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Core Power Control of the fast nuclear reactors with estimation of the delayed neutron precursor density using Sliding Mode method
Nuclear Engineering and Design 296 (2016) 1–8
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Core Power Control of the fast nuclear reactors with estimation of the delayed neutron precursor density using Sliding Mode method G.R. Ansarifar ∗ , M.N. Nasrabadi, R. Hassanvand Department of Nuclear Engineering, Faculty of Advanced Sciences & Technologies, University of Isfahan, Darvaze Shiraz, Isfahan 81746-73441, Iran
h i g h l i g h t s • • • • •
We present a S.M.C. system based on the S.M.O for control of a fast reactor power. A S.M.O has been developed to estimate the density of delayed neutron precursor. The stability analysis has been given by means Lyapunov approach. The control system is guaranteed to be stable within a large range. The comparison between S.M.C. and the conventional PID controller has been done.
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 15 October 2015 Accepted 17 October 2015 M. Instrumentation and control
a b s t r a c t In this paper, a nonlinear controller using sliding mode method which is a robust nonlinear controller is designed to control a fast nuclear reactor. The reactor core is simulated based on the point kinetics equations and one delayed neutron group. Considering the limitations of the delayed neutron precursor density measurement, a sliding mode observer is designed to estimate it and finally a sliding mode control based on the sliding mode observer is presented. The stability analysis is given by means Lyapunov approach, thus the control system is guaranteed to be stable within a large range. Sliding Mode Control (SMC) is one of the robust and nonlinear methods which have several advantages such as robustness against matched external disturbances and parameter uncertainties. The employed method is easy to implement in practical applications and moreover, the sliding mode control exhibits the desired dynamic properties during the entire output-tracking process independent of perturbations. Simulation results are presented to demonstrate the effectiveness of the proposed controller in terms of performance, robustness and stability. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Considering the reducing fossil energy resources, pollution, the greenhouse gas and the economic efficiency, nuclear power is taken into consideration. Since thermal reactors are able to use such a small percentage of uranium fuel, the design, construction and operation of fast reactors is underway (Driscoll et al., 1979). By greatly simplifying the nuclear fuel cycle, fast reactors such as travelling wave reactor could improve the cost, safety, social acceptability, and long term sustainability of nuclear energy as a source of emissions-free electricity (Weaver et al., 2009). On
∗ Corresponding author. Tel.: +98 03117934392; fax: +98 03117934242. E-mail address: [email protected] (G.R. Ansarifar). http://dx.doi.org/10.1016/j.nucengdes.2015.10.015 0029-5493/© 2015 Elsevier B.V. All rights reserved.
the other hand, to use a nuclear power, reactor energy must be controlled. According to the reactor period dependency to the neutron lifetime and the crucial importance of this quantity in reactor power, it is necessary to have control on the reactor power (Hetrick, 1965). Due to the very short life time of prompt neutrons in the fast nuclear reactor, it is difficult to control its power. Therefore, the delayed neutrons which are produced indirectly by precursors must be considered to increase the reactor period. Besides, it seems that a simple and high performance control system is needed. A successful strategy to control uncertain nonlinear systems is Sliding Mode Control (SMC). The sliding mode controller is an attractive robust control algorithm because of its inherent insensitivity and robustness to plant uncertainties and external disturbances (Choi, 1999; Furat and Eker, 2012). In this paper, a SMC system is designed to control a fast nuclear reactor based on the point kinetics equations with one group of the delayed neutrons.
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Since the measurement of the delayed neutron precursor density is practically difficult and it should be measured to design the controller, a sliding mode observer which has the robust characteristics facing the external disturbances and parameters uncertainties is proposed based on the reactor power measurement to estimate the densities of delayed neutron precursors. Finally, a sliding mode control system based on the sliding mode observer is presented for controlling the fast nuclear reactor core power. Simulation results are provided to show the effectiveness of the proposed control system.
Table 1 Values of constants used for control analysis and simulations (Walter et al., 2012; Kim and Taiwo, 2012). Parameter
Values
Parameter
Values
ˇ M(MW/◦ C) P0 (MW) ff Tin(◦ C)
0.00334 0.085 1000 0.92 355
(1/s) l(s) Gr A (¢) B (¢)
0.0337 0.00000038 0.0405 −73 −41
◦
C ¢/ C
◦
c MWs/ C
−0. 4
f
0.06
◦ MWs/◦ C
0. 1802
˝ MW/ C
0.447
2. The reactor core model 2.1. The point kinetics equations Reactor kinetics equations for both fast and thermal reactors are identical. Point kinetics approximations can be used more effectively for fast reactors than for thermal reactors because fast reactors are more tightly coupled neutronically. Tighter coupling implies that the neutron flux is more nearly separable in space and time, which is a necessary condition for point kinetics approximations to be valid (Walter et al., 2012). To simulate the nuclear reactor core, point kinetics equations with one group of the delayed neutrons is used (Hetrick, 1965). The model assumes feedback from lumped fuel and coolant temperatures. The normalized model, with respect to an equilibrium condition, based on point kinetics equations with one delayed neutron group is as follows: ˇ −ˇ dnr nr + Cr = dt l l
(1)
dcr = nr − Cr dt
(2)
where nr , , cr , , l and ˇ refer to relative (normalized) power, reactivity, relative precursor density, decay constant of precursors, prompt neutron lifetime and effective delayed neutron fraction, respectively. Reactor temperatures vary as a function of power generated in fuel and heat transfer from (or to) the system. The reactor power can be represented as P (t) = nr P0
(3)
where P(t) is reactor power at time t (MW), and P0 is the nominal power (MW). 2.2. Fast reactor reactivity equation
P
(t) − 1 B + ıTin C + r Fr (t) r
dr = Gr zr dt
(4) (5)
A = ˛D TF (0) B = ˛D + ˛Na + ˛Axi + 2
XMC XAC
+ 2˛Crod (aLCRD − bLVHP )
˛Radi
T
2.3. Thermal-hydraulics model of the reactor core The thermal-hydraulics model of the reactor core can be represented with the following equations dTf dt
=
ff P0 f
nr −
˝ ˝ ˝ T + Tout + T f f 2f 2f in
(6)
2M + ˝ 2M − ˝ (1 − ff )P0 ˝ dTout nr + Tf − Tout + Tin = c c 2c 2c dt (7)
One of the most important parameters of the dynamics behaviour of the reactor is reactivity (Hetrick, 1965). Due to the fast neutron spectra in fast reactors, fast reactors reactivity equation is very different from thermal reactors, and appears as follows (Tentner et al., 2010): = [Pr (t) − 1] A +
where Pr (t) is the normalized reactor power, Fr (t) is the normalized core coolant flow, A, B, C is the reactivity feedback parameters, ˛D is the doppler coefficient (pcm/◦ C), ˛Na is the sodium density coefficient (pcm/◦ C), ˛Axi is the fuel axial exp ansion coefficient (pcm/◦ C), ˛Radi is the core radial expansion coefficient (pcm/◦ C), ˛Crod is the control rod driveline expansion coefficient (pcm/m), XMC is the grid plate to core midplane distance (m), XAC is the grid plate to above − core load plane distance (m), a is the thermal expansion coefficient of the control rod driveline (1/◦ C), b is the thermal expansion coefficient of the vessel wall (1/◦ C), LCRD is the length of control rod driveline in contact with the hot pool (m), LVHP is the length of vessel wall in contact with the hot pool (m), LVCP is the length of vessel wall in contact with the cold pool (m), is the reactor reactivity change, r is the control rods reactivity (pcm), Gr is the reactivity worth of rod per unit length (pcm/cm), zr is the control input, control rod speed (cm/s), TF (0) is the steady state Temperature difference, fuel to coolant, Tc (0) is the steady state Coolant temperature rise, inlet to outlet.
