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Proceedings ArticleDOI

Robust model predictive control for an uncertain smart thermal grid

01 Jun 2016-pp 1195-1200

TL;DR: The focus of this paper is on modeling and control of Smart Thermal Grids in which the uncertainties in the demand and/or supply are included and it is shown that the robust MPC approach successfully keeps the supply-demand balance in the STG while satisfying the constraints of the production units in the presence of uncertainty in the heat demand.

AbstractThe focus of this paper is on modeling and control of Smart Thermal Grids (STGs) in which the uncertainties in the demand and/or supply are included. We solve the corresponding robust model predictive control (MPC) optimization problem using mixed-integer-linear programming techniques to provide a day-ahead prediction for the heat production in the grid. In an example, we compare the robust MPC approach with the robust optimal control approach, in which the day-ahead production plan is obtained by optimizing the objective function for entire day at once. There, we show that the robust MPC approach successfully keeps the supply-demand balance in the STG while satisfying the constraints of the production units in the presence of uncertainties in the heat demand. Moreover, we see that despite the longer computation time, the performance of the robust MPC controller is considerably better than the one of the robust optimal controller.

Topics: Robust control (66%), Model predictive control (61%), Robustness (computer science) (57%), Control theory (54%), Optimal control (54%)

Summary (2 min read)

Introduction

  • The authors solve the corre- sponding robust model predictive control (MPC) optimization problem using mixed-integer-linear programming techniques to provide a day-ahead prediction for the heat production in the grid.
  • Moreover, the authors see that despite the longer computation time, the performance of the robust MPC controller is considerably better than the one of the robust optimal controller.
  • As about half of a neighborhood’s electricity consumption is typically used for thermal purposes [1], introducing STG neighborhoods could have substantial benefits, such as: 1) less transport of energy, less energy loss, and lower transportation costs, and 2) using the produced heat at the neighborhood level as an energy source to avoid wasting heat.
  • The authors consider robust MPC for STGs in the presence of uncertainties in the grid to provide a day-ahead heat production plan for the thermal grid.
  • The uncertainties in the network can be due to the uncertainty in the demand and/or in the production because of using different resources such as solar energy or biogas.

II. SMART THERMAL GRIDS

  • Each of these greenhouses is considered as an agent and the full information of each agent, such as the production resources, the demand request for the next day, etc., is assumed to be available to the whole network.
  • In addition to the local heat generation, there are one or more external parties that can provide heat to the network.
  • The authors consider all the external parties as one single agent.
  • To model the physical system, the authors discretize the system with sampling time of one hour.
  • Let Hexchi j denotes the exchanged heat between two adjacent greenhouses i and j.

III. MODEL PREDICTIVE CONTROL FOR STGS

  • The authors aim is to reduce the overall production costs of the network while providing the network’s required heat under different operational constraints such as the limits for the generators and the buffers.
  • The control objective will be focused on demand response [9], [17], which is the ability of domestic net-consumption of heat to respond to real-time1 electricity prices.
  • 1The real-time electricity price is the one that varies almost every 15 minutes in the electricity market on the exact day of the electricity production.
  • The cost of importing heat by greenhouse j at time step k is Cimp(HimpEx j(k)) = HimpEx j(k) ·HbuyingEx(k), (13) where HbuyingEx(k) is the price that greenhouse j pays for buying heat from external parties at time step k.
  • Accordingly, the authors can rewrite (15) as a linear equation by introducing new binary and continuous 2As mentioned by experts at Eneco, a Dutch utility company and their project sponsor, boilers do not require a time-on/off constraints.

IV. SOLVING THE WORST-CASE MPC

  • At the beginning of each time step k, the controller measures the system state of the previous step.
  • Then, using the information regarding the demand and the energy price, the controller determines the decision variables PGCHP j,HGBoil j,HimpEx j,µ stop u j , and µstartu j .
  • To this end, the authors solve the inner optimization problem first.
  • Note that the available mp-MILP algorithms are not very efficient when the size of the vector of parameters and the prediction horizon Np increases.

V. EXAMPLE

  • The authors solve robust MPC optimization problem to obtain a day-ahead prediction for the heat production plan for a small network of greenhouses and they compare the results with the ones obtained using robust optimal control approach.
  • To solve the optimization problem (28)-(29), the authors chose M = 500 different uncertainty vectors e to obtain a 0.95% confidence level with accuracy error of 1% and they use the MILP solver from IBM CPLEX.
  • The first plot of Figure 3 shows the heat demand of each greenhouse for one day.
  • Here also, the CHPs are mainly used during the hours that the electricity price is quite high and they can also sell the extra electricity in the market .
  • Moreover, robust MPC is a better control choice than robust optimal control although it requires more computation time.

VI. CONCLUSIONS

  • The authors have considered control of a typical smart thermal grid, namely a network of greenhouses, under uncertainties in demand and/or response.
  • The authors assumed the uncertainty to be bounded and hence, a worst-case MPC optimization problem was solved.
  • Since both the cost function and the constraints are linear, the optimization problem was formulated as a mixed-integer linear programming (MILP) problem; the authors have discussed three approaches to solve the obtained optimization problem.
  • In a case study, the authors compared the MPC approach with the optimal control approach to obtain a day-ahead production plan for a sample network of greenhouses.
  • The efficient control approach in this case is a distributed model predictive control approach in which the agents can only have partial information about the network.

