinear 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) Nonlinear 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
RL 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)
Nonlinear 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
Generatorside converter control: Maximum Power Point Tracking (MPPT) Fieldoriented control (FOC) of SCIG Gridside converter control: dclink 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)
Nonlinear 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
Generatorside converter control: Maximum Power Point Tracking (MPPT) Fieldoriented control (FOC) of SCIG Gridside converter control: dclink 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)
Nonlinear 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.
Generatorside converter control inputs: mds , mqs Gridside converter control inputs: mdg , mqg
George Konstantopoulos (The University of Sheeld)
Nonlinear 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
Nonlinear 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 oscillatorbased nonlinear controller for the dutyratio inputs as in the case of the dc/dc boost converter The limits of the dutyratio signals are not independent: m
2 2 ds + mqs ≤ 1,
George Konstantopoulos (The University of Sheeld)
m
2 2 dg + mqg ≤ 1
Nonlinear 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 oscillatorbased nonlinear controller for the dutyratio inputs as in the case of the dc/dc boost converter The limits of the dutyratio signals are not independent: m
2 2 ds + mqs ≤ 1,
George Konstantopoulos (The University of Sheeld)
m
2 2 dg + mqg ≤ 1
Nonlinear 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 oscillatorbased nonlinear controller for the dutyratio inputs as in the case of the dc/dc boost converter The limits of the dutyratio signals are not independent: m
2 2 ds + mqs ≤ 1,
George Konstantopoulos (The University of Sheeld)
m
2 2 dg + mqg ≤ 1
Nonlinear Control in Power Systems
Wind power system control Generatorside 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 nonnegative constants
George Konstantopoulos (The University of Sheeld)
Nonlinear 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 dutyratio 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)
Nonlinear Control in Power Systems
Realtime 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]
Nonlinear 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)
Nonlinear Control in Power Systems
Power network  microgrid
Is nonlinear control theory important in this case? Basic control structure: Droop control (nonlinear structure of P and Q ) Accurate load modelling (linear, nonlinear) → Linearisation and local stability is not enough
George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Power network  microgrid
Is nonlinear control theory important in this case? Basic control structure: Droop control (nonlinear structure of P and Q ) Accurate load modelling (linear, nonlinear) → Linearisation and local stability is not enough
George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear 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/nonlinear plant + nonlinear controller =⇒Is the closedloop system stable? George Konstantopoulos (The University of Sheeld)
Nonlinear 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/nonlinear plant + nonlinear controller =⇒Is the closedloop system stable? George Konstantopoulos (The University of Sheeld)
Nonlinear 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)
Nonlinear Control in Power Systems
BDC operation RMS voltage dynamics Eqi
frequency dynamics zqi
starting point
Eqi 0 desired steadystate equilibrium point
φi Vi ȝj*
ȅ
starting point
1
θi
aȖ E * i
Ei
ȝj*
ȅ
rotating with angular velocity equal to θɺi
aȖ
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)
Nonlinear Control in Power Systems
Realtime simulations (nonlinear 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)
Nonlinear Control in Power Systems
Realtime simulations (nonlinear load) BDC Eq1: [200V/div] E1: [200V/div]
zq1: [1/div]
z1: [1/div]
George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Gridfriendly 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)
Nonlinear 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
Nonlinear 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 eldexcitation 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)
Nonlinear 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 eldexcitation 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)
Nonlinear Control in Power Systems
Realtime simulation results Ps: [50W/div]
Qs: [500Var/div]
fgfn: [0.1Hz/div]
ffn: [0.1Hz/div]
Time: [5s/div]
Time: [5s/div]
va: [10V/div] E: [0.5pu/div]
VVg: [0.5V/div]
iga: [10A/div]
Time: [5s/div]
George Konstantopoulos (The University of Sheeld)
Time: [20ms/div]
Nonlinear Control in Power Systems
Completely Autonomous Power Systems (CAPS)
Big step for guaranteeing stability of CAPS! George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Closedloop system stability A fundamental problem in control systems is that closedloop 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 (zerogain 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)
Nonlinear Control in Power Systems
Closedloop system stability A fundamental problem in control systems is that closedloop 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 (zerogain 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)
Nonlinear Control in Power Systems
Closedloop system stability A fundamental problem in control systems is that closedloop 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 (zerogain 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)
Nonlinear Control in Power Systems
Singlephase 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
Nonlinear 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
Outerloop PI controller for voltage regulation Innerloop 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)
Nonlinear Control in Power Systems
Current limiting nonlinear controller Current limiting nonlinear 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)
Nonlinear 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]
Nonlinear Control in Power Systems
Outline Introduction Model of some typical Power & Energy Systems Nonlinear control theory Dissipative Hamiltonian form Oscillatorbased nonlinear controller Bounded integral controller (BIC)
Applications
DC/DC boost converter (energy storage, PV) Wind power Parallel operation of inverters Gridfriendly inverters (Synchronverters): CAPS Rectiers with limited current
Summary George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Summary Stability and advanced operation of power systems require: accurate system modelling (average analysis → dissipative Hamiltonian structure) nonlinear control design (parameterfree) Oscillatorbased nonlinear controller Bounded integral controller
Nonlinear system analysis Passivity analysis zerogain property of the controller convergence to the desired equilibrium George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Control applications Power converters Renewable energy systems (wind, solar, etc.) Parallel operation of inverters Gridfriendly inverters (Synchronverters) many more...
Nonlinear control theory can change the way power systems have been treated so far!
George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems
Control applications Power converters Renewable energy systems (wind, solar, etc.) Parallel operation of inverters Gridfriendly inverters (Synchronverters) many more...
Nonlinear control theory can change the way power systems have been treated so far!
George Konstantopoulos (The University of Sheeld)
Nonlinear Control in Power Systems