# Nonlinear Fuzzy Model Predictive Control for a PWR Nuclear Power Plant

Authors: Xiangjie Liu; Mengyue Wang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Reliable power and temperature control in pressurized water reactor (PWR) nuclear power plant is necessary to guarantee high efficiency and plant safety. Since the nuclear plants are quite nonlinear, the paper presents nonlinear fuzzy model predictive control (MPC), by incorporating the realistic constraints, to realize the plant optimization. T-S fuzzy modeling on nuclear power plant is utilized to approximate the nonlinear plant, based on which the nonlinear MPC controller is devised via parallel distributed compensation (PDC) scheme in order to solve the nonlinear constraint optimization problem. Improved performance compared to the traditional PID controller for a TMI-type PWR is obtained in the simulation.


---

## Body

## 1. Introduction

In nuclear power plants, steam energy is produced in the nuclear reactor from continuous fission of the atoms of the fuel. The steam is then used to drive the turbine and generator to produce electricity.Since burning fossil fuels in thermal power plant can have severe environment problem, the development of nuclear power plants should be encouraged. For example in China, although the supercritical and ultrasupercritical technique, as well as integrated gasification combined cycle (IGCC) plants, have been well developed, the coal consumption still reaches about 3.91 million tons in 2012, which comprised over 70% of national primary energy. Fine particles in the air measuring less than 2.5 micrometres reached 993 micrograms per cubic meter in Beijing on 12 January 2013, compared with the limit of 25 published by the WHO guidelines.Since nuclear power plants are the complex and nonlinear systems, it is a great challenge to control the power and temperature of the nuclear reactor, especially when wide-range power variations occur in the load following condition. Thus, various advanced control schemes have appeared during the past two decades, for example, the observer-based optimal state feedback assisted control [1, 2], the linear quadratic Gaussian with loop transfer recovery control [3, 4], the nonlinear control approach [5, 6], and the neural network/fuzzy approaches [7, 8].Modern nuclear power plants should respond to the load demand on the power grid, which demand high plant operation performance, subject to various kinds of constraints. Meanwhile, nuclear safety and radioactive pollution prevention have long been much concerned problem. It is therefore extremely important to reach the economical and safe operation to maximize the thermal efficiency of the nuclear power plant. Under these aims, model predictive control (MPC) shows its obvious advantage, since it is an advanced model-based control scheme. It performs an optimization procedure to calculate optimal control actions at every sampling period based on an explicit process model subject to process input, output, and state constraints. So far, MPC has been well constituted for thermal power plant control [9–11] and also in nuclear power plant water-level control [12–14]. Paper [15] reports a MPC application in a three-dimensional nuclear reactor analysis code for nuclear reactor power.Nuclear power plants are generally nonlinear, due to the frequent changes of the operating point right across the whole operation range. In general, the nonlinear model predictive control (NMPC) also online solves an optimization problem, by using the sequential quadratic program (SQP). The resulting nonlinear programming problems are usually nonconvex, and the online computational burden is generally large. Since the nonlinearities vary with operating power levels, as well as ageing effects [5, 16], fuzzy models are used to express the plant dynamic. MPCs are realized in a parallel distributed compensation (PDC) scheme; that is, for each local model, a MPC is designed. The overall controller, which is nonlinear in nature, is similarly constructed by combining local controllers via fuzzy inference. Simulations on the three-mile island- (TMI-) type pressurized water reactor (PWR) show the effectiveness of the proposed method.


## 2. The Plant Description

As shown in Figure1, nuclear fission reaction in the reactor releases tremendous energy, which is transferred from the reactor to the steam generator by the coolant, heating the water in the steam generator to generate abundant steam. Then the turbine is driven by steam to generate electricity. The main objective is to control the power and temperature of the reactor by inserting or elevating the control rods. Conventionally, reactor control system controls the average temperature of the reactor core coolant to track the reference temperature which is proportional to the turbine load, in order to guarantee well matching between the reactor power level and the load demand at the same time. This temperature control strategy, considering not only high thermal efficiency but also technical limits and economical and safe operation, is widely used in modern PWR power plants.Figure 1
PWR and steam generation process.However, the plant is complex and highly nonlinear whose parameters vary with the operation conditions. Meanwhile, realistic constraints in the system may lead to actuator saturation. The performance of the conventional controllers is often unsatisfactory. Besides, unpredictable disturbances in the temperature measurement system will induce fluctuations on the measuring values of coolant temperature, which may lead to the abnormal control rod action and thus must be considered in the controller design. The proposed Takagi-Sugeno fuzzy modeling method is used to approximate the nonlinear plant, based on which the nonlinear MPC controller is devised via PDC scheme in order to solve the nonlinear and constraint problems in the reactor power and temperature control system.