c
(0)
2
C = ˛D + ˛Na + ˛Axi + ˛Radi + ˛Crod · (aLCRD − bLVHP − bLVCP )
where ff is the fraction of reactor power deposited in the fuel,
M is the mass flow rate times heat capacity of Na MW/◦ C , f is the fuel mass times specific heat MWs/◦ C , c is ◦
˝ is the heat transfer between fuel and coolant MW/◦ C ,
the coolant mass times specific heat MWs/ C , Tf is the fuel temperature (◦ C), Tout is the Temperature of coolant leaving the core (◦ C), Tin is the Temperature of coolant entering the core (◦ C). The parameter values of the reactor model are given in Table 1 as follows: In Table 1, kinetics parameters for the fast spectrum neutrons such as prompt neutron lifetime and effective delayed neutron fraction were taken from Walter et al. (2012) which are consistent for all the fast nuclear reactors and the integral reactivity parameters A, B, C and detailed reactivity coefficients of fast reactors depend to the nuclear reactor type and fuel type which in this paper they were taken from Kim and Taiwo (2012) for ABR-1000 with oxide fuel which is a fast nuclear reactor. Therefore, kinetics parameters and reactivity parameters are consistent to each other.
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
Eq. (6) describes the rate of change of fuel temperature, which is equal to the heat energy produced in reactor minus the heat energy transferred to the coolant, whole divided by the heat capacity of fuel. Heat capacity of fuel is obtained by multiplying the mass of fuel by its specific heat at the desired operating temperature. Eq. (7) describes the rate of change of coolant temperature leaving the reactor core which is the difference of heat energy transferred from fuel to coolant and amount of heat energy transferred from coolant to the secondary circuit in steam generator whole divided by the heat capacity of the coolant. Heat capacity of coolant is obtained by multiplying the mass of coolant with its specific heat at the desired operating conditions of temperature and pressure. 3. Preliminary
3.1. Sliding mode control Sliding mode control is a variable structure control system. Since it has systematic design procedure, it is one of the most powerful solutions for any practical control design and is a successful control method for nonlinear system (Furat and Eker, 2012). A sliding mode is a motion on a discontinuity set of a dynamic system and is characterized by a feedback control law and a decision rule known as switching function. The sliding mode controller is an attractive robust control algorithm because of its inherent insensitivity and robustness to plant uncertainties and external disturbances (Herrmann et al., 2001). Considering a general nonlinear system x˙ (t) = f (x) + g (x) u
(8)
y = h (x)
where f, g, and h are sufficiently smooth functions. Let e (t) = yd − y be the tracking error. Furthermore, a stable switching surface s(t) in the state-space Rr can be defined by the scalar equation s (t) = 0, where s (t) =
d dt
r−1
+m
(9)
e (t)
where r is a relative degree and m is a strictly positive constant which defines the bandwidth of the error dynamic. Remark 1: The relative degree r for the nonlinear system (8) is the integer for which the following equations hold (Isidori, 1989):
Lg Lfk h (x) = 0,
k
Lg Lfr−1 h (x)
= / 0
(10)
s˙ = − tanh
s
(12)
ϕ
3.2. Sliding mode observer The actual state x(t) may not be measurable, so we will need to design an observer and use the observed state for feedback. Based on the work done by Lyon-Berger, applying the state observer indicated that not only is it usable in observing and controlling the system, but also in detecting the faults in dynamic system. This is due to the fact that the entire observer designs are based on the mathematical model of the system. The disturbances, dynamic system uncertainties and non-liner factors cause a big challenge in practical applications. Therefore, designing the robust observer with good function has been considered recently and several developed observers have been presented. The sliding mode observer is considered to be robust compared to the conventional observers. The sliding mode observer is based on sliding mode principle. Similar to sliding mode control design, the sliding mode observer design procedure consists of performing the following two steps. First, design the manifold s(y, t) such that the estimation error trajectories restricted to s(y, t) have the desired stable dynamics. Second, the observer gain is determined to drive the estimation error trajectories to s(y, t) and maintain it on the set, once intercepted, for all subsequent time or in other words, sliding mode observer works by minimizing the error between plant model and the observer model by using a switching function. The observer gain is adjusted by the error such that the plant output matches with observer output and error surface moves towards minimum (Misawa, 1998). Because of simplicity, a non-liner system has been considered which is described as follows:
x˙ = f (x, u, t)
(13)
y = g (x, u, t)
Sliding Mode Observer (SMO) is defined for non-liner system as follows (Drakunov and Utkin, 1995):
·
where Lp h (x) = ∂h (x) /∂x · p for p = f, g is the Lie derivative of the function h(x) (Utkin, 1977). The tracking problem amounts to remaining on the switching surface s(t) for the rest of time. The sliding mode control design is then choosing the control input in such a way to satisfy the following attractive equation: 1 d T s s ≤ − |s| 2 dr
In order to achieve the sliding surface, the control rule needs to be discontinuous along the switching surface s(t). Since switching is not spontaneous and the S value is not exact, the variable structure control application is not completed and chattering occurs in the vicinity of the sliding surface and thus it is possible to excite the dynamic high frequencies. Therefore, it should be reduced and deleted to be able to apply the variable structure controller well (Slotine and Sastry, 1983). Here, in order to reduce chattering, boundary layer around sliding surface strategy will beused(Dywer and Sira-Ramirez, 1988). In sliding mode control tanh s/ϕ will be used instead of sign(s) as follows:
where ϕ is the width of the boundary layer.