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Robust Model Predictive Control for an Uncertain Smart Thermal Grid
Samira S. Farahani
, Zofia Lukszo
⇤⇤
, Tam
´
as Keviczky
⇤⇤⇤
, Bart De Schutter
⇤⇤⇤
, Richard M. Murray
Abstract The focus of this paper is on modeling and control
of Smart Thermal Grids (STGs) in which the uncertainties in
the demand and/or supply are included. We solve the corre-
sponding robust model predictive control (MPC) optimization
problem using mixed-integer-linear programming techniques to
provide a day-ahead prediction for the heat production in the
grid. In an example, we compare the robust MPC approach with
the robust optimal control approach, in which the day-ahead
production plan is obtained by optimizing the objective function
for entire day at once. There, we show that the robust MPC
approach successfully keeps the supply-demand balance in the
STG while satisfying the constraints of the production units in
the presence of uncertainties in the heat demand. Moreover, we
see that despite the longer computation time, the performance
of the robust MPC controller is considerably better than the
one of the robust optimal controller.
I. INTRODUCTION
Smart Thermal Grids (STGs) can contribute to obtaining
sustainable energy systems by guaranteeing a reliable heating
supply to various customers by using renewable energy
sources such as solar or geothermal energy. STGs are best ap-
plicable to neighborhoods with small-scale utility companies
and independent users. As about half of a neighborhood’s
electricity consumption is typically used for thermal purposes
[1], introducing STG neighborhoods could have substantial
benefits, such as: 1) less transport of energy, less energy loss,
and lower transportation costs, and 2) using the produced
heat at the neighborhood level as an energy source to avoid
wasting heat.
Considering the complexities of such systems, mainly due
to uncertainty in demand and supply as well as the large size
of the networks, smart energy systems need to be managed
and controlled in an automated way in order to increase the
efficiency for both producers and consumers. To this end,
model predictive control (MPC) [15] has been proved to be
a useful tool in both simulations and real-life applications
[7], [12], [13].
We consider robust MPC for STGs in the presence of
uncertainties in the grid to provide a day-ahead heat pro-
duction plan for the thermal grid. The uncertainties in the
network can be due to the uncertainty in the demand and/or
in the production because of using different resources such
as solar energy or biogas. Although the control aspects of
thermal energy have been studied implicitly in the context of
Samira S. Farahani and Richard M. Murray are with the Department
of Control and Dynamical Systems, California Institute of Technology,
Pasadena, CA. farahani@caltech.edu
⇤⇤
Zofia Lukszo is with the Energy and Industry Division, Delft University
of Technology, Delft, the Netherlands
⇤⇤⇤
Tam
´
as Keviczky and Bart De Schutter are with the Delft Center for
Systems and Control, Delft University of Technology, Delft, the Netherlands
Combined Heat and Power (CHP) systems or general smart
grids, using distributed MPC and other similar agent-based
control approaches [9], [14], the explicit implementation of
the controller for STG systems requires careful investigation
due to the structural differences between STGs and other
types of smart grids such as smart electric grids. In the
context of smart thermal grids and MPC, to the authors’
best knowledge, this paper will be the first attempt that
studies robust MPC for STGs. Hence, the novelty of our
contribution is not in the approach but is in the application
of this approach. To model the network, we use a mixed
logical dynamical (MLD) model and we assume that the
uncertainty is bounded within a polyhedral set. Hence, the
worst-case MPC optimization problem can be recast as a
Mixed-Integer-Linear Programming (MILP) problem which
can be solved efficiently using the available algorithms. In
the end, in an example, we compare this approach with the
robust optimal control approach in which the optimal input
is calculated at once for the entire simulation hours (24 hours
related to the day-ahead prediction).
II. SMART THERMAL GRIDS
We consider a regional network of greenhouses, which is a
typical example of a thermal grid. Each of these greenhouses
is considered as an agent and the full information of each
agent, such as the production resources, the demand request
for the next day, etc., is assumed to be available to the whole
network. Each agent is facilitated with an CHP system and
a boiler and hence, is capable of local production of heat
and electricity that can be used by the same agent or be
exported/sold to the network. We assume that the agents can
only trade heat among each other and the electricity will
be bought or sold to the electricity market only. Moreover,
each agent has a buffer system to store heat and either
to use it internally or to sell it to the other agents in the
network. In addition to the local heat generation, there are
one or more external parties that can provide heat to the
network. We consider all the external parties as one single
agent. The greenhouses are connected to each other and to
the external suppliers by several pipes of different sizes.
Moreover, to adjust the input and output heat to and from
the greenhouses, there are several heat exchangers located
outside the greenhouses.
To model the physical system, we discretize the system
with sampling time of one hour. The time step counter is
denoted by k. For the sake of compactness, the definition of
model parameters are presented in Table I.
As indicated by experts, the heat exchangers do not add
additional costs to the heat production and hence, they can
2016 European Control Conference (ECC)
http://www.cds.caltech.edu/~murray/papers/far+16-ecc.pdf