## 3. T-S Fuzzy Modeling

The reactor model considered is the point kinetics with one delayed neutron group and also coolant and fuel temperature feedback. It is typical for a TMI-type PWR at the middle of the fuel cycle rated at 2500 MW [2, 7]:
(1)
d
d
t
n
r
=
δ
ρ
-
β
Λ
n
r
+
λ
c
r
,
d
d
t
c
r
=
λ
n
r
-
λ
c
r
,
d
d
t
T
f
=
f
f
P
a
0
μ
f
n
r
-
Ω
μ
f
T
f
d
d
t
T
f
=
+
Ω
2
μ
f
T
l
+
Ω
2
μ
f
T
e
,
d
d
t
T
l
=
(
1
-
f
f
)
P
a
0
μ
c
n
r
+
Ω
μ
c
T
f
d
d
t
T
l
=
-
(
2
M
+
Ω
)
2
μ
c
T
l
+
(
2
M
-
Ω
)
2
μ
c
T
e
,
d
d
t
δ
ρ
r
=
G
r
z
r
,
δ
ρ
=
δ
ρ
r
+
α
f
(
T
f
-
T
f
0
)
+
α
c
2
(
T
l
-
T
l
0
)
+
α
c
2
(
T
e
-
T
e
0
)
.The symbols in the above equations are demonstrated in the Nomenclature section.Define(2)
δ
n
r
=
n
r
-
n
r
0
,
δ
c
r
=
c
r
-
c
r
0
,
δ
T
f
=
T
f
-
T
f
0
,
δ
T
l
=
T
l
-
T
l
0
,
δ
ρ
r
=
ρ
r
-
ρ
r
0
,
where the symbol δ indicates a deviation about an equilibrium point. n
r
0, c
r
0, T
f
0, T
l
0, and ρ
r
0 correspond to the values of n
r, c
r, T
f, T
l, and ρ
r at an equilibrium point, respectively.Choose the state, input, and output vector, respectively, as follows:(3)
x
=
[
δ
n
r
δ
c
r
δ
T
f
δ
T
l
δ
ρ
r
]
T
,
u
=
z
r
,
y
=
[
δ
n
r
δ
T
l
]
T
.The model can be linearized at the equilibrium point using the perturbation theory which is valid only forδ
n
r
≪
n
r
0. The linear model can be written as the following state space form:
(4)
x
˙
=
A
x
+
B
u
,
y
=
C
x
.
The corresponding matrices A, B, and C, are
(5)
A
=
[
-
β
Λ
β
Λ
n
r
0
i
α
f
Λ
n
r
0
i
α
c
2
Λ
n
r
0
i
Λ
λ
-
λ
0
0
0
f
f
P
0
a
μ
f
0
-
Ω
μ
f
Ω
2
μ
f
0
(
1
-
f
f
)
P
0
a
μ
c
0
Ω
μ
c
-
2
M
+
Ω
2
μ
c
0
0
0
0
0
0
]
,
B
=
[
0
0
0
0
G
r
i
]
T
,
C
=
[
1
0
0
0
0
0
0
0
1
0
]
,
where n
r
0
i is the power level and G
r
i is the control rod worth for the ith (
i
=
1,2
,
…
,
9
) operating point which is defined in Table 1. The linear model (4) is defined at each operating point (
n
r
0
i
,
G
r
i
) and the parameters in (6) are functions of the operating point [7, 17]. The remaining constant parameters are shown in Table 2. Consider the following:
(6)
α
f
(
n
r
0
)
=
(
n
r
0
-
4.24
)
×
1
0
-
5
δ
k
k
/
C
°
,
α
c
(
n
r
0
)
=
(
-
4.0
n
r
0
-
17.3
)
×
1
0
-
5
δ
k
k
/
C
°
,
μ
c
(
n
r
0
)
=
(
160
9
n
r
0
+
54.022
)
MW
/
C
°
,
Ω
(
n
r
0
)
=
(
5
3
n
r
0
+
4.9333
)
MW
/
C
°
,
