Model Predictive Control of Power Systems - Supé · PDF file1/36 Gilney Damm Model Predictive Control of Power Systems. ... Why Model Predictive Control Basis of MPC Plan Première

Model Predictive Control of Power Systems

Gilney Damm

ICT Labs Smart Energy Summer School

Berlin 2013

1/36 Gilney Damm Model Predictive Control of Power Systems

MotivationFoundations of MPC

MPC

Why Model Predictive ControlBasis of MPC

Plan

Première partie I

Introduction

1 MotivationWhy Model Predictive ControlBasis of MPC

2 Foundations of MPC

3 MPCMPC and LQRDecentralized MPCHVDC Example



MPC


Why Model Predictive Control

MPC is largely used in industrySimple conceptsIntuitiveIntroduce the concept of optimality from the conception phaseEase integration of constraints and limitations

Several applications in Power SystemsVoltage controlFrequency controlReconfiguration of Transmission lines → Network CongestionIntegration of energy markets and weather forecast



MPC


Why Model Predictive Control

MPC is largely used in industrySimple conceptsIntuitiveIntroduce the concept of optimality from the conception phaseEase integration of constraints and limitations

Several applications in Power SystemsVoltage controlFrequency controlReconfiguration of Transmission lines → Network CongestionIntegration of energy markets and weather forecast



MPC


Basic ideaCompute at each time instant the sequence of future controlmoves that will make the future predicted controlled variablesto best follow the reference over a finite horizon and takinginto account the control effort.Only the first element of the sequence is used and thecomputation is done again at the next sampling time.



MPC


Basic ideaCompute at each time instant the sequence of future controlmoves that will make the future predicted controlled variablesto best follow the reference over a finite horizon and takinginto account the control effort.Only the first element of the sequence is used and thecomputation is done again at the next sampling time.



MPC


Basic ideaExplicit use of a model to predict output.Compute the control moves minimizing an objective function.Receding horizon strategy - use of a sliding time window (timehorizon moves towards the future)The algorithms mainly differ in the type of model andobjective function used.



MPC


Basic ideaExplicit use of a model to predict output.Compute the control moves minimizing an objective function.Receding horizon strategy - use of a sliding time window (timehorizon moves towards the future)The algorithms mainly differ in the type of model andobjective function used.



MPC


At sampling time t the future control sequence is compute sothat the future sequence of predicted output y(t + k/t) alonga horizon N follows the future references as best as possible.The first control signal is used and the rest disregarded.The process is repeated at the next sampling instant t + 1.



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Plan

Première partie I

Introduction






MPC

State Space MPC

Considering the standard linear system :

x(t + 1) = Ax(t) + Bu(t)

y(t) = Cx(t)

We can define an incremental model in the state space. Taking anew input signal given by :

∆u(t) = u(t)− u(t − 1).

We can rewrite the system as :[x(t + 1)

u(t)

]=

[A B0 I

] [x(t)

u(t − 1)

]+

[BI

]∆u(t)

y(t) =[

C 0] [ x(t)

u(t − 1)

]12/36 Gilney Damm Model Predictive Control of Power Systems


MPC

State Space MPC

Considering the standard linear system :

x(t + 1) = Ax(t) + Bu(t)

y(t) = Cx(t)

We can define an incremental model in the state space. Taking anew input signal given by :

∆u(t) = u(t)− u(t − 1).

We can rewrite the system as :[x(t + 1)

u(t)

]=

[A B0 I

] [x(t)

u(t − 1)

]+

[BI

]∆u(t)

y(t) =[

C 0] [ x(t)

u(t − 1)

]12/36 Gilney Damm Model Predictive Control of Power Systems


MPC

Defining a new state vector

x(t) =[

x(t) u(t − 1)]T

The system can be rewritten as :

x(t + 1) = Mx(t) + Γ ∆u(t)

y(t) = Qx(t)

And we can obtain a prediction of the future outputs in a recursiveway as :

y(t + j) = QM j x(t) +

j−1∑i=0

QM j−i−1Γ∆u(t + i)



MPC

Defining a new state vector

x(t) =[

x(t) u(t − 1)]T

The system can be rewritten as :

x(t + 1) = Mx(t) + Γ ∆u(t)

y(t) = Qx(t)

And we can obtain a prediction of the future outputs in a recursiveway as :

y(t + j) = QM j x(t) +

j−1∑i=0

QM j−i−1Γ∆u(t + i)



MPC

This prediction needs a good state measurement x(t). If the statevector is not accessible, it is necessary to include an observer :

x(t|t) = x(t|t − 1) + K (ym(t)− y(t|t − 1))

where ym(t) is the measured output.The predictions along the time horizon are given by :

y =

y(t + 1|t)y(t + 2|t)

...y(t + N|t)

=

QMx(t) + QΓ ∆u(t)

QM2x(t) +∑1

i=0 QM1−iΓ∆u(t + i)...