Parameters Symbol Unit
Transportation cost per MW C
trans
e
The energy content of gas for CHP start up g
start
MW
Electrical efficiency of the CHP unit h
e
-
Thermal efficiency of the CHP unit h
th
-
Thermal efficiency of the boiler h
Boil
-
Turnaround efficiency of the buffer unit h
Buf
-
Fuel price per MW F
price
e
CHP maintenance cost per MW C
CHP
e
CHP fixed start up cost C
fix
e
Buffer capacity of each greenhouse B
C
MWh
Minimum heat production capacity for unit u U
¯
u
MWh
Maximum heat production capacity for unit u
¯
U
u
MWh
TABLE I
be left out from the network model. The fuel energy content
(gas in our case) used by a CHP unit at greenhouse j at time
step k in MW can be specified as [16]
g
CHP j
(k)=
P
G
CHP
j
(k)
h
e
= H
G
CHP
j
(k) ·
1
h
th
, (1)
where P
G
CHP
j
(k) and H
G
CHP
j
(k) are respectively the electrical
power and the heat generated by the CHP unit of greenhouse
j at time step k in MW. Similarly, for a boiler, we have [10]
g
Boil j
(k)=H
G
Boil
j
(k) ·
1
h
Boil
, (2)
where g
Boil j
(k) is defined similarly to g
CHP j
(k) and H
G
Boil
j
(k)
is the heat generated by the boiler of greenhouse j at time
step k in MW.
If the CHP or boiler unit are operating at greenhouse j, the
thermal power can vary at each time step between a certain
minimum and maximum for both the CHP and the boiler as
U
¯
CHP j
H
G
CHP
j
(k)
¯
U
CHP j
8 k, j (3)
U
¯
Boil j
H
G
Boil
j
(k)
¯
U
Boil j
8 k, j. (4)
Moreover, in the case that the production units, i.e., the boiler
and the CHP, of greenhouse j produce more heat than is
demanded by the greenhouse itself, the heat can be stored
in a buffer to be used at other hours or to be used by other
greenhouses in the network. We assume that each greenhouse
j can only send or receive heat to or from its immediate
neighbors, respectively. Let H
exchij
denotes the exchanged
heat between two adjacent greenhouses i and j. The buffer
state of greenhouse j can then be defined as
B
S j
(k)=B
S j
(k 1)+h
Buf
H
G
CHP
j
(k)+H
G
Boil
j
(k)
H
D j
(k)+H
impEx j
(k)+
Â
i2f
j
(1 a
ij
)H
exchij
, (5)
where H
D j
(k) denotes the heat demand of greenhouse j
at time step k, H
impEx j
(k) denotes the imported heat by
greenhouse j from an external party at time step k, f
j
is the set of neighbors of greenhouse j, and a
ij
denotes
the percentage of heat loss due to transportation between
greenhouse i and j. Moreover, we have capacity constraints
for the buffer and constraint for the amount of heat imported
from external parties or exchanged between two neighbors
due to for instance pipe or network capacity.
There is also an additional constraint for the transported
heat among the greenhouses in order to make sure that the
supply-demand balance is satisfied at each time step,
0 B
S j
(k) B
C j
8 k, j (6)
0 H
impEx j
(k)
¯
U
impEx j
8 k, j (7)
H
exchij
= H
exch ji
8 j, i 2 f
j
(8)
U
¯
exchij
H
exchij
¯
U
exchij
(9)
where
¯
U
impEx j
is the maximum possible heat import from
external parties and U
¯
exchij
and
¯
U
exchij
are minimum and
maximum amount of heat that can be exchanged between two
adjacent neighbors. Note that H
exchij
takes both positive and
negative values indicating the imported heat by greenhouse
j from greenhouse i and exporting heat from greenhouse i
to greenhouse j, respectively.
Figure 1 illustrates the energy flow between one green-
house and the network of greenhouses, as well as the heat-
producing external parties and the energy retailers.
CHP$
Boiler$
Buffer$
electricity$$
demand$
heat$$
demand$
energy$retailers$
(gas$and$electricity)$
network$of$
greenhouses$
j
g
CHP
j
g
Boil
j
H
Boil
j
H
CHP
j
H
D
ji
H
exch
j
P
CHP
j
P
D
j
P
exp
j
P
imp
external$
par>es$
j
H
impEx
Fig. 1. Energy flow between greenhouse j, network of greenhouses, heat
producing external parties, and energy retailers.
Remark 1: The connection between each greenhouse j
and the external parties in Figure 1 does not reflect the
physical connection and is only an indicator for the heat
flow.
III. MODEL PREDICTIVE CONTROL FOR STGS
Our aim is to reduce the overall production costs of the
network while providing the network’s required heat under
different operational constraints such as the limits for the
generators and the buffers. To this end, we intend to develop
an advanced control approach that is suitable for practical
applications. The control objective will be focused on de-
mand response [9], [17], which is the ability of domestic
net-consumption of heat to respond to real-time
1
electricity
prices. In this paper by “real-time” electricity prices we mean
the hourly varying supply tariff, which is equal to the hourly
day-ahead prices of the electricity market.
1
The real-time electricity price is the one that varies almost every
15 minutes in the electricity market on the exact day of the electricity
production.