In this section, a brief description of the relevant theory and control design algorithms used in the development of the fast nuclear reactor controller is given.
3
xˆ = f xˆ , u, t + k (y − y) + yˆ = g (x, u, t)
sign y − yˆ
(14)
In this case, as control designing, the tanh y − yˆ /ϕ is used instead of switching function in order to prevent the chattering phenomenon in sliding mode observer design.
(11)
where is a strictly positive constant which determines the desired reaching time to the sliding surface. The attractive Eq. (11) is also called sliding condition which implies that the distance to the sliding surface decreases along all system trajectories. Furthermore, the sliding condition makes the sliding surface an invariant set, that is, once a system trajectory reaches the surface, it remains on it for the rest of time. In addition, for any initial condition, the sliding surface is reached within a finite time.
4. Sliding mode observer design to estimate the precursor density Since the delayed neutron precursors densities in reality cannot be measured in nuclear reactors, an observer is needed to estimate the immeasurable values. In this section, a sliding mode observer which has the robust characteristics facing the external disturbances and parameters uncertainties is proposed based on the reactor power measurement.
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First, according to the point kinetics equations and observer structure (14), the reactor core power is used as an output and the immeasurable values are estimated as follows: .
nˆ r =
−ˇ ˇ nˆ r + cˆr + k1 nr − nˆ r + l l
1
tanh
nr − nˆ r
ϕ
(15)
Relative density of delayed neutron precursor is estimated by following equation:
.
ˆ + cˆ = nˆ r − ˆcr + k2 nr − nr
2 tanh
nr − nˆ r
(16)
ϕ
In order to prove that the sliding mode observer can provide convergent state observation, Lyapunov synthesis can be used (Slotine, 1984). Using the estimation error of states, consider the Lyapunovlike function: V=
1 2 enr + ec2r 2
(17)
In this section, stability of the designed Sliding Mode Control system based on the Sliding Mode observer with control law (23) is analyzed using the Lyapunov synthesis. Consider the Lyapunov-like function: V=
1 2 s (t) 2
(24)
where s(t) is the desirable sliding surface (21). Using control law (23), derivative of the Lyapunov-like function (24) is identified as follows:
enr = nˆ r − nr
(18)
ecr = cˆr − cr
Considering the Lyapunov-like function (17), its derivative is identified as follows: V˙ = enr e˙ nr + ecr e˙ cr
(19)
Now by calculating e˙ nr & e˙ cr using the previous equations with choosing appropriate observer gains and substituting them into (19), yields the following: V˙ = −k1 en2r −
1 enr
tanh
e nr
ϕ
− ec2r
(20)
which shows the negative-definiteness of V˙ and confirms that the sliding mode observer ensures the bounded ness of the states and can provide the convergence of observed states to the their actual values. Therefore, enr & ecr → 0. 5. Sliding mode Controller design based on the observer for core power of fast reactor According to point kinetics equations and considering control rod speed as the control input, relative degree of the reactor system is 2 and therefore, at the first step of the sliding mode controller design, the desirable sliding surface is represented as follow: s (t) = e˙ (t) + me (t)
s
ˇ V˙ = s − e˙ cr − tanh l
(25)
ϕ
According to Eq. (20) and stability of the sliding mode observer, the convergence of states estimation error to zero can be satisfied and hence e˙ cr → 0. Therefore, from Eq. (25) we can get the following: V˙ = −s · tanh
where
5.1. Stability analysis of the designed sliding mode control system
s ≤0
(26)
Eq. (26) implies that the Lyapunov function (24) is bounded and hence s(t) is bounded. However, the convergence of s(t) to zero cannot be established. To show the convergence of s(t), we can use Barbalat’s lemma (Jean-Jacques et al., 1991). Towards this end, consider: V¨ = 2 tanh
s ϕ
tanh
s ϕ
+
s ϕ
1 + tanh2
s
(27)
ϕ
It can be seen that V¨ is bounded since s(t) is bounded. Since V is bounded, V˙ is bounded and V¨ is also bounded, we can infer from the Barbalat’s lemma that V˙ → 0 as t→ ∞ and hence s → 0. This proves that the control law (23) satisfies the existence condition of the sliding mode in the system on the sliding surface (21) and ensures the perfect tracking of the desired reactor power. Based on the applicable inputs in nuclear power plants, designed sliding mode control system is applied to the reactor and results are indicated in the next section. 6. Simulation results The simulation is performed on the reactor model described in Section 2 using MATLAB software. An applicable input is used to evaluate the performance of the controller. In this case, the objective is to follow the reference power in reactor. Figs. 1–4 show the performance of the SMC system for 100% → 50% → 100% demand power level change with the rate of
(21)
1.4
e (t) = nr − nrd
(22)
The sliding surface is reached in a finite time tr = s (0) / and
the system’s trajectory stays on the manifold thereafter where s(0) is an initial value of a sliding surface and > 0. Now, using the sliding surface, control input is chosen to satisfy the attractive Eq. (11) using Eq. (12) as follows: zr =
l nr Gr
n¨ rd − me˙ − tanh
1 − [n˙ r (A + B)] Gr
.
−ˇ s (t) 1 − n˙ r − ˇCˆ r l l
Reactor Relative Power
Actual
where e (t) is tracking error of the desired relative power as:
Desired
1.2 1 0.8 0.6 0.4 0.2 0
0
100
200
300
400
500 600 Time (s)
(23) Fig. 1. Reactor relative power.
700
800
900
1000
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8 -3
1.4
x 10
Actual
Reactor Relative Power
Reactor Reactivity(dk/k)
5
4
3
2
1
1 0.8 0.6 0.4 0.2 0
0
100
200
300
400
500 600 Time (s)
700
800
900
Desired
1.2
0
-1
5
0
100
200
1000
300
400
500 600 Time (s)
700
800
900
800
900
1000
Fig. 5. Reactor relative power.
Fig. 2. Total reactivity of the core. -3
5
x 10
-3
x 10
Reactor Reactivity (dk/k)
Control Rod Reactivity (dk/k)
5 4 3 2 1
4
3
2
1
0
0 -1
0
100
200
300
400
-1
500 600 Time (s)
700
1000
Fig. 6. Total reactivity of the core.
-2
0
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 3. Control rod reactivity.