M
(
n
r
0
)
=
(
28.0
n
r
0
+
74.0
)
MW
/
C
°
.Table 1
The definition of the operating points.
G
r
n
r
0
0.1
0.5
1.0
0.029
Point 3
Point 4
Point 5
0.0145
Point 2
Point 1
Point 6
0.0070
Point 9
Point 8
Point 7Table 2
The constant parameters in the system matrices.
λ
0.15 s−1
β
0.006019
f
f
0.92
Λ
0.0001 s
μ
f
23.6 MWs/°C
P
a
0
2500.0 MWThe sampling period is selected to be 1 s, and then nine discrete-time local models of (4) can be described as follows:
(7)
x
(
k
+
1
)
=
G
i
x
(
k
)
+
H
i
u
(
k
)
,
y
(
k
)
=
C
i
x
(
k
)
00000000000
(
i
=
1,2
,
…
,
9
)
,
where(8)
G
1
=
[
5.42
×
1
0
-
1
4.51
×
1
0
-
1
-
1.40
×
1
0
-
2
-
3.59
×
1
0
-
2
3.75
×
1
0
1
1.12
×
1
0
-
3
9.99
×
1
0
-
1
-
1.16
×
1
0
-
5
-
2.97
×
1
0
-
5
3.09
×
1
0
-
2
6.54
×
1
0
-
1
2.17
×
1
0
-
1
9.91
×
1
0
-
1
-
1.62
×
1
0
-
2
1.80
×
1
0
1
2.40
×
1
0
-
2
7.92
×
1
0
-
3
6.63
×
1
0
-
4
9.85
×
1
0
-
1
6.58
×
1
0
-
1
0
0
0
0
1
]
,
G
2
=
[
5.47
×
1
0
-
1
4.52
×
1
0
-
1
-
3.11
×
1
0
-
3
-
6.60
×
1
0
-
3
7.51
1.13
×
1
0
-
3
9.99
×
1
0
-
1
-
2.60
×
1
0
-
6
-
5.50
×
1
0
-
6
6.19
×
1
0
-
3
6.56
×
1
0
-
1
2.17
×
1
0
-
1
9.97
×
1
0
-
1
-
2.22
×
1
0
-
3
3.61
2.70
×
1
0
-
2
8.93
×
1
0
-
3
8
.
45
×
1
0
-
4
9.86
×
1
0
-
1
1.48
×
1
0
-
1
0
0
0
0
1
]
,
G
6
=
[
5.47
×
1
0
-
1
4.52
×
1
0
-
1
-
2.43
×
1
0
-
3
-
7.94
×
1
0
-
3
7.51
×
1
0
1
1.13
×
1
0
-
3
9.99
×
1
0
-
1
-
2.00
×
1
0
-
6
-
6.60
×
1
0
-
6
6.19
×
1
0
-
2
6.56
×
1
0
-
1
2.17
×
1
0
-
1
9.96
×
1
0
-
1
-
2.58
×
1
0
-
3
3.61
×
1
0
1
2.11
×
1
0
-
2
6.96
×
1
0
-
3
8.74
×
1
0
-
4
9.85
×
1
0
-
1
1.16
0
0
0
0
1
]
,
G
1
=
G
4
=
G
8
,
G
2
=
G
3
=
G
9
,
G
5
=
G
6
=
G
7
,
H
1
=
[
2.99
×
1
0
-
3
1.60
×
1
0
-
6
9.14
×
1
0
-
4
3.33
×
1
0
-
5
1.45
×
1
0
-
4
]
,
H
4
=
[
2.89
×
1
0
-
4
2.00
×
1
0
-
7
8.83
×
1
0
-
5
3.60
×
1
0
-
6
7.00
×
1
0
-
5
]
,
H
8
=
[
1.20
×
1
0
-
2
6.30
×
1
0
-
6
3.66
×
1
0
-
3
1.17
×
1
0
-
4
2.90
×
1
0
-
4
]
,
H
1
=
H
2
=
H
6
,
H
3
=
H
4
=
H
5
,
H
7
=
H
8
=
H
9
,
C
1
=
[
1
0
0
0
0
0
0
0
1
0
]
,
C
1
=
C
2
=
C
3
=
C
4
=
C
5
=
C
6
=
C
7
=
C
8
=
C
9
.According to Table1, power level n
r
0 and control rod worth G
r are selected as the premise variables. n
r
0 and G
r are defined in region [
0,1
] and region [0, 0.029], respectively. n
r
0 and G