QMN2 x(t) +∑N−1

i=0 QMN−1−iΓ∆u(t + i)



MPC

This prediction needs a good state measurement x(t). If the statevector is not accessible, it is necessary to include an observer :

x(t|t) = x(t|t − 1) + K (ym(t)− y(t|t − 1))

where ym(t) is the measured output.The predictions along the time horizon are given by :

y =

y(t + 1|t)y(t + 2|t)

...y(t + N|t)

=

QMx(t) + QΓ ∆u(t)

QM2x(t) +∑1

i=0 QM1−iΓ∆u(t + i)...

QMN2 x(t) +∑N−1

i=0 QMN−1−iΓ∆u(t + i)



MPC

that may be expressed in the vectorial form :

y = F x(t) + Hu

where u = [∆u(t) ∆u(t + 1) . . .∆u(t + N − 1)]T

H is a block lower triangular matrix with nonnull elements definedby Hij = QM i−jΓand matrix F defined by :

F =

QMQM2

...QMN



MPC

The cost function to be minimized may be given by :

J = (Hu + F x(t)− w)T (Hu + F x(t)− w) + λuTu

that, in the non constrained case, is minimized by the control law :

u = (HTH + λI )−1HT (w − F x(t))

As a receding horizon strategy is used, only the first element ofthe control sequence, ∆u(t), is sent to the plant and all thecomputation is repeated at the next sampling time.If the prediction horizon is infinity and there are no constraints,the predictive controller becomes the well-known linearquadratic regulator (LQR). The optimal control sequence isgenerated by a static state feedback law where the feedbackgain matrix is computed via the solution of an algebraicRiccati equation.



MPC




u = (HTH + λI )−1HT (w − F x(t))




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u = (HTH + λI )−1HT (w − F x(t))




MPC

MPC and LQRDecentralized MPCHVDC Example

Plan

Première partie I

Introduction






MPC


LQR based MPC

Lets consider again the systems :

x(t + 1) = Ax(t) + Bu(t)

with known initial condition x(0).The objective is to find the control sequenceu(0), u(1), . . . , u(N − 1) that drives the process from the initial tothe final state minimizing the cost given by :

J = x(N)TQNx(N) +N−1∑k=0

x(k)TQkx(k) + u(k)Rku(k)

where Qk = QTk ≥ 0 et Rk = RT

k > 0



MPC


To obtain the control sequence the problem is solved in reverseorder. Let us define I ∗1 as the optimal cost of the last stage (frominitial state x(N1) to the final x(N)) :

I ∗1 (x(N − 1)) = minu(N−1)

x(N)TQNx(N) + u(N − 1)RN−1u(N − 1)

Which can be solved explicitly :

u(N − 1) = −(BTQNB + R)−1BTQNA x(N − 1) = KN−1 x(N − 1)

The control action is a linear feedback of the state vector.The last stage cost is then given by :

I ∗1 = (Ax + BKN−1x)TQN(Ax + BKN−1x) + xTKTN−1RN−1KN−1x



MPC



I ∗1 (x(N − 1)) = minu(N−1)

x(N)TQNx(N) + u(N − 1)RN−1u(N − 1)







MPC



I ∗1 (x(N − 1)) = minu(N−1)

x(N)TQNx(N) + u(N − 1)RN−1u(N − 1)







MPC


Defining :

PN−1 = (A + BKN−1)TQN(A + BKN−1) + KTN−1RN−1KN−1

So I ∗1 can be written as a quadratic form of the state :

I ∗1 = x(N − 1)TPN−1x(N − 1)

This procedure can be extended, leading to :

u(k) = Kk x(k) = −(BTPk+1B + R)−1BTPk+1A x(k)

The symmetric semidefinite matrix Pk is given by :

Pk = ATPk+1A + ATPk+1BKk + Qk

This is called the Discrete-time Riccati equation which can besolved recursively from a final point PN = QN at instant N.