The control strategy that is proposed here for demand
response is Model Predictive Control (MPC). The control
objective is to minimize the total heat production costs,
which includes the variable costs of the network related to
the heat production as well as the earnings. Without loss of
generality, we assume that the network is owned by a single
owner and hence, all greenhouses cooperate with each other
in order to keep the total heat generation costs of the network
as low as possible. This means that they try to generate as
much heat as possible in order to satisfy the heat demand
of the network and buy as less as possible from the external
parties. The total heat production cost function of greenhouse
j at time step k can be defined as
C(P
G
CHP
j
(k),H
G
Boil
j
(k),H
impEx j
(k),µ
start
CHP j
(k)) (10)
= C
G
(P
G
CHP
j
(k), H
G
Boil
j
(k)) +C
O
(P
G
CHP
j
(k))
+C
imp
(H
impEx j
(k)) +C
start
(µ
start
CHP j
(k)) E
P
(P
CHP, j
(k)).
The heat generation cost for each greenhouse depends
on the amount of fuel that is used. Therefore, considering
equations (1) and (2), it can be defined as
C
G
(P
G
CHP
j
(k), H
G
Boil
j
(k)) =
g
CHP j
(k)+g
Boil j
(k)
F
price
.
(11)
For each CHP, there will also be an additional cost, namely,
the operation cost, which is defined for each greenhouse j
at time step k as
C
O
(P
G
CHP
j
(k)) = P
G
CHP
j
(k) ·C
CHP
. (12)
The import cost matters when the greenhouse needs to buy
heat from an external party, in the case that the generated heat
by the greenhouse itself and the amount that is imported from
other greenhouses in the network is less than its demand. The
cost of importing heat by greenhouse j at time step k is
C
imp
(H
impEx j
(k)) = H
impEx j
(k) · H
buyingEx
(k), (13)
where H
buyingEx
(k) is the price that greenhouse j pays for
buying heat from external parties at time step k. We assume
that the taxes and the transportation cost are included in
H
buyingEx
(k). Moreover, there are fixed start-up costs and
fuel-based start-up costs for a CHP unit of greenhouse j,
which can be calculated as [8]
C
start
(µ
start
CHP j
(k)) = µ
start
CHP j
(k)
C
fix
+ g
start
· F
price
, (14)
where µ
start
uj
is a binary variable such that µ
start
uj
(k)=1 if unit
u (CHP or boiler) of greenhouse j is started for production
of energy at time step k and µ
start
uj
(k)=0 otherwise. The
second part of the production cost is related to the electricity
earnings obtained from selling electricity to the electricity
market. The selling price is variable and is different every
hour. The electricity earnings of greenhouse j at time step k
can be written as
E
P
(P
G
CHP
j
(k)) =
8
>
>
<
>
>
:
P
G
CHP
j
(k) P
D j
(k)
P
selling
(k)
if P
G
CHP
j
(k) P
D j
(k)
0 if P
G
CHP
j
(k) < P
D j
(k)
(15)
where P
D j
(k) indicates the electricity demand of greenhouse
j at time step k and P
selling
(k) is the selling price of electricity
at time step k. Note that since we assume cooperation
between the greenhouses, there are no heat earnings while
the greenhouses exchange heat among each other.
Therefore, considering (10), the cost function J(k) at time
step k over the prediction horizon N
p
is defined as
J =
N
p
1
Â
l=0
n
Â
j=1
C
P
G
CHP
j
(k + l),H
G
Boil
j
(k + l),
H
impEx j
(k + l), µ
start
CHP j
(k + l)
. (16)
This cost function will be minimized subject to the con-
straints on different components of the systems. Some of
these constraints have been presented in the previous section.
In addition to those, we need extra constraints related to on-
off states of the CHP and boiler [9]. We define µ
stop
u, j
as
a binary variable such that µ
stop
u, j
(k)=1 if unit u (CHP or
boiler) of greenhouse j is shut down at time step k and 0
otherwise. Moreover, we define the binary variable v
u j
(k)
for each production unit u, of greenhouse j at time step
k such that v
u j
(k)=1 if unit u operates and 0 otherwise.
Therefore, the capacity constraints for the heat production,
i.e., equations (3)-(4) can be rewritten as
U
¯
u j
·v
u j
(k) H
G
u
j
(k)
¯
U
u j
·v
u j
(k) 8 k, j (17)
Moreover, the following equations link the above binary
variables [8], [9]:
v
u j
(k)v
u j
(k 1)=µ
start
u, j
(k)µ
stop
u, j
(k) 8 j,k (18)
µ
start
u, j
(k)+µ
stop
u, j
(k) 1 8 j , k (19)
v
CHP j
(k)
Â
kU
¯
on
CHP j
<`k
µ
start
CHP, j
(`) 8 j,k (20)
1 v
CHP j
(k)
Â
kU
¯
of f
CHP j
<`k
µ
stop
CHP, j
(`) 8 j,k (21)
The last two constraints are related to the minimum time-on
and time-off constraints for each CHP unit
2
where U
¯
on
CHP j
and U
¯
of f
CHP j
are minimum required on and off time steps
respectively.
In order to obtain a linear system with continuous and bi-
nary variables, we apply the mixed logical dynamical (MLD)
formalism [3], which allows the transformation of logical
statements involving continuous variables into mixed-integer
linear inequalities. Accordingly, we can rewrite (15) as a
linear equation by introducing new binary and continuous
2
As mentioned by experts at Eneco, a Dutch utility company and our
project sponsor, boilers do not require a time-on/off constraints.