r are divided into three fuzzy sets, denoted, respectively, as “low (L),” “medium (M),” and “high (H).” The corresponding univariate membership functions of n
r
0 or G
r are defined by triangle membership functions, as shown in Figure 2. Thus, the space is divided into nine fuzzy subspaces, given as follows:Figure 2
The division of fuzzy space and the definition of univariate membership functions.R
(
i
)
:
(
i
=
1
,
2
,
…
,
9
)
(9)
IF
z
1
(
k
)
is
F
1
i
and
z
2
(
k
)
is
F
2
i
THEN
x
i
(
k
+
1
)
=
G
i
x
i
(
k
)
+
H
i
u
i
(
k
)
y
i
(
k
)
=
C
i
x
i
(
k
)
,
where R
(
i
) denotes the ith fuzzy inference rule. z
j
(
k
)
(
j
=
1,2
) are premise variables for n
r
0 and G
r; F
j
i denotes the fuzzy set partitioned for the jth premise variable, which is valid in the ith rule. (G
i
,
H
i
,
C
i) represents the consequent ith linear model deduced at the ith equilibrium point in (7).By using the center of gravity defuzzification method [10, 18], the final output of the nonlinear T-S model with nine fuzzy rules is
(10)
x
(
k
+
1
)
=
∑
i
=
1
9
μ
-
i
x
i
(
k
+
1
)
=
∑
i
=
1
9
μ
-
i
(
G
i
x
i
(
k
)
+
H
i
u
i
(
k
)
)
y
(
k
)
=
∑
i
=
1
9
μ
-
i
y
i
(
k
)
=
∑
i
=
1
9
μ
-
i
C
i
x
i
(
k
)
,
where
(11)
μ
-
i
=
μ
i
∑
i
=
1
9
μ
i
,
(12)
μ
i
=
∏
j
=
1
2
α
F
j
i
(
z
j
(
k
)
)
.Theα
F
j
i
(
z
j
(
k
)
) denotes the univariate membership function of premise variable z
j
(
k
) with respect to the fuzzy set F
j
i. μ
i is the multivariate membership function obtained by the product of the outputs of the two univariate membership functions, as shown in Figure 3.Figure 3
Multivariate membership functions in the fuzzy subspace 1.Four rules are activated each time instant. At point A in Figure2, these four rules are as follows:R
1: if G
r
(
k
) is M and n
r
0
(
k
)
is
M,
then
x
1
(
k
+
1
)
=
G
1
x
1
(
k
)
+
H
1
u
1
(
k
)
y
1
(
k
)
=
C
1
x
1
(
k
).
R
6: if G
r
(
k
) is M and n
r
0
(
k
) is H,
then
x
6
(
k
+
1
)
=
G
6
x
6
(
k
)
+
H
6
u
6
(
k
)
y
6
(
k
)
=
C
6
x
6
(
k
).
R
7: if G
r
(
k
) is H and n
r
0
(
k
) is H,
then
x
7
(
k
+
1
)
=
G
7
x
7
(
k
)
+
H
7
u
7
(
k
)
y
7
(
k
)
=
C
7
x
7
(
k
).
R
8: if G
r
(
k
) is H and n
r
0
(
k
) is M,
then
x
8
(
k
+
1
)
=
G
8
x
8
(
k
)
+
H
8
u
8
(
k
)
y
8
(
k
)
=
C
8
x
8
(
k
).The membership functions are shown in Figure4.Figure 4
Multivariate membership functions activated at point A.