MPC


Defining :



I ∗1 = x(N − 1)TPN−1x(N − 1)








MPC


Defining :



I ∗1 = x(N − 1)TPN−1x(N − 1)








MPC


Infinite Horizon

As the controller is a linear feedback of the state, if this one is notavailable, the use of a state estimator or observer is required tocompute the control action.If the observer is a Kalman Filter, then it gives rise to thewell-known control strategy called Linear Quadratic Gaussian(LQG).If we can assume that the terminal time is infinitely far in thefuture, we will obtain a constant feedback gain matrix, which canbe calculated considering that Pk → P∞ ≥ 0.Matrix P∞ is computed using the Discrete-time Riccati equation :

P∞ = ATP∞A + ATP∞BK∞ + Q



MPC


Infinite Horizon

As the controller is a linear feedback of the state, if this one is notavailable, the use of a state estimator or observer is required tocompute the control action.If the observer is a Kalman Filter, then it gives rise to thewell-known control strategy called Linear Quadratic Gaussian(LQG).If we can assume that the terminal time is infinitely far in thefuture, we will obtain a constant feedback gain matrix, which canbe calculated considering that Pk → P∞ ≥ 0.Matrix P∞ is computed using the Discrete-time Riccati equation :

P∞ = ATP∞A + ATP∞BK∞ + Q



MPC


Infinite Horizon

Now the control action becomes the constant state feedback law :

u(k) = K∞x(k) = −(BTP∞B + R)−1BTP∞A x(k)

It can be proven that this is a stabilizing control law using theLyapunov function :

V (x(k)) = x(k)TP∞x(k)

Using the incremental state space model with

x(t) =[

x(t) u(t − 1)]T



MPC


Infinite Horizon

Now the control action becomes the constant state feedback law :

u(k) = K∞x(k) = −(BTP∞B + R)−1BTP∞A x(k)

It can be proven that this is a stabilizing control law using theLyapunov function :

V (x(k)) = x(k)TP∞x(k)

Using the incremental state space model with

x(t) =[

x(t) u(t − 1)]T



MPC


From Centralized to Decentralized MPC

Consider the LTI model

x(t + 1) = Ax(t) + Bu(t)

y(t) = Cx(t)

u = [u′1 u′2 · · · u′M ]′ ∈ Rm

x = [x ′1 x ′2 · · · x ′M ]′ ∈ Rn

y = [y ′1 y ′2 · · · y ′M ]′ ∈ Rz

A =

A11 A12 · · · A1M...

.

.

.. . .

.

.

.Ai1 Ai2 · · · AiM...

.

.

.. . .

.

.

.AM1 AM2 · · · AMM

B =

B11 B12 · · · B1M...

.

.

.. . .

.

.

.Bi1 Bi2 · · · BiM...

.

.

.. . .

.

.

.BM1 BM2 · · · BMM

C =

C11 0 · · · 00 C22 · · · 0...

.

.

.. . .

.

.

.0 · · · · · · CMM



MPC




x(t + 1) = Ax(t) + Bu(t)

y(t) = Cx(t)

u = [u′1 u′2 · · · u′M ]′ ∈ Rm

x = [x ′1 x ′2 · · · x ′M ]′ ∈ Rn

y = [y ′1 y ′2 · · · y ′M ]′ ∈ Rz

A =

A11 A12 · · · A1M...

.

.

.. . .

.

.

.Ai1 Ai2 · · · AiM...

.

.

.. . .

.

.

.AM1 AM2 · · · AMM

B =

B11 B12 · · · B1M...

.

.

.. . .

.

.

.Bi1 Bi2 · · · BiM...

.

.

.. . .

.

.

.BM1 BM2 · · · BMM

C =

C11 0 · · · 00 C22 · · · 0...

.

.

.. . .

.

.

.0 · · · · · · CMM



MPC




x(t + 1) = Ax(t) + Bu(t)

y(t) = Cx(t)

u = [u′1 u′2 · · · u′M ]′ ∈ Rm

x = [x ′1 x ′2 · · · x ′M ]′ ∈ Rn

y = [y ′1 y ′2 · · · y ′M ]′ ∈ Rz

A =

A11 A12 · · · A1M...

.

.

.. . .

.

.

.Ai1 Ai2 · · · AiM...

.

.

.. . .

.

.

.AM1 AM2 · · · AMM

B =

B11 B12 · · · B1M...

.

.

.. . .

.

.

.Bi1 Bi2 · · · BiM...

.

.

.. . .