auxiliary variables. In this way, the system dynamics and the
constraints are formulated as mixed-integer linear equations
and hence, we solve a mixed-integer linear programming
(MILP) problem. Note that this control approach is a central-
ized one, which means while the overall production cost of
the network is minimized, each individual greenhouse may
not have the optimal cost at each time step.
IV. SOLVING THE WORST-CASE MPC
At the beginning of each time step k, the controller mea-
sures the system state of the previous step. In our case, the
state variables are B
S j
, v
CHP j
, and v
Boil j
. At each time step k,
we assume that the previous value of these variables is known
or measured. Then, using the information regarding the
demand and the energy price, the controller determines the
decision variables P
G
CHP
j
,H
G
Boil
j
,H
impEx j
, µ
stop
u j
, and µ
start
u j
.
We choose 24 time steps, i.e., k = 1,...,24, corresponding
to the 24 hours in one day.
We also assume that there is an uncertainty in the heat
demand H
D
, i.e., H
D
(k)=H
D,pred
(k)+e(k) where H
D,pred
(k)
is the predicted heat demand for the greenhouses at time step
k. We gather the uncertainty for time steps k,...,k +N
p
1 in
the vector ˜e(k)=[e
T
(k),...,e
T
(k +N
p
1)]
T
2 E where E =
{ ˜e(k) :
˜
S ˜e(k) ˜q} is a bounded polyhedral set. Accordingly,
we can define the worst-case MPC optimization problem as
min
˜u(k)
max
˜e(k)2E
J( ˜u(k), ˜e(k)) (22)
s.t. P(k) ˜u(k)+Q(k) ˜e(k)+q(k) 0 (23)
where J is the cost function, ˜u(k) is the vector of decision
variables containing both continuous and binary variables
as well as the continuous and binary auxiliary variables
obtained from the MLD model (defined similarly to ˜e(k)),
P(k), Q(k) are inequality constraint matrices and q(k) is the
inequality constraint constant vector, all defined according
to the constraints (6)-(9) and (17)-(21). Since both the cost
function J and the constraints are piecewise affine in ˜u(k),
we can solve the optimization problem (22)-(23) as an MILP
problem.
Remark 2: Solving an MILP for the robust MPC de-
sign does not scale well with the complexity (size) of the
model/system, and disturbance set representation. This issue
can be avoided by either decreasing the number of binary
variables by choosing a small-enough prediction horizon N
p
,
or by relaxing the constraints on the binary variables to
obtain an LP optimization problem instead. Generally, MILP
complexity grows exponentially as the number of binary
variables increases.
The first approach we apply to solve our MILP op-
timization problem is multi-parametric MILP (mp-MILP)
optimization. To this end, we solve the inner optimization
problem first. For a given ˜u(k), the optimization problem
max
˜e(k)
J( ˜u(k), ˜e(k)) (24)
s.t.
˜
S ˜e(k) ˜q (25)
P(k) ˜u(k )+Q(k)˜e(k)+q(k) 0
can be solved as an mp-MILP problem, in which ˜u(k) is the
parameter, using the algorithm in [6].
Let ˜e
( ˜u(k)) = argmax
˜e(k)
J( ˜u(k), ˜e(k)) denote the solution
of the mp-MILP problem (24)-(25), which is a piecewise-
affine function in ˜u(k) (see [4], [11]). Hence, the outer
optimization problem, i.e.,
min
˜u(k)
J( ˜u(k), ˜e
( ˜u(k))) (26)
s.t. P(k) ˜u(k)+Q(k) ˜e
( ˜u(k)) + q(k) 0 (27)
can be solved as an MILP optimization problem using the
available MILP solvers that are based on e.g. branch-and-
bound or cutting plane algorithms [2].
Note that the available mp-MILP algorithms are not
very efficient when the size of the vector of param-
eters and the prediction horizon N
p
increases. There-
fore, we now discuss alternative approaches to mp-
MILP. One approach is to use Monte Carlo simulation
to eliminate the inner optimization problem as follows.
Let ˜e
(1)
(k),..., ˜e
(M)(k)
denote M different noise realiza-
tions belonging to the polyhedral set E and let t(k)=
max
˜e
(1)
(k),..., ˜e
(M)(k)
(J( ˜u(k), ˜e
(1)
(k)), ··· , J( ˜u(k), ˜e
(M)
(k))). The
optimization problem (22)-(23) can be then rewritten as
min
˜u(k),t(k)
t(k) (28)
s.t. t(k) J( ˜u(k), ˜e
(1)
(k)) (29)
···
t(k) J( ˜u(k), ˜e
(M)
(k))
P(k) ˜u(k )+Q(k)˜e
(1)
(k)+q(k) 0
···
P(k) ˜u(k )+Q(k)˜e
(M)
(k)+q(k) 0
which can be solved as an MILP optimization problem.
Another approach is to use the Farkas’ lemma [5] to obtain
an MILP optimization problem that is equivalent to (22)-(23).
Let J( ˜u, ˜e)=C
T
1
˜u(k)+C
T
2
˜e(k) where C
1
and C
2
are vectors of
coefficients. For a fixed value of t, we can rewrite (22)-(23)
as
min
˜u(k)
C
T
1
˜u(k)+t (30)
s.t. C
T
2
˜e(k) t 8 ˜e(k) : S ˜e(k) ˜q (31)
P(k) ˜u(k )+Q(k)˜e(k)+q(k) 0 (32)
Now, using Farkas’ lemma, we can rewrite the constraints as
C
T
2
˜e(k)t = µ
T
(S ˜e(k) ˜q)+a (33)
P
i
(k) ˜u(k )+Q
i
(k) ˜e(k )+q
i
(k)=b
i
+ L
i
(S ˜e(k) ˜q) (34)
i = 1,...,m
L, µ 0, a,b 0 (35)
where µ and L are Lagrange multipliers such that µ is a
vector of size ˜q, L is a matrix of size m n
˜q
and n
˜q
is the
size of vector ˜q. Let m denote the number of constraints in
(23), b is a vector of length m, and a is a scalar. So we can
rewrite (30)-(32) as
min
˜u,µ,L,a,b
C
T
1
˜u +t (36)