15%/min. The desired power is reached quickly, with no overshoot and oscillation. Based on Eq. (18), Fig. 4 demonstrates the performance of the sliding mode observer. Also, in Fig. 1, the maximum difference between the actual and desired relative power values is less than ∼0.5%. Figs. 5–8 illustrate the SMC system behaviour under the parameters uncertainties and external disturbance. All system parameters are perturbed by ±30% from their nominal values. The results show
that the robustness and stability have indeed been achieved. Also, it is observed that the sliding mode observer is satisfactory in the presence of the parameters uncertainties and disturbance. In Fig. 5, the maximum difference between the actual and desired relative power values is less than ∼2%. As shown in Figs. 4 and 8, indeed, the behaviour of the estimation error depends to the situation of the states relative to the observer sliding surface: so = nr − nˆ r . At any initial condition, estimation error of the relative power moves to the observer sliding surface, s. In the other words, at the first nˆ r should reach to nr : nˆ r → nr and so → 0, thereafter cˆr → cr gradually.
Control Rod Reactivity (dk/k)
Estimation Error of Relative Precursor Density
-3
5
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
100
200
300
400
500 600 Time (s)
Fig. 4. Estimation error of Cr .
700
800
900
1000
x 10
4 3 2 1 0 -1 -2
0
100
200
300
400
500 600 Time (s)
Fig. 7. Control rod reactivity.
700
800
900
1000
Estimation Error of Relative Precursor Density
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G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
0.1
0.08
0.06
0.04
0.02
0
200
400
600
800
1000 1200 Time (s)
1400
1600
1800
2000
Fig. 8. Estimation error of Cr with uncertainties & disturbance.
Therefore, the estimation error of the precursor density increase at the first ∼50 s until nˆ r → nr , then decrease as time because so → 0, and sliding mode observer is stable as proved using the Lyapunov synthesis. Besides, dynamical variations of the precursor density are slower than variations of the relative power. Also,
in the controller process, nr → nrd as shown in Figs. 1 and 5. Figs. 5 and 8 illustrate the SMC system based on the observer behaviour under the parameters uncertainties and external disturbance, therefore, maximum difference between the actual and desired relative power in Fig. 5 and precursor density estimation error in Fig. 8 are greater than that of in Figs. 1 and 4, respectively. Considering the lack of published data regarding a control system for the fast nuclear reactors, performance of the designed sliding mode control system is compared with a conventional industrial control system. A comparison of the S.M.C. and the conventional PID controller with optimal gains is depicted in Fig. 9. Faster response and better tracking of the desired power can be observed for the S.M.C. This comparison is performed on the reactor model for 50% → 80% → 60% → 80% → 50% demand power level change with the rate of 18%/min. Also, in the presence of the parameters uncertainties and disturbance, a comparison of the S.M.C. and the conventional PID controller with optimal gains is depicted in Fig. 10. Fig. 10 demonstrates a significant improvement in the desired core power tracking and an increased ability in disturbance rejection for the S.M.C. Besides, simplicity of the controller structure and design procedure and lower computational cost are remarkable as compared with the conventional PID controller.
0.9 Sliding Mode Control Desired PID Control
Reactor Relative Power
0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 9. Comparison of the proposed controller and Conventional PID Controller.
Reactor Relative Power
0.9 Sliding Mode Control Desired PID Control
0.8
0.7
0.6
0.5
0.4
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 10. Comparison of the proposed controller and Conventional PID Controller with uncertainties &disturbance.
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
7. Conclusions
7
0.9
In this paper, a Sliding Mode Control (SMC) system has been presented for core power control of the fast nuclear reactor to improve the load following capability. The reactor core was simulated based on the point kinetics equations and one delayed neutron group. In order to deal with shortcomings arising from system states unavailability, a sliding mode observer has been developed to estimate the density of delayed neutron precursor which is difficult to measure in practice. Stability of the designed Sliding Mode Control system based on the Sliding Mode observer has been analyzed using the Lyapunov synthesis. The main advantage of the S.M.C. technique is its inherent insensitivity and robustness to plant uncertainties and external disturbances. This approach provides a high-performance controller on the system. Simulation results demonstrated that clear asymptotic output tracking was insensitive to the external disturbance and parameters uncertainties. Besides, it was observed that the sliding mode observer was satisfactory in the presence of the parameters uncertainties and disturbance. The comparison between S.M.C. and the conventional PID controller showed a significant improvement in the desired core power tracking with S.M.C. system and an increased ability in disturbance rejection.
Reactor Relative Power
0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
0
100
200
300
400
500 600 Time (sec)
700
800
900
1000
Fig. 11. Reactor relative power using S.M.C. with sign(s).
A.2. Derivation of Eq. (23) According to the desirable sliding surface (21) and considering control rod speed zr as the control input, to derive the sliding mode controller the attractive Eq. (11) using Eq. (12) should be satisfied. Therefore, at the first, we achieve the first-time derivative of sliding surface (21) as follows:
Appendix A. s˙ = me˙ + e¨ , In this section, derivation of Eqs. (20) and (23) are given as follows: A.1. Derivation of Eq. (20)
−ˇ l
+ tanh
enr +
e nr
−
ϕ
ˇ ec + ecr − k2 ecr − k1 enr l r
1 enr
−
2 ecr
− ec2r
(28)
Now, to ensure negative-definiteness of Eq. (28) we can choose observer gains as:
− ˇ
k1
l
k2 = +
1
1,
0≺
(29)
ˇ l 2
10−2
Therefore, based on the above observer gains, Eq. (20) is achieved. Considering equilibrium vector of system in the state-space as:
Xe =
enr ecr
1
0,
ϕ
= me˙ +
−ˇ l
n˙ r +
˙ ˇ nr + c˙ r − n¨ rd l l
(33)
A.3. Comparison between S.M.C. with sign function and S.M.C. with tanh function Also, in order to further analysis, performance of the S.M.C. with sign function has been compared with that of the S.M.C. with tanh function. performance of the S.M.C. with sign function is presented for 80% → 50% → 80% demand power level change with the rate of 10%/min as Fig. 11. Also, the performance of the S.M.C. with tanh function is presented for 80% → 50% → 80% demand power level change as Fig. 12. Smooth response and better tracking of the desired power can be observed for the S.M.C. with tanh function rather than S.M.C. with sign function. Also, computation time is very larger for S.M.C. with sign function.
0.9
=0
(30)
It can be seen that every term of Eq. (20) is negative for Xe = / 0 because: k1 0,
s
Now, using Eqs. (4) and (5) and estimated values c r from observer, control input is chosen as Eq. (23).
en2r 0 ⇒ −k1 en2r ≺ 0 enr · tanh(
enr ) 0⇒− ϕ
1 enr
tanh(
enr )≺0 ϕ
(31)
0 ⇒ −ec2r ≺ 0 Therefore, Eq. (20) is negative for Xe = / 0 and is zero for Xe = 0. Hence, negative-definiteness of Eq. (20) is ensured.
