Robust Randomized Model Predictive Control for Energy ...

Robust Randomized Model Predictive Control forEnergy Balance in Smart Thermal Grids

Vahab Rostampour, Tamas Keviczky

Delft University of TechnologyDelft Center of Systems and Control

15th European Control ConferenceJune 29 - July 1, 2016

Aalborg, Denmark

V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 1 / 20

Smart Thermal Grids: Conceptual Representation

Bank Buildings Mega Markets

University



External Party


University



External Party

Heat Storage Systems

· ON/OFF Status

· Dynamical Model


University



External Party

Boiler· ON/OFF Status· Set-Point Model


University



External Party

Micro-CHP· ON/OFF Status· Set-Point Model


University



External Party

External PartyHeat Exchange

· Pipe Line· Set-Point Model· Export/Import Thermal Energy· Loss of Thermal Energy


University



External Party

Power and Heat Demand

· Hourly Based Demand· Complex Dynamical Model· Desired Building Temperature


University




University

External Party

UncertainPower and Heat

Demand

· Hourly Based Demand· Complex Dynamical Model· Desired Building Temperature· Weather Condition Effects


Outline

1 Mathematical Model

2 Stochastic Control

3 Simulation Study

4 Conclusions


Building Demand Generator

Heating Energy Demand

Cooling Energy Demand

Building Demand Energy Generator Based on Desired Comfort Service

and Weather Conditions

Environmental Variables Specific Weather

Realization

Desired Building

Temperatures

Demand Profile Generator:

• Complete and detailed building dynamical model• Desired building temperature (local controller unit)• In a specific weather realization, deterministic demand profiles are generated


Building Demand Generator





Environmental Variables

Desired Building

Temperatures

Weather Conditions

Uncertain

Demand Profile Generator:

• Complete and detailed building dynamical model• Desired building temperature (local controller unit)• In uncertain weather conditions, uncertain demand profiles are generated


Building Comfort Service





Environmental Variables

Desired Building

Temperatures

Weather Conditions

Uncertain

Building Control Unit

Optimal Energy Management

Building Control Unit:

• Main components: Boiler, HP, HE, micro-CHP, Buffer Storage• ON/OFF status together with production schedule as decisions• Thermal energy balance for dynamical systems


Mathematical Model

Define xk to be imbalance error between production and building energy demand

xk+1 = Axk +B uk + wkuk xk

wkstochastic disturbancewith unknown set

Our objective: design a state feedback control policy that aims at:• Keeping room temperatures between comfortable limits• Minimizing building operational cost and energy consumption• Taking into account uncertain weather conditions, operational constraints


Constrained Control Problem

Finite horizon open loop control problem for each agent:

minimize(uk ,yk)M

k=1

J (xk , uk) := E

[ M∑k=0

x>k Qxk +M−1∑k=0

u>k Ruk

], Q � 0 , R � 0

subject to fk(xk , uk , yk) ≤ 0 , yk ∈ {0, 1}

xk ∈ X , k = 0, 1, · · · ,M ⇒ hard constraints

Challenges:

1 Control policy parametrization to obtain a less conservative formulation2 Probabilistic interpretation of robustness feature of hard constraints3 Handling mixed-integer optimization together with stochastic programming


Step 1: Control Policy Parametrization


wk

uk = gk(xk, xk−1, xk−2, · · · , x0)

state feedback control policy




wk

uk =k−1∑j=0

θk,jwj + γk xk −Axk−1 −B uk−1

wk−1

Affine feedback policy in the reconstructed (and possibly saturated) noise




wk

uk =k−1∑j=0

θk,jwj + γkwk−1

Control input and state variables depend linearly on the parameters: θk,j , γk


Step 2: Control Problem Formulation


wk

uk =k−1∑j=0


wk−1

∆

Robust approach:• Constraints must be satisfied for every disturbance realization in ∆

Disturbance realizations are treated equally likely (hard constraints)• Intractable problem formulation due to the unknown disturbance set ∆


Step 2: Control Problem Formulation


wk

uk =k−1∑j=0


wk−1

Pr = ε

∆

Chance constrained approach:• Constraints must be satisfied for most disturbance realizations except for a

set of probability ≤ ε (soft constraints)• Nonconvex optimization problem and in general hard to solve


Step 2: Randomized Approximation

minη∈H

J(η)

s.t. Pr [h(η, δ) ≤ 0] ≥ 1− ε

optimization variables uncertainty parametersδ = {wk|k = 1, · · · ,M}η = {uk|k = 1, · · · ,M}

The following randomized approximation that only relies on data can providea (conservative) solution to the chance constrained problem:

minη∈H

J (η)

s.t. h(η, δ(i)) ≤ 0 , i = 1, 2, · · · ,N

N is the number of required disturbance realizations that one needs to generate.This approach provides a solution guaranteed to be probabilistically fulfilling thechance constraints1.