s.t. C
2
= S
T
µ (37)
t = µ
T
˜q a (38)
Q
i
(k)=S
T
L
i
i = 1,...,m (39)
P
i
(k) ˜u(k )+q
i
(k)=b
i
L
i
˜qi= 1,...,m (40)
L, µ 0, a,b 0 (41)
which is an MILP optimization problem. This problem will
be solved for different values of t and then, we choose the
optimal ˜u that have the minimum objective function C
T
1
˜u +t
for all these values of t. Note that since in our model for
the STG, the cost function is not explicitly dependent on
˜e(k), we can solve the optimization problem (36)-(41) by
eliminating t and the constraints (37) and (38).
V. E X A M P L E
In this section, we solve robust MPC optimization problem
to obtain a day-ahead prediction for the heat production plan
for a small network of greenhouses and we compare the
results with the ones obtained using robust optimal control
approach. In this case study, we consider two greenhouses
and an external producer. Each of the greenhouses has a CHP
unit, a boiler, and a buffer. The aim is to minimize the heat
production cost of the network while satisfying the network
constraints and the supply-demand balance.
Fig. 2. Physical topology of the thermal network of the case study
We consider the cost function (16) and we an uncertain
heat demand H
D j
(k)=H
D,pred j
(k)+e
j
(k) where H
D,pred j
(k)
is the predicted heat demand for greenhouse j 2{1,2} at
time step k and e
j
(k) denotes the uncertainty such that
|e
j
(k)|1. We also have U
¯
on
CHP j
= 5 and U
¯
of f
CHP j
= 3.
The cost function is minimized subject to the constraints
(6)-(9) and (17)-(21). Using MPC with a prediction horizon
N
p
= 15, at each time step we have 285 control variables
(150 binary variables including the auxiliary variables from
the MLD model), an uncertainty vector of size 30, and 660
inequality constraints. In the optimal control approach since
we optimize the system for 24 hours, we have 456 control
variables (240 binary variables), an uncertainty vector of size
48, and 1056 inequality constraints. For both approaches, we
apply the three different methods explained in Section IV.
Even though we picked a very small network to be able to
use mp-MILP approach, solving the mp-MILP optimization
problem using the MPT toolbox seems to be very inefficient
for a problem of this size. Even for a small prediction
horizon such as N
p
= 3, the computation time is so long that
makes this approach infeasible for this example. Therefore,
we only use the Monte Carlo approach and the equivalent
reformulation of min-max problem based on Farkas’ lemma
(cf. Section IV) to obtain the day-ahead heat production plan,
which is solved in Matlab R2014b on a 2.6 GHz Intel Core
i5 processor. To solve the optimization problem (28)-(29),
we chose M = 500 different uncertainty vectors e to obtain
a0.95% confidence level with accuracy error of 1% and we
use the MILP solver from IBM CPLEX. The computation
time and the total heat production cost of the network during
24 hours are given in Table II. The total production costs
Control Solution Time Production Cost
Approach Approach (s) (e)
Robust MPC Monte Carlo 3180 22576
Farkas’ lemma 116 22325
Robust Optimal Control Monte Carlo 207 23190
Farkas’ lemma 13 22708
TABLE II
COMPUTATION TIME AND TOTAL PRODUCTION COST FOR THE NETWORK
of this network for the case that the greenhouses do not
exchange heat among each other and only buy from the
external heat producers is 30194e. The choice between the
MPC approach and the optimal control approach is a matter
of trade-off between the computation time and the production
cots. The MPC computation time can be reduced by choosing
a smaller prediction horizon N
p
. In this example we have
chosen N
p
= 15 to show that even for a large prediction
horizon, this approach is still computationally feasible.
Fig. 3. Heat and electricity production plan of the thermal network using
robust MPC approach vs. robust optimal control (OC) approach.
The optimization results using robust optimal control ver-
sus the robust MPC, solved using the reformulation technique
based on Farkas’ lemma, are shown in Figure 3. The first
plot of Figure 3 shows the heat demand of each greenhouse
for one day. The second and third plots show the amount
of electricity and heat that needs to be generated by the
CHP and boiler units at each greenhouse, respectively. The
forth plot shows the amount of imported heat from the