## 4. The Nonlinear Fuzzy MPC

### 4.1. The Local Linear MPC

The local linear model (7) can be expressed as follows:
(13)
x
(
k
)
=
G
x
(
k
-
1
)
+
H
u
(
k
-
1
)
,
y
(
k
-
1
)
=
C
x
(
k
-
1
)
.Subtracting (13) from (7) can result in
(14)
Δ
x
(
k
+
1
)
=
G
Δ
x
(
k
)
+
H
Δ
u
(
k
)
,
Δ
y
(
k
)
=
C
Δ
x
(
k
)
,
where Δ
x
(
k
)
=
x
(
k
)
-
x
(
k
-
1
), Δ
u
(
k
)
=
u
(
k
)
-
u
(
k
-
1
), and Δ
y
(
k
)
=
y
(
k
)
-
y
(
k
-
1
).Hence,Δ
y
(
k
+
1
)
=
C
Δ
x
(
k
+
1
)
=
C
G
Δ
x
(
k
)
+
C
H
Δ
u
(
k
).Choosing a new state variable vectorx
′
(
k
)
=
[
Δ
x
(
k
)
T
y
(
k
)
T
]
T can result in
(15)
[
Δ
x
(
k
+
1
)
y
(
k
+
1
)
]
=
[
G
0
C
G
I
]
[
Δ
x
(
k
)
y
(
k
)
]
+
[
H
C
H
]
Δ
u
(
k
)
,
y
(
k
)
=
[
0
I
]
[
Δ
x
(
k
)
y
(
k
)
]
.Denote (15) as the following normal form:
(16)
x
′
(
k
+
1
)
=
G
′
x
′
(
k
)
+
H
′
u
(
k
)
,
y
(
k
)
=
C
′
x
′
(
k
)
,
where x
′
(
k
)
∈
R
n
′
(
n
′
=
n
+
p
), G
′
=
[
G
0
C
G
I
], H
′
=
[
H
C
H
], and C
′
=
[
0
I
]. n and p are, respectively, the number of the state variables and the output variables.The predictive control is obtained by minimizing the following cost function:(17)
J
=
∥
Y
-
r
s
∥
Q
2
+
∥
Δ
U
∥
R
2
.The control is obtained by minimizing the cost function at each time instantk.Define(18)
Y
=
[
y
^
(
k
+
1
∣
k
)
T
y
^
(
k
+
2
∣
k
)
T
⋯
y
^
(
k
+
N
P
∣
k
)
T
]
T
,
r
s
=
[
r
(
k
+
1
)
T
r
(
k
+
2
)
T
⋯
r
(
k
+
N
P
)
T
]
T
,
Δ
U
=
[
Δ
u
(
k
)
Δ
u
(
k
+
1
)
⋯
Δ
u
(
k
+
N
c
-
1
)
]
T
,
Q
=
[
Q
(
1
)
0
⋯
0
0
Q
(
2
)
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
Q
(
N
p
)
]
,
R
=
[
R
(
0
)
0
⋯
0
0
R
(
1
)
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
R
(
N
c
-
1
)
]
,
where Q and R are the weighting matrices for the prediction error and the control, respectively, N
p and N
c are, respectively, the costing horizon for the prediction error and the control, and y
^
(
k
+
j
∣
k
)
T and r
(
k
+
j
)
T
(
j
=
1,2
,
…
,
N
p
) are the optimum jth step ahead prediction of the system output on data up to time instant k and the jth step ahead reference trajectory, respectively. Δ
U is the vector composed of the future control increments which is obtained by computing the derivative of J, expressed as follows:
(19)
Δ
U
=
(
R
+
Φ
T
Q
Φ
)
-
1
Φ
T
Q
(
r
s
-
F
x
(
k
)
)
,
where(20)
F
=
[
C
′
G
′
C
′
G
′
2
C
′
G
′
3
⋮
C
′
G
′
N
p
]
,
Φ
=
[
C
′
H
′
0
0
⋯
0
C
′
G
′
H
′
C
′
H
′
0
⋯
0
C
′
G
′
2
H
′
C
′
G
′
H
′
C
′
H
′
⋯
0
⋮
⋮
⋮
⋱
⋮
C
′
G
′
N
p
-
1
H
′
C
′
G
′
N
p
-
2
H
′
C
′
G
′
N
p
-
3
H
′
⋯
C
′
G
′
N
p
-
N
c
H
′
]
.The optimizedN
c steps ahead control Δ
U is computed, and only the first step ahead control Δ
u
(
k
) is implemented, using a receding horizon principle [10], giving
(21)
Δ
u
(
k
)
=
[
1
0
⋯
0
]
︸
N
c
Δ
U
;
further, u
(
k
) can be obtained by
(22)
u
(
k
)
=
u
(
k
-
1
)
+
Δ
u
(
k
)
.The input constraint is expressed as follows:(23)
U
min
⁡
≤
U
≤
U
max
⁡
.
U can be expressed by Δ
U as follows:
(24)
[
u
(
k
)
u
(
k
+
1
)
u
(
k
+
2
)
⋮
u
(
k
+
N
c
-
1
)
]
=
[
1
1
1
⋮
1
]
u
(
k
-
1
)
+
[
1
0
0
⋯
0
1
1
0
⋯
0
1
1
1
⋯
0
⋮
⋮
⋮
⋱
⋮
1
1
1
⋯
1
]
[
Δ
u
(
k
)
Δ
u
(
k
+
1
)
Δ
u
(
k
+
2
)
⋮
Δ
u
(
k
+
N
c
-
1
)
]
.Define(25)
C
1
=
[
1
1
1
⋮
1
]
,
C
2
=
[
1
0
0
⋯
0
1
1
0
⋯
0
1
1
1
⋯
0
⋮
⋮
⋮
⋱
⋮
1
1
1
⋯
1
]
,
ψ
=
[
-
C
2
C
2
]
,
γ
=
[
-
U
min
⁡
+
C
1
u
(
k
-
1
)
U
max
⁡
-
C
1
u
(
k
-
1
)
]
,
and constraint (23) can be rewritten as follows:
(26)
ψ
Δ
U
≤
γ
.Hence, the constrained optimization problem can be described as(27)
min
⁡
J
=
∥
Y
-
r
s
∥
Q
2
+
∥
Δ
U
∥
R
2
s
.
t
.
000
ψ
Δ
U
≤
γ
which can be solved by quadratic programming (QP) method.