1[Calafiore, Campi, TAC, 2005]V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 9 / 20

Step 2: Randomized Approximation & Constraint Removal

Advantages:

• Only relies on the data• Reformulation is a convex

optimization problemDisadvantages:

• Convex reformulation isusually computationallydemanding

• Still conservativeperformance with respect tothe desired level of violation

*N

optimization direction

One way, to improve performance of the solution, is by using constraint removaltechniques such as greedy algorithm2,etc.

2[Campi, Garatti, OTA, 2010]V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 10 / 20


Advantages:





*1N





Advantages:





*2N




Step 3: Nonconvex Randomized Approximation

• Mixed IntegerProgram

minη∈H,y

J (η)

s.t. Pr [h(η, y, δ) ≤ 0] ≥ 1− εy ∈ {0, 1}M

y :(vector of integer variables

along the horizon length)

Can be reformulated via robust (worst-case) programming as follows:

• Worst Case Program• hj(η, δ) := h(η, yj , δ)

minη∈H

J (η)

s.t. maxj∈{1,··· ,2M}

Pr [hj(η, δ) ≤ 0] ≥ 1− ε

Using randomized approximation, we need to generate at least 2M N disturbancerealizations to provide a solution guaranteed to be chance constrained feasible.This leads to intractable optimization formulation3.

3[Esfahani, Sutter, et al., TAC, 2015]V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 11 / 20

Step 3: Robust Randomized OptimizationInstead we provide two-step approach4 in a receding horizon setting:

1 Determining a bounded set that contains 1− ε portion of ∆:minγ

M−1∑k=0

γk − γk

s.t. Pr[δi,k ∈ [γk , γk ] , ∀k

]≥ 1− ε

2 Solving the robust counterpart of problem w.r.t. the bounded set γ∗:

(1)(2)

( )N

4[Margellos, Rostampour, et al., ECC, 2013]V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 12 / 20

Step 3: Robust Randomized OptimizationInstead we provide two-step approach4 in a receding horizon setting:

1 Determining a bounded set that contains 1− ε portion of ∆:minγ

M−1∑k=0

γk − γk

s.t. Pr[δi,k ∈ [γk , γk ] , ∀k

]≥ 1− ε

minγ

M−1∑k=0

γk − γk

s.t. δji,k ∈ [γk , γk ] ,

{∀k∀j

2 Solving the robust counterpart of problem w.r.t. the bounded set γ∗:

minη∈H,y

J (η)

s.t. h(η, y, γo) + h(η, y, γworst) ≤ 0y ∈ {0, 1}M

(1)(2)

( )N

*

4[Margellos, Rostampour, et al., ECC, 2013]V. Rostampour, T. Keviczky (TUD) RRMPC for Energy Balance in STGs June 29 - July 1 (ECC’16) 12 / 20

Three-Agent Case Study

• Day-ahead control problem• Economical cost function• Operational constraints• Uncertain energy demand• Unit commitment problem• Production scheduling problem

households, greenhousessmart thermal grid example

-CHP -CHP

-CHP

23exh

32exh

31exh 13

exh

1sh

21exh

3sh

2sh

1b

h

3 3,d d

h p

12exh

3b

h

2b

h

1im

hExternal Party

2im

h

3im

h

Line 3 Line 1

Line 21 1,d d

h p

2 2,d d

h p

Mixed-Integer Chance-Constrained Linear Optimization Problem


Proposed Approach

Resulting optimization problem for each agent:

minimize(uk ,yk)M

k=1

J (xk , uk)

subject to fk(xk , uk , yk) ≤ 0 , yk ∈ {0, 1} , k = 0, 1, · · · ,M

Pr{xk ∈ X} ≥ 1− ε ⇒ chance constraints

Theoretical features/contributions of proposed framework:

• Unified framework to solve mixed-integer stochastic optimization problems• Robustness features of constraints in a relaxed probabilistic setting based on

randomization of the constraints• A-priori probabilistic guarantee on the feasibility of the optimal solution of

the problem


Comparison: Benchmark Approach

Forecast Weather

Realization

UncertainWeather

Conditions

Optimal Unit Status

Second Step Optimization:Production Scheduling Problem (Chance Constrained Problem )

First Step Optimization:Unit Commitment Problem (Mixed-Integer Program)


Simulation Results: Relative Cost Improvement


Simulation Results: ON/OFF Status of Boilers


Simulation Results: Imbalance Error Trajectories


Conclusions

Contributions:

• Centralized control problem formulation for a SmartThermal Grid

• Affine Uncertainty Feedback Policy with chance constraint formulation• Convex Reformulation of the proposed stochastic constrained control• A-priori Probabilistic Feasibility Certificate for a mixed-integer

chance-constrained program in a receding horizon scheme

Next Steps:

• Developing a more realistic Building Demand Profile Generator by usinga more detailed dynamical model

• Developing a new scheme to Distribute Computations in the developedframework among the agents


Robust Randomized Model Predictive Control forEnergy Balance in Smart Thermal Grids

Vahab Rostampour, Tamas Keviczky

Delft University of TechnologyDelft Center of Systems and Control

15th European Control ConferenceJune 29 - July 1, 2016

Aalborg, Denmark


Robust Randomized Model Predictive Control for Energy ...

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