Citations
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Posted Content
Abstract: This paper presents an energy management framework for building climate comfort (BCC) systems interconnected in a grid via aquifer thermal energy storage (ATES) systems in the presence of two types of uncertainty (private and common). ATES can be used either as a heat source (hot well) or sink (cold well) depending on the season. We consider the uncertain thermal energy demand of individual buildings as a private uncertainty source and the uncertain common resource pool (ATES) between neighbors as a common uncertainty source. We develop a large-scale stochastic hybrid dynamical model to predict the thermal energy imbalance in a network of interconnected BCC systems together with mutual interactions between their local ATES. We formulate a finite-horizon mixed-integer quadratic optimization problem with multiple chance constraints at each sampling time, which is in general a non-convex problem and difficult to solve. We then provide a computationally tractable framework by extending the so-called robust randomized approach and offering a less conservative solution for a problem with multiple chance constraints. A simulation study is provided to compare completely decoupled, centralized and move-blocking centralized solutions. We also present a numerical study using a geohydrological simulation environment (MODFLOW) to illustrate the advantages of our proposed framework.

26 citations


Journal ArticleDOI
TL;DR: A large-scale stochastic hybrid dynamical model to predict the thermal energy imbalance in a network of interconnected BCC systems together with mutual interactions between their local ATES and a computationally tractable framework is provided.
Abstract: This paper presents an energy management framework for building climate comfort (BCC) systems interconnected in a grid via aquifer thermal energy storage (ATES) systems in the presence of two types of uncertainty (private and common). ATES can be used either as a heat source (hot well) or sink (cold well) depending on the season. We consider the uncertain thermal energy demand of individual buildings as a private uncertainty source and the uncertain common resource pool (ATES) between neighbors as a common uncertainty source. We develop a large-scale stochastic hybrid dynamical model to predict the thermal energy imbalance in a network of interconnected BCC systems together with mutual interactions between their local ATES. We formulate a finite-horizon mixed-integer quadratic optimization problem with multiple chance constraints at each sampling time, which is in general a non-convex problem and difficult to solve. We then provide a computationally tractable framework by extending the so-called robust randomized approach and offering a less conservative solution for a problem with multiple chance constraints. A simulation study is provided to compare completely decoupled, centralized and move-blocking centralized solutions. We also present a numerical study using a geohydrological simulation environment (MODFLOW) to illustrate the advantages of our proposed framework.

21 citations


Cites methods from "Robust model predictive control for..."

  • ...STGs with uncertain thermal energy demands have been considered in [27], where a MPC strategy...

    [...]


Journal ArticleDOI
Abstract: Future district heating systems (DHS) will be supplied by renewable sources, most of which are limited in temperature and flow rate. Therefore, operational optimization of DHS is required to maximize the use of renewable sources and minimize (fossil) peak loads. In this paper, we present a robust and fast model-predictive control approach to use the thermal mass of buildings as a daily storage without violating temperature constraints. The novelty of this paper includes two elements. First, the focus on an operational control strategy that explicitly accounts for temperature-limited renewable sources, like a geothermal source. Secondly, the optimization problem is formulated as a (nearly) convex optimization problem, which is required for adoption of model-predictive control in practice. The examples show that the peak heating demand can be reduced by 50%, if the thermal inertia of the buildings is used and the heating setpoints are adapted. Furthermore, the operational optimization finds the proper balance between benefits of pre-heating using renewable sources with limited capacity and costs of additional heat losses due to pre-heating.

5 citations


Journal ArticleDOI
01 Jun 2021-Energy
Abstract: District heating networks transport thermal energy from one or more sources to a plurality of consumers. Lowering the operating temperatures of district heating networks is a key research topic to reduce energy losses and unlock the potential of low-temperature heat sources, such as waste heat. With an increasing share of uncontrolled heat sources in district heating networks, control strategies to coordinate energy supply and network operation become more important. This paper focuses on the modeling, control, and optimization of a low-temperature district heating network, presenting a case study with a high share of waste heat from high-performance computers. The network consists of heat pumps with temperature-dependent characteristics. In this paper, quadratic correlations are used to model temperature characteristics. Thus, a mixed-integer quadratically-constrained program is presented that optimizes the operation of heat pumps in combination with thermal energy storages and the operating temperatures of a pipe network. The network operation is optimized for three sample days. The presented optimization model uses the flexibility of the thermal energy storages and thermal inertia of the network by controlling its flow and return temperatures. The results show savings of electrical energy consumption of 1.55%–5.49%, depending on heat and cool demand.