### 4.2. The Nonlinear Fuzzy MPC

Based on the T-S fuzzy model, the controller is devised via the parallel distributed control (PDC) scheme [19, 20]. In this scheme, the overall controller, which is naturally nonlinear, is a fuzzy combination of each individual linear controller designed based on the corresponding local model. Here, a nonlinear MPC controller, consisting of nine local MPC controllers, is constituted as shown in Figure 5. For each local linear model, a conventional MPC controller is designed independently. Consequently, the final control is the weighted sum of the control obtained from nine local MPC controllers. Consider the following:
(28)
u
(
k
)
=
∑
i
=
1
9
μ
-
i
u
i
(
k
)
,
where u
i
(
k
) is the control of the ith local predictive controller and μ
-
i is defined previously in (11). Since fuzzy logics dominate the switching between local controllers, the resulting control can be smooth by properly selecting the weights μ
-
i [10].Figure 5
The schematic diagram of fuzzy MPC system.


## 4.1. The Local Linear MPC

The local linear model (7) can be expressed as follows:
(13)
x
(
k
)
=
G
x
(
k
-
1
)
+
H
u
(
k
-
1
)
,
y
(
k
-
1
)
=
C
x
(
k
-
1
)
.Subtracting (13) from (7) can result in
(14)
Δ
x
(
k
+
1
)
=
G
Δ
x
(
k
)
+
H
Δ
u
(
k
)
,
Δ
y
(
k
)
=
C
Δ
x
(
k
)
,
where Δ
x
(
k
)
=
x
(
k
)
-
x
(
k
-
1
), Δ
u
(
k
)
=
u
(
k
)
-
u
(
k
-
1
), and Δ
y
(
k
)
=
y
(
k
)
-
y
(
k
-
1
).Hence,Δ
y
(
k
+
1
)
=
C
Δ
x
(
k
+
1
)
=
C
G
Δ
x
(
k
)
+
C
H
Δ
u
(
k
).Choosing a new state variable vectorx
′
(
k
)
=
[
Δ
x
(
k
)
T
y
(
k
)
T
]
T can result in
(15)
[
Δ
x
(
k
+
1
)
y
(
k
+
1
)
]
=
[
G
0
C
G
I
]
[
Δ
x
(
k
)
y
(
k
)
]
+
[
H
C
H
]
Δ
u
(
k
)
,
y
(
k
)
=
[
0
I
]
[
Δ
x
(
k
)
y
(
k
)
]
.Denote (15) as the following normal form:
(16)
x
′
(
k
+
1
)
=
G
′
x
′
(
k
)
+
H
′
u
(
k
)
,
y
(
k
)
=
C
′
x
′
(
k
)
,
where x
′
(
k
)
∈
R
n
′
(
n
′
=
n
+
p
), G
′
=
[
G
0
C
G
I
], H
′
=
[
H
C
H
], and C
′
=
[
0
I
]. n and p are, respectively, the number of the state variables and the output variables.The predictive control is obtained by minimizing the following cost function:(17)
J
=
∥
Y
-
r
s
∥
Q
2
+
∥
Δ
U
∥
R
2
.The control is obtained by minimizing the cost function at each time instantk.Define(18)
Y
=
[
y
^
(
k
+
1
∣
k
)
T
y
^