.

.

.BM1 BM2 · · · BMM

C =

C11 0 · · · 00 C22 · · · 0...

.

.

.. . .

.

.

.0 · · · · · · CMM



MPC


Cost Functions

General cost function

Ji =∞∑

k=t

xi (k)TQixi (k) + ui (k)Riui (k)

Centralized MPC

minu J =∑

j=1...M

$iJi

x(k + 1|t) = Ax(k|t) + Bu(k|t)

k ≥ t ui (k|t) ∈ Ωi

Decentralized MPC

minui Ji

xi (k + 1|t) = Aii xi (k|t) + Biiui (k|t)




MPC


Cost Functions


Ji =∞∑

k=t


Centralized MPC

minu J =∑

j=1...M

$iJi

x(k + 1|t) = Ax(k|t) + Bu(k|t)


Decentralized MPC

minui Ji





MPC


Cost Functions


Ji =∞∑

k=t


Centralized MPC

minu J =∑

j=1...M

$iJi

x(k + 1|t) = Ax(k|t) + Bu(k|t)


Decentralized MPC

minui Ji





MPC


Distributed MPC

Communicating MPC

minui Ji


+∑j 6=i

[Aijxp−1j (k|t) + Bijup−1

j (k|t)]


Cooperating MPC

minui $iui +∑i 6=j

$jJj (up−1j )

xi (k + 1|t)⇒ iterative inside

a sampling period



MPC


Distributed MPC

Communicating MPC

minui Ji


+∑j 6=i


j (k|t)]


Cooperating MPC


$jJj (up−1j )


a sampling period



MPC


Distributed MPC

Communicating MPC

minui Ji


+∑j 6=i


j (k|t)]


Cooperating MPC


$jJj (up−1j )


a sampling period



MPC


Four Area Frequency Stability

At 5s there is a 25% load increase in area 2 and asimultaneous 25% load drop in area 3Control horizon N = 20 in each case



MPC





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Secondary Power Flow Control of an Multipoint HVDCnetwork

Figure: Multiterminal HVDC system with three nodes.



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Electricity Price and Wind

The electricity price in the electricity market for a week (Source :CRE).

Figure: Energy Price

Figure: a) Wind speed b) Wind power



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Constraints

PL(k) = PW (k)− PS(k) ∀k

PW (k) the power generated by the wind farmPS(k) is the power absorbed/supplied to the storage devicePL(k) is the load demand

J = max( ∑Np

k=1(PW (k)− PS(k)) · p(k))

subject to

Emin ≤ E (k) ≤ Emax (a)PS(k) ≤ PW (k) (b)−PW ,nom ≤ PS(k) (c)

Where E (k) is the stored energy at instant k and it is defined as :

E (k + 1) = E (k) + PS(k) · T ∀k



MPC


Constraints

PL(k) = PW (k)− PS(k) ∀k

PW (k) the power generated by the wind farmPS(k) is the power absorbed/supplied to the storage devicePL(k) is the load demand

J = max( ∑Np

k=1(PW (k)− PS(k)) · p(k))

subject to

Emin ≤ E (k) ≤ Emax (a)PS(k) ≤ PW (k) (b)−PW ,nom ≤ PS(k) (c)

Where E (k) is the stored energy at instant k and it is defined as :

E (k + 1) = E (k) + PS(k) · T ∀k



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Simulation 24 h prevision



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Simulation 24 h prevision



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Questions ?



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Bibliographie1 E.F. Camacho and C. Bordons, “Model Predictive Control”,

Springer- Verlag, 20042 E. F. Camacho, “MPC :An Introductory Survey”, Paris 20093 Eduardo F. Camacho y Carlos Bordons, “Control Predictivo :

Pasado, Presente y Futuro”4 James B. Rawlings, “An overview of distributed model

predictive control (MPC)”, IFAC Workshop : Hierarchical andDistributed Model Predictive Control, Algorithms andApplications, Milano, Italy, August 28, 2011

5 Aswin N. Venkat, Ian A. Hiskens, James B. Rawlings, andStephen J. Wright, “Distributed MPC Strategies WithApplication to Power System Automatic Generation Control”,IEEE Trans. Cont. Syst. Tec., vol. 16, No. 6, Nov. 2008


Documents

Model Predictive Control of Power Systems - Supé · PDF file1/36 Gilney Damm Model Predictive Control of Power Systems. ... Why Model Predictive Control Basis of MPC Plan Première