5 citations


01 Jan 2018
TL;DR: A cooperative multi-agent system (MAS) hierarchical model predictive control (HMPC) implementation is presented as smart controller for 4GDH networks, and as alternative approach to improving TNO’s HeatMatcher (HM) algorithm with the proposed algorithm of PowerMatchers (PM) that relies on locational marginal pricing (LMP).
Abstract: In The Netherlands, the current heat energy system accounts for 44% of the primary energy usage and relies almost entirely on fossil fuels such as natural gas. To meet the Paris Climate Agreement goals, 4th Generation District Heating (4GDH) networks are expected as sustainable heat energy system solution. The concept relies on optimally matching the heat energy supply of sources such as waste heat, combined heat and power (CHP) plants and geothermal energy, with the demand of consumers such as households or greenhouses, whilst using the flexibility of buffers such as aquifer thermal energy storage (ATES) systems. In this thesis, a cooperative multi-agent system (MAS) hierarchical model predictive control (HMPC) implementation is presented as smart controller for 4GDH networks, and as alternative approach to improving TNO’s HeatMatcher (HM) algorithm with the proposed algorithm of PowerMatcher (PM) that relies on locational marginal pricing (LMP). The model predictive control (MPC) approach is chosen mainly due to the advantage that it optimizes over a prediction horizon or time span, instead of a single time step. This allows to take into account heat energy demand predictions, time-based constraints, and the inherent dynamic characteristics of 4GDH systems such as buffer flexibility and the variable time delay present in the heat energy exchange. The centralized model predictive control (CMPC) control problem is formulated as a deterministic, MAS, mixed-integer quadratic programming (MIQP) optimization problem and is subsequently distributed based on the Optimal Exchange Problem formulation using the alternating direction method of multipliers (ADMM). Hybrid system modelling theory is applied to model the agents’ subsystems and a simplified heat energy exchange model with constant time delay is assumed. The latter was chosen as decoupled thermal and hydraulic equations proved to be non-linear in the valve positions and mass flow, iterative due to the friction factor and the Reynolds number, and dependent on a variable spatial sampling to accurately track the thermal propagation through the network. The CMPC and HMPC algorithms are applied on an academic initial design case study to test desired controller behaviour under perfect heat demand prediction, and on a more realistic case study of the WarmCO2 heat grid involving 5 greenhouses with non-perfect heat demand predictions during a summer and winter scenario. The initial case study confirms that both algorithms perform as desired with the exception of a small shortcoming of the hybrid modelling, and that they are similar in their optimal solutions as expected. The same holds for the WarmCO2 case study. However, it also showcases that the deterministic optimization can become infeasible due to the time delay modelling and non-perfect heat demand predictions, and that therefore a stochastic optimization approach is preferred. Furthermore, good quality local optimal solutions of the NP-hard problem could be found within a relatively short computing time limit, using the heuristic methods of the Gurobi solver. And lastly, the importance of developing a non-cooperative MPC algorithm to accurately represent the individual optimization goals of different stakeholders.

3 citations


Cites background or methods from "Robust model predictive control for..."

  • ...A deterministic demand profile without uncertainty is modelled, meaning that the optimization method also does not have to take into account any uncertain demand predictions during optimization, as is done in [23], [24] and [25]....

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  • ...In [19] and [24], the profit from selling electricity generated by the CHP is also included....

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  • ...In [23] and [24], a MAS MPC control framework is presented that additionally focuses on dealing with the uncertainty in heat energy demand [25], whereas in this thesis a straightforward deterministic heat demand is assumed [26]....

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  • ...Also, the mixed-integer constraints found in [23], [24], [29] for the production capacity, ramping and minimum required on/off times are slightly adjusted with the help of hybrid modelling theory found in [30]....

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  • ...For CHP specifically, this can include the coupling between heat and electricity production [24]....

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TL;DR: A standard formulation of Predictive Control is presented, with examples of step response and transfer function formulations, and a case study of robust predictive control in the context of MATLAB.
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TL;DR: The flexible constraint handling capabilities of MPC are shown to be a significant advantage in the context of the overall operating objectives of the process industries and the 1-, 2-, and ∞-norm formulations of the performance objective are discussed.
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TL;DR: A unique vision for the future of smart transmission grids is presented in which their major features are identified and each smart transmission grid is regarded as an integrated system that functionally consists of three interactive, smart components.
Abstract: A modern power grid needs to become smarter in order to provide an affordable, reliable, and sustainable supply of electricity. For these reasons, considerable activity has been carried out in the United States and Europe to formulate and promote a vision for the development of future smart power grids. However, the majority of these activities emphasized only the distribution grid and demand side leaving the big picture of the transmission grid in the context of smart grids unclear. This paper presents a unique vision for the future of smart transmission grids in which their major features are identified. In this vision, each smart transmission grid is regarded as an integrated system that functionally consists of three interactive, smart components, i.e., smart control centers, smart transmission networks, and smart substations. The features and functions of each of the three functional components, as well as the enabling technologies to achieve these features and functions, are discussed in detail in the paper.

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  • ...Because of this capability, STG implementation could contribute to a further decrease in carbon emissions, improved energy efficiency, and renewable energy implementation [7], [18]....

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Frequently Asked Questions (2)
Q1. What are the contributions in "Robust model predictive control for an uncertain smart thermal grid" ?

The focus of this paper is on modeling and control of Smart Thermal Grids ( STGs ) in which the uncertainties in the demand and/or supply are included. The authors solve the corresponding robust model predictive control ( MPC ) optimization problem using mixed-integer-linear programming techniques to provide a day-ahead prediction for the heat production in the grid. There, the authors show that the robust MPC approach successfully keeps the supply-demand balance in the STG while satisfying the constraints of the production units in the presence of uncertainties in the heat demand. Moreover, the authors see that despite the longer computation time, the performance of the robust MPC controller is considerably better than the one of the robust optimal controller. 

In a case study, the authors compared the MPC approach with the optimal control approach to obtain a day-ahead production plan for a sample network of greenhouses. Moreover, in future work, the authors will also consider the constraints of the physical network ’ s model to be able to take the dependencies and the possible delays into account. An alternative scenario to the centralized control architecture is that each greenhouse tries to maximize its own benefit and hence, they will sell heat to the other greenhouses in the network.