Reactor Relative Power
V˙ = enr
(32)
According to the point kinetics equations and using Eqs. (12) and (32) can be written as follows: − tanh
Considering Eq. (19), by calculating e˙ nr & e˙ cr and substituting into (19), yields the following:
e = nr − nrd ⇒ s˙ = me˙ + n¨ r − n¨ rd
0.8 0.7 0.6 0.5 0.4
0
200
400
600
800
1000
Time(sec)
Fig. 12. Reactor relative power using S.M.C. with tanh s/ .
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References Choi, H.H., 1999. On the existence of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties. Automatica 37, 1707–1715. Drakunov, S.V., Utkin, V., 1995 Dec. Sliding mode observers, tutorial. In: Proc. 34th IEEE Conference on Decision and Control (CDC), Dec 1995, New Orleans, LA, pp. 3376–3378. Driscoll, M.J., et al., 1979. An evaluation of the breed/burn fast reactor concept. In: MITNE-229. Dywer, T.A.W., Sira-Ramirez, H., 1988. Variable structure control of spacecraft attitude maneuvers. J. Guid. Control Dynam. 11, 262–270. Furat, M., Eker, I., 2012. Experimental Evaluation of Sliding-mode Control Techniques. C¸ukurova Univ. J. Fac. Eng. Archit. 27 (1), 23–37. Herrmann, G., et al., 2001. A robust sliding mode output tracking control for a class of relative degree zero and non-minimum phase plants: a chemical process application. Int. J. Control 72, 1194–1209. Hetrick, David, 1965. Dynamic of Nuclear Reactor. The University of Chicago Press, Chicago. Isidori, A., 1989. Nonlinear Control Systems. Springer, New York, NY.
Slotine, Jean-Jacques E., Li, Weiping, 1991. Applied Nonlinear Control. Prentice Hall, Upper Saddle River, NJ. Kim, T.K., Taiwo, T.A., February 2012. Numerical Benchmark Results for 1000 MWth Sodium-cooled Fast Reactor. Argonne National Laboratory, Korea. Misawa, A., 1998. Nonlinear State Estimation Using Sliding Observers. Massachusetts Institute of Technology, PhD Thesis. Slotine, J.J., Sastry, S.S., 1983. Tracking control of nonlinear systems using sliding surfaces with application to robot manipulators. Int. J. Control 38, 465–492. Slotine, J.J., 1984. Sliding controller design for non-linear systems. Int. J. Control 40 (2), 421–434. Tentner, A.M., et al., March 2010. Severe Accident Approach—Final Report Evaluation of Design Measures for Severe Accident Prevention and Consequence Mitigation, ANL-GENIV-128. Utkin, V.I., 1977. Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22, 212–222. Walter, A.E., Todd, D.R., Tsvetkov, P.V., 2012. Fast Spectrum Reactors. Pergamon Press, Springer, Amazon. Weaver, K.D., Ahlfeld, C., Gilleland, J., Whitmer, C., Zimmerman, G., 2009. Extending the nuclear fuel cycle with traveling-wave reactors. In: Proceedings of Global 2009, Paris, France.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Core Power Control of the fast nuclear reactors with estimation of the delayed neutron precursor density using Sliding Mode method G.R. Ansarifar ∗ , M.N. Nasrabadi, R. Hassanvand Department of Nuclear Engineering, Faculty of Advanced Sciences & Technologies, University of Isfahan, Darvaze Shiraz, Isfahan 81746-73441, Iran
h i g h l i g h t s • • • • •
We present a S.M.C. system based on the S.M.O for control of a fast reactor power. A S.M.O has been developed to estimate the density of delayed neutron precursor. The stability analysis has been given by means Lyapunov approach. The control system is guaranteed to be stable within a large range. The comparison between S.M.C. and the conventional PID controller has been done.
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 15 October 2015 Accepted 17 October 2015 M. Instrumentation and control
a b s t r a c t In this paper, a nonlinear controller using sliding mode method which is a robust nonlinear controller is designed to control a fast nuclear reactor. The reactor core is simulated based on the point kinetics equations and one delayed neutron group. Considering the limitations of the delayed neutron precursor density measurement, a sliding mode observer is designed to estimate it and finally a sliding mode control based on the sliding mode observer is presented. The stability analysis is given by means Lyapunov approach, thus the control system is guaranteed to be stable within a large range. Sliding Mode Control (SMC) is one of the robust and nonlinear methods which have several advantages such as robustness against matched external disturbances and parameter uncertainties. The employed method is easy to implement in practical applications and moreover, the sliding mode control exhibits the desired dynamic properties during the entire output-tracking process independent of perturbations. Simulation results are presented to demonstrate the effectiveness of the proposed controller in terms of performance, robustness and stability. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Considering the reducing fossil energy resources, pollution, the greenhouse gas and the economic efficiency, nuclear power is taken into consideration. Since thermal reactors are able to use such a small percentage of uranium fuel, the design, construction and operation of fast reactors is underway (Driscoll et al., 1979). By greatly simplifying the nuclear fuel cycle, fast reactors such as travelling wave reactor could improve the cost, safety, social acceptability, and long term sustainability of nuclear energy as a source of emissions-free electricity (Weaver et al., 2009). On
∗ Corresponding author. Tel.: +98 03117934392; fax: +98 03117934242. E-mail address: [email protected] (G.R. Ansarifar). http://dx.doi.org/10.1016/j.nucengdes.2015.10.015 0029-5493/© 2015 Elsevier B.V. All rights reserved.
the other hand, to use a nuclear power, reactor energy must be controlled. According to the reactor period dependency to the neutron lifetime and the crucial importance of this quantity in reactor power, it is necessary to have control on the reactor power (Hetrick, 1965). Due to the very short life time of prompt neutrons in the fast nuclear reactor, it is difficult to control its power. Therefore, the delayed neutrons which are produced indirectly by precursors must be considered to increase the reactor period. Besides, it seems that a simple and high performance control system is needed. A successful strategy to control uncertain nonlinear systems is Sliding Mode Control (SMC). The sliding mode controller is an attractive robust control algorithm because of its inherent insensitivity and robustness to plant uncertainties and external disturbances (Choi, 1999; Furat and Eker, 2012). In this paper, a SMC system is designed to control a fast nuclear reactor based on the point kinetics equations with one group of the delayed neutrons.
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Since the measurement of the delayed neutron precursor density is practically difficult and it should be measured to design the controller, a sliding mode observer which has the robust characteristics facing the external disturbances and parameters uncertainties is proposed based on the reactor power measurement to estimate the densities of delayed neutron precursors. Finally, a sliding mode control system based on the sliding mode observer is presented for controlling the fast nuclear reactor core power. Simulation results are provided to show the effectiveness of the proposed control system.