(
k
+
2
∣
k
)
T
⋯
y
^
(
k
+
N
P
∣
k
)
T
]
T
,
r
s
=
[
r
(
k
+
1
)
T
r
(
k
+
2
)
T
⋯
r
(
k
+
N
P
)
T
]
T
,
Δ
U
=
[
Δ
u
(
k
)
Δ
u
(
k
+
1
)
⋯
Δ
u
(
k
+
N
c
-
1
)
]
T
,
Q
=
[
Q
(
1
)
0
⋯
0
0
Q
(
2
)
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
Q
(
N
p
)
]
,
R
=
[
R
(
0
)
0
⋯
0
0
R
(
1
)
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
R
(
N
c
-
1
)
]
,
where Q and R are the weighting matrices for the prediction error and the control, respectively, N
p and N
c are, respectively, the costing horizon for the prediction error and the control, and y
^
(
k
+
j
∣
k
)
T and r
(
k
+
j
)
T
(
j
=
1,2
,
…
,
N
p
) are the optimum jth step ahead prediction of the system output on data up to time instant k and the jth step ahead reference trajectory, respectively. Δ
U is the vector composed of the future control increments which is obtained by computing the derivative of J, expressed as follows:
(19)
Δ
U
=
(
R
+
Φ
T
Q
Φ
)
-
1
Φ
T
Q
(
r
s
-
F
x
(
k
)
)
,
where(20)
F
=
[
C
′
G
′
C
′
G
′
2
C
′
G
′
3
⋮
C
′
G
′
N
p
]
,
Φ
=
[
C
′
H
′
0
0
⋯
0
C
′
G
′
H
′
C
′
H
′
0
⋯
0
C
′
G
′
2
H
′
C
′
G
′
H
′
C
′
H
′
⋯
0
⋮
⋮
⋮
⋱
⋮
C
′
G
′
N
p
-
1
H
′
C
′
G
′
N
p
-
2
H
′
C
′
G
′
N
p
-
3
H
′
⋯
C
′
G
′
N
p
-
N
c
H
′
]
.The optimizedN
c steps ahead control Δ
U is computed, and only the first step ahead control Δ
u
(
k
) is implemented, using a receding horizon principle [10], giving
(21)
Δ
u
(
k
)
=
[
1
0
⋯
0
]
︸
N
c
Δ
U
;
further, u
(
k
) can be obtained by
(22)
u
(
k
)
=
u
(
k
-
1
)
+
Δ
u
(
k
)
.The input constraint is expressed as follows:(23)
U
min
⁡
≤
U
≤
U
max
⁡
.
U can be expressed by Δ
U as follows:
(24)
[
u
(
k
)
u
(
k
+
1
)
u
(
k
+
2
)
⋮
u
(
k
+
N
c
-
1
)
]
=
[
1
1
1
⋮
1
]
u
(
k
-
1
)
+
[
1
0
0
⋯
0
1
1
0
⋯
0
1
1
1
⋯
0
⋮
⋮
⋮
⋱
⋮
1
1
1
⋯
1
]
[
Δ
u
(
k
)
Δ
u
(
k
+
1
)
Δ
u
(
k
+
2
)
⋮
Δ
u
(
k
+
N
c
-
1
)
]
.Define(25)
C
1
=
[
1
1
1
⋮
1
]
,
C
2
=
[
1
0
0
⋯
0
1
1
0
⋯
0
1
1
1
⋯
0
⋮
⋮
⋮
⋱
⋮
1
1
1
⋯
1
]
,
ψ
=
[
-
C
2
C
2
]
,
γ
=
[
-
U
min
⁡
+
C
1
u
(
k
-
1
)
U
max
⁡
-
C
1
u
(
k
-
1
)
]
,
and constraint (23) can be rewritten as follows:
(26)
ψ
Δ
U
≤
γ
.Hence, the constrained optimization problem can be described as(27)
min
⁡
J
=
∥
Y
-
r
s
∥
Q
2
+
∥
Δ
U
∥
R
2
s
.
t
.
000
ψ
Δ
U
≤
γ
which can be solved by quadratic programming (QP) method.