Table 1 Values of constants used for control analysis and simulations (Walter et al., 2012; Kim and Taiwo, 2012). Parameter
Values
Parameter
Values
ˇ M(MW/◦ C) P0 (MW) ff Tin(◦ C)
0.00334 0.085 1000 0.92 355
(1/s) l(s) Gr A (¢) B (¢)
0.0337 0.00000038 0.0405 −73 −41
◦
C ¢/ C
◦
c MWs/ C
−0. 4
f
0.06
◦ MWs/◦ C
0. 1802
˝ MW/ C
0.447
2. The reactor core model 2.1. The point kinetics equations Reactor kinetics equations for both fast and thermal reactors are identical. Point kinetics approximations can be used more effectively for fast reactors than for thermal reactors because fast reactors are more tightly coupled neutronically. Tighter coupling implies that the neutron flux is more nearly separable in space and time, which is a necessary condition for point kinetics approximations to be valid (Walter et al., 2012). To simulate the nuclear reactor core, point kinetics equations with one group of the delayed neutrons is used (Hetrick, 1965). The model assumes feedback from lumped fuel and coolant temperatures. The normalized model, with respect to an equilibrium condition, based on point kinetics equations with one delayed neutron group is as follows: ˇ −ˇ dnr nr + Cr = dt l l
(1)
dcr = nr − Cr dt
(2)
where nr , , cr , , l and ˇ refer to relative (normalized) power, reactivity, relative precursor density, decay constant of precursors, prompt neutron lifetime and effective delayed neutron fraction, respectively. Reactor temperatures vary as a function of power generated in fuel and heat transfer from (or to) the system. The reactor power can be represented as P (t) = nr P0
(3)
where P(t) is reactor power at time t (MW), and P0 is the nominal power (MW). 2.2. Fast reactor reactivity equation
P
(t) − 1 B + ıTin C + r Fr (t) r
dr = Gr zr dt
(4) (5)
A = ˛D TF (0) B = ˛D + ˛Na + ˛Axi + 2
XMC XAC
+ 2˛Crod (aLCRD − bLVHP )
˛Radi
T
2.3. Thermal-hydraulics model of the reactor core The thermal-hydraulics model of the reactor core can be represented with the following equations dTf dt
=
ff P0 f
nr −
˝ ˝ ˝ T + Tout + T f f 2f 2f in
(6)
2M + ˝ 2M − ˝ (1 − ff )P0 ˝ dTout nr + Tf − Tout + Tin = c c 2c 2c dt (7)
One of the most important parameters of the dynamics behaviour of the reactor is reactivity (Hetrick, 1965). Due to the fast neutron spectra in fast reactors, fast reactors reactivity equation is very different from thermal reactors, and appears as follows (Tentner et al., 2010): = [Pr (t) − 1] A +
where Pr (t) is the normalized reactor power, Fr (t) is the normalized core coolant flow, A, B, C is the reactivity feedback parameters, ˛D is the doppler coefficient (pcm/◦ C), ˛Na is the sodium density coefficient (pcm/◦ C), ˛Axi is the fuel axial exp ansion coefficient (pcm/◦ C), ˛Radi is the core radial expansion coefficient (pcm/◦ C), ˛Crod is the control rod driveline expansion coefficient (pcm/m), XMC is the grid plate to core midplane distance (m), XAC is the grid plate to above − core load plane distance (m), a is the thermal expansion coefficient of the control rod driveline (1/◦ C), b is the thermal expansion coefficient of the vessel wall (1/◦ C), LCRD is the length of control rod driveline in contact with the hot pool (m), LVHP is the length of vessel wall in contact with the hot pool (m), LVCP is the length of vessel wall in contact with the cold pool (m), is the reactor reactivity change, r is the control rods reactivity (pcm), Gr is the reactivity worth of rod per unit length (pcm/cm), zr is the control input, control rod speed (cm/s), TF (0) is the steady state Temperature difference, fuel to coolant, Tc (0) is the steady state Coolant temperature rise, inlet to outlet.
c
(0)
2
C = ˛D + ˛Na + ˛Axi + ˛Radi + ˛Crod · (aLCRD − bLVHP − bLVCP )
where ff is the fraction of reactor power deposited in the fuel,
M is the mass flow rate times heat capacity of Na MW/◦ C , f is the fuel mass times specific heat MWs/◦ C , c is ◦
˝ is the heat transfer between fuel and coolant MW/◦ C ,
the coolant mass times specific heat MWs/ C , Tf is the fuel temperature (◦ C), Tout is the Temperature of coolant leaving the core (◦ C), Tin is the Temperature of coolant entering the core (◦ C). The parameter values of the reactor model are given in Table 1 as follows: In Table 1, kinetics parameters for the fast spectrum neutrons such as prompt neutron lifetime and effective delayed neutron fraction were taken from Walter et al. (2012) which are consistent for all the fast nuclear reactors and the integral reactivity parameters A, B, C and detailed reactivity coefficients of fast reactors depend to the nuclear reactor type and fuel type which in this paper they were taken from Kim and Taiwo (2012) for ABR-1000 with oxide fuel which is a fast nuclear reactor. Therefore, kinetics parameters and reactivity parameters are consistent to each other.