## 4.2. The Nonlinear Fuzzy MPC

Based on the T-S fuzzy model, the controller is devised via the parallel distributed control (PDC) scheme [19, 20]. In this scheme, the overall controller, which is naturally nonlinear, is a fuzzy combination of each individual linear controller designed based on the corresponding local model. Here, a nonlinear MPC controller, consisting of nine local MPC controllers, is constituted as shown in Figure 5. For each local linear model, a conventional MPC controller is designed independently. Consequently, the final control is the weighted sum of the control obtained from nine local MPC controllers. Consider the following:
(28)
u
(
k
)
=
∑
i
=
1
9
μ
-
i
u
i
(
k
)
,
where u
i
(
k
) is the control of the ith local predictive controller and μ
-
i is defined previously in (11). Since fuzzy logics dominate the switching between local controllers, the resulting control can be smooth by properly selecting the weights μ
-
i [10].Figure 5
The schematic diagram of fuzzy MPC system.


## 5. The Application to the PWR Nuclear Power Plant

In using the proposed fuzzy MPC, the reference temperature function of power level is defined as follows:(29)
T
l
0
=
290
+
24.51
n
r
0
.
Figure 6 shows this relationship.Figure 6
Reference temperature function of power level.Assume that the maximum speed of control rod is 0.2 fraction of core length per second. Hence, the constraint on the control is considered as-
0.2
≤
u
≤
0.2 which is involved in the optimization algorithm of MPC.Define(30)
Q
=
q
×
[
I
2
×
2
0
⋯
0
0
I
2
×
2
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
I
2
×
2
]
N
p
×
N
p
and R
=
r
×
I
N
c
×
N
c, where q and r are the two coefficients. Assume the control rod worth G
r to be 0.0145, the parameters of the nine local controllers are listed in Table 3.Table 3
The parameters of the nine local controllers.
Parameters
i
th controller
1
2
3
4
5
6
7
8
9
q
1
1
1
1
1
1
1
1
1
r
0.07
0.01
0.02
0.07
0.1
0.1
0.1
0.05
0.1
N
p
380
430
400
340
290
300
325
400
460
N
c
50
45
45
50
60
60
60
40
30The performance of the nonlinear fuzzy MPC can be evaluated in three cases.Case A.
Full power operation, that is,100
%
→
90
%
→
100
% step changes in power level. Comparison of the performance between the proposed fuzzy MPC and the conventional PID controller is presented in Figures 7(a) and 7(b). The temperature and the power stabilize to their respect set points in an acceptable time, whereas the nonlinear fuzzy MPC reduces the fluctuation obviously. In real-time nuclear power control, inserting the control rod can slow down the release of energy in the reactor so that the temperature and power level are decreased, while lifting the control rod speed will raise the temperature and power level.Comparison of the performance between the nonlinear fuzzy MPC and the conventional PID controller (100
%
→
90
%
→
100
% power level change), (a) coolant temperature (unit: °C), (b) power level, and (c) control rod speed.
(a)
(b)
(c)Case B.
Low power operation, that is,20
%
→
10
%
→
20
% step changes in power level. Comparison of the performance between the proposed controller and the conventional PID controller is presented in Figures 8(a) and 8(b), showing the obvious advantage of the proposed fuzzy MPC. It is noticed that the response time is longer than that in the full power operation and the resulting fluctuation is more severe.
In both cases, the constraint on the control rod speed can be handled by the proposed fuzzy MPC effectively, while the PID controller cannot achieve it, as shown in Figures7(c) and 8(c).Comparison of the performance between the nonlinear fuzzy controller and the conventional PID controller (20
%
→
10
%
→
20
% power level change), (a) coolant temperature (unit: °C), (b) power level, and (c) control rod speed.
(a)
(b)
(c)Case C.
Wide-range load following, that is,100
%
→
10
%
→
100
% ramp changes in power level with 5%/min rate, as shown in Figure 9. With the constraint handling on the control rod speed by the fuzzy MPC, satisfactory tracking performance is obtained for both temperature and power.The performance for the nonlinear fuzzy controller (100
%
→
10
%
→
100
% ramp changes in power level with 5%/min rate), (a) coolant temperature (unit:°C), (b) power level, and (c) control rod speed.
(a)
(b)
(c)


## 6. Conclusions

The paper constituted a nonlinear MPC controller by incorporating fuzzy modeling technique. Feasible optimal solutions have been acquired under plant nonlinearity and constraints. The proposed nonlinear fuzzy MPC is simulated in power and temperature control for a three-mile island- (TMI-) type PWR. Better performance is obtained while it is compared with the PID controller. Load following is accomplished under constraints on the control rod.



---

*Source: 908526-2014-06-02.xml*