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Eq. (6) describes the rate of change of fuel temperature, which is equal to the heat energy produced in reactor minus the heat energy transferred to the coolant, whole divided by the heat capacity of fuel. Heat capacity of fuel is obtained by multiplying the mass of fuel by its specific heat at the desired operating temperature. Eq. (7) describes the rate of change of coolant temperature leaving the reactor core which is the difference of heat energy transferred from fuel to coolant and amount of heat energy transferred from coolant to the secondary circuit in steam generator whole divided by the heat capacity of the coolant. Heat capacity of coolant is obtained by multiplying the mass of coolant with its specific heat at the desired operating conditions of temperature and pressure. 3. Preliminary
3.1. Sliding mode control Sliding mode control is a variable structure control system. Since it has systematic design procedure, it is one of the most powerful solutions for any practical control design and is a successful control method for nonlinear system (Furat and Eker, 2012). A sliding mode is a motion on a discontinuity set of a dynamic system and is characterized by a feedback control law and a decision rule known as switching function. The sliding mode controller is an attractive robust control algorithm because of its inherent insensitivity and robustness to plant uncertainties and external disturbances (Herrmann et al., 2001). Considering a general nonlinear system x˙ (t) = f (x) + g (x) u
(8)
y = h (x)
where f, g, and h are sufficiently smooth functions. Let e (t) = yd − y be the tracking error. Furthermore, a stable switching surface s(t) in the state-space Rr can be defined by the scalar equation s (t) = 0, where s (t) =
d dt
r−1
+m
(9)
e (t)
where r is a relative degree and m is a strictly positive constant which defines the bandwidth of the error dynamic. Remark 1: The relative degree r for the nonlinear system (8) is the integer for which the following equations hold (Isidori, 1989):
Lg Lfk h (x) = 0,
k
Lg Lfr−1 h (x)
= / 0
(10)
s˙ = − tanh
s
(12)
ϕ
3.2. Sliding mode observer The actual state x(t) may not be measurable, so we will need to design an observer and use the observed state for feedback. Based on the work done by Lyon-Berger, applying the state observer indicated that not only is it usable in observing and controlling the system, but also in detecting the faults in dynamic system. This is due to the fact that the entire observer designs are based on the mathematical model of the system. The disturbances, dynamic system uncertainties and non-liner factors cause a big challenge in practical applications. Therefore, designing the robust observer with good function has been considered recently and several developed observers have been presented. The sliding mode observer is considered to be robust compared to the conventional observers. The sliding mode observer is based on sliding mode principle. Similar to sliding mode control design, the sliding mode observer design procedure consists of performing the following two steps. First, design the manifold s(y, t) such that the estimation error trajectories restricted to s(y, t) have the desired stable dynamics. Second, the observer gain is determined to drive the estimation error trajectories to s(y, t) and maintain it on the set, once intercepted, for all subsequent time or in other words, sliding mode observer works by minimizing the error between plant model and the observer model by using a switching function. The observer gain is adjusted by the error such that the plant output matches with observer output and error surface moves towards minimum (Misawa, 1998). Because of simplicity, a non-liner system has been considered which is described as follows:
x˙ = f (x, u, t)
(13)
y = g (x, u, t)
Sliding Mode Observer (SMO) is defined for non-liner system as follows (Drakunov and Utkin, 1995):
·
where Lp h (x) = ∂h (x) /∂x · p for p = f, g is the Lie derivative of the function h(x) (Utkin, 1977). The tracking problem amounts to remaining on the switching surface s(t) for the rest of time. The sliding mode control design is then choosing the control input in such a way to satisfy the following attractive equation: 1 d T s s ≤ − |s| 2 dr
In order to achieve the sliding surface, the control rule needs to be discontinuous along the switching surface s(t). Since switching is not spontaneous and the S value is not exact, the variable structure control application is not completed and chattering occurs in the vicinity of the sliding surface and thus it is possible to excite the dynamic high frequencies. Therefore, it should be reduced and deleted to be able to apply the variable structure controller well (Slotine and Sastry, 1983). Here, in order to reduce chattering, boundary layer around sliding surface strategy will beused(Dywer and Sira-Ramirez, 1988). In sliding mode control tanh s/ϕ will be used instead of sign(s) as follows:
where ϕ is the width of the boundary layer.
In this section, a brief description of the relevant theory and control design algorithms used in the development of the fast nuclear reactor controller is given.
3
xˆ = f xˆ , u, t + k (y − y) + yˆ = g (x, u, t)
sign y − yˆ
(14)
In this case, as control designing, the tanh y − yˆ /ϕ is used instead of switching function in order to prevent the chattering phenomenon in sliding mode observer design.
(11)
where is a strictly positive constant which determines the desired reaching time to the sliding surface. The attractive Eq. (11) is also called sliding condition which implies that the distance to the sliding surface decreases along all system trajectories. Furthermore, the sliding condition makes the sliding surface an invariant set, that is, once a system trajectory reaches the surface, it remains on it for the rest of time. In addition, for any initial condition, the sliding surface is reached within a finite time.
4. Sliding mode observer design to estimate the precursor density Since the delayed neutron precursors densities in reality cannot be measured in nuclear reactors, an observer is needed to estimate the immeasurable values. In this section, a sliding mode observer which has the robust characteristics facing the external disturbances and parameters uncertainties is proposed based on the reactor power measurement.
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First, according to the point kinetics equations and observer structure (14), the reactor core power is used as an output and the immeasurable values are estimated as follows: .
nˆ r =
−ˇ ˇ nˆ r + cˆr + k1 nr − nˆ r + l l
1
tanh
nr − nˆ r
ϕ
(15)
Relative density of delayed neutron precursor is estimated by following equation:
.
ˆ + cˆ = nˆ r − ˆcr + k2 nr − nr
2 tanh
nr − nˆ r
(16)
ϕ
In order to prove that the sliding mode observer can provide convergent state observation, Lyapunov synthesis can be used (Slotine, 1984). Using the estimation error of states, consider the Lyapunovlike function: V=
1 2 enr + ec2r 2
(17)
In this section, stability of the designed Sliding Mode Control system based on the Sliding Mode observer with control law (23) is analyzed using the Lyapunov synthesis. Consider the Lyapunov-like function: V=
1 2 s (t) 2
(24)
where s(t) is the desirable sliding surface (21). Using control law (23), derivative of the Lyapunov-like function (24) is identified as follows:
enr = nˆ r − nr
(18)
ecr = cˆr − cr
Considering the Lyapunov-like function (17), its derivative is identified as follows: V˙ = enr e˙ nr + ecr e˙ cr
(19)
Now by calculating e˙ nr & e˙ cr using the previous equations with choosing appropriate observer gains and substituting them into (19), yields the following: V˙ = −k1 en2r −
1 enr
tanh
e nr
ϕ
− ec2r
(20)
which shows the negative-definiteness of V˙ and confirms that the sliding mode observer ensures the bounded ness of the states and can provide the convergence of observed states to the their actual values. Therefore, enr & ecr → 0. 5. Sliding mode Controller design based on the observer for core power of fast reactor According to point kinetics equations and considering control rod speed as the control input, relative degree of the reactor system is 2 and therefore, at the first step of the sliding mode controller design, the desirable sliding surface is represented as follow: s (t) = e˙ (t) + me (t)
s
ˇ V˙ = s − e˙ cr − tanh l
(25)
ϕ
According to Eq. (20) and stability of the sliding mode observer, the convergence of states estimation error to zero can be satisfied and hence e˙ cr → 0. Therefore, from Eq. (25) we can get the following: V˙ = −s · tanh
where
5.1. Stability analysis of the designed sliding mode control system
s ≤0
(26)
Eq. (26) implies that the Lyapunov function (24) is bounded and hence s(t) is bounded. However, the convergence of s(t) to zero cannot be established. To show the convergence of s(t), we can use Barbalat’s lemma (Jean-Jacques et al., 1991). Towards this end, consider: V¨ = 2 tanh
s ϕ
tanh
s ϕ
+
s ϕ
1 + tanh2
s
(27)
ϕ
It can be seen that V¨ is bounded since s(t) is bounded. Since V is bounded, V˙ is bounded and V¨ is also bounded, we can infer from the Barbalat’s lemma that V˙ → 0 as t→ ∞ and hence s → 0. This proves that the control law (23) satisfies the existence condition of the sliding mode in the system on the sliding surface (21) and ensures the perfect tracking of the desired reactor power. Based on the applicable inputs in nuclear power plants, designed sliding mode control system is applied to the reactor and results are indicated in the next section. 6. Simulation results The simulation is performed on the reactor model described in Section 2 using MATLAB software. An applicable input is used to evaluate the performance of the controller. In this case, the objective is to follow the reference power in reactor. Figs. 1–4 show the performance of the SMC system for 100% → 50% → 100% demand power level change with the rate of
(21)
1.4
e (t) = nr − nrd
(22)
The sliding surface is reached in a finite time tr = s (0) / and
the system’s trajectory stays on the manifold thereafter where s(0) is an initial value of a sliding surface and > 0. Now, using the sliding surface, control input is chosen to satisfy the attractive Eq. (11) using Eq. (12) as follows: zr =
l nr Gr
n¨ rd − me˙ − tanh
1 − [n˙ r (A + B)] Gr
.
−ˇ s (t) 1 − n˙ r − ˇCˆ r l l
Reactor Relative Power
Actual
where e (t) is tracking error of the desired relative power as:
Desired
1.2 1 0.8 0.6 0.4 0.2 0
0
100
200
300
400
500 600 Time (s)
(23) Fig. 1. Reactor relative power.
700
800
900
1000
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8 -3
1.4
x 10
Actual
Reactor Relative Power
Reactor Reactivity(dk/k)
5
4
3
2
1
1 0.8 0.6 0.4 0.2 0
0
100
200
300
400
500 600 Time (s)
700
800
900
Desired
1.2
0
-1
5
0
100
200
1000
300
400
500 600 Time (s)
700
800
900
800
900
1000
Fig. 5. Reactor relative power.
Fig. 2. Total reactivity of the core. -3
5
x 10
-3
x 10
Reactor Reactivity (dk/k)
Control Rod Reactivity (dk/k)
5 4 3 2 1
4
3
2
1
0
0 -1
0
100
200
300
400
-1
500 600 Time (s)
700
1000
Fig. 6. Total reactivity of the core.
-2
0
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 3. Control rod reactivity.
15%/min. The desired power is reached quickly, with no overshoot and oscillation. Based on Eq. (18), Fig. 4 demonstrates the performance of the sliding mode observer. Also, in Fig. 1, the maximum difference between the actual and desired relative power values is less than ∼0.5%. Figs. 5–8 illustrate the SMC system behaviour under the parameters uncertainties and external disturbance. All system parameters are perturbed by ±30% from their nominal values. The results show
that the robustness and stability have indeed been achieved. Also, it is observed that the sliding mode observer is satisfactory in the presence of the parameters uncertainties and disturbance. In Fig. 5, the maximum difference between the actual and desired relative power values is less than ∼2%. As shown in Figs. 4 and 8, indeed, the behaviour of the estimation error depends to the situation of the states relative to the observer sliding surface: so = nr − nˆ r . At any initial condition, estimation error of the relative power moves to the observer sliding surface, s. In the other words, at the first nˆ r should reach to nr : nˆ r → nr and so → 0, thereafter cˆr → cr gradually.
Control Rod Reactivity (dk/k)
Estimation Error of Relative Precursor Density
-3
5
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
100
200
300
400
500 600 Time (s)
Fig. 4. Estimation error of Cr .
700
800
900
1000
x 10
4 3 2 1 0 -1 -2
0
100
200
300
400
500 600 Time (s)
Fig. 7. Control rod reactivity.
700
800
900
1000
Estimation Error of Relative Precursor Density
6
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
0.1
0.08
0.06
0.04
0.02
0
200
400
600
800
1000 1200 Time (s)
1400
1600
1800
2000
Fig. 8. Estimation error of Cr with uncertainties & disturbance.
Therefore, the estimation error of the precursor density increase at the first ∼50 s until nˆ r → nr , then decrease as time because so → 0, and sliding mode observer is stable as proved using the Lyapunov synthesis. Besides, dynamical variations of the precursor density are slower than variations of the relative power. Also,
in the controller process, nr → nrd as shown in Figs. 1 and 5. Figs. 5 and 8 illustrate the SMC system based on the observer behaviour under the parameters uncertainties and external disturbance, therefore, maximum difference between the actual and desired relative power in Fig. 5 and precursor density estimation error in Fig. 8 are greater than that of in Figs. 1 and 4, respectively. Considering the lack of published data regarding a control system for the fast nuclear reactors, performance of the designed sliding mode control system is compared with a conventional industrial control system. A comparison of the S.M.C. and the conventional PID controller with optimal gains is depicted in Fig. 9. Faster response and better tracking of the desired power can be observed for the S.M.C. This comparison is performed on the reactor model for 50% → 80% → 60% → 80% → 50% demand power level change with the rate of 18%/min. Also, in the presence of the parameters uncertainties and disturbance, a comparison of the S.M.C. and the conventional PID controller with optimal gains is depicted in Fig. 10. Fig. 10 demonstrates a significant improvement in the desired core power tracking and an increased ability in disturbance rejection for the S.M.C. Besides, simplicity of the controller structure and design procedure and lower computational cost are remarkable as compared with the conventional PID controller.
0.9 Sliding Mode Control Desired PID Control
Reactor Relative Power
0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 9. Comparison of the proposed controller and Conventional PID Controller.
Reactor Relative Power
0.9 Sliding Mode Control Desired PID Control
0.8
0.7
0.6
0.5
0.4
100
200
300
400
500 600 Time (s)
700
800
900
1000
Fig. 10. Comparison of the proposed controller and Conventional PID Controller with uncertainties &disturbance.
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
7. Conclusions
7
0.9
In this paper, a Sliding Mode Control (SMC) system has been presented for core power control of the fast nuclear reactor to improve the load following capability. The reactor core was simulated based on the point kinetics equations and one delayed neutron group. In order to deal with shortcomings arising from system states unavailability, a sliding mode observer has been developed to estimate the density of delayed neutron precursor which is difficult to measure in practice. Stability of the designed Sliding Mode Control system based on the Sliding Mode observer has been analyzed using the Lyapunov synthesis. The main advantage of the S.M.C. technique is its inherent insensitivity and robustness to plant uncertainties and external disturbances. This approach provides a high-performance controller on the system. Simulation results demonstrated that clear asymptotic output tracking was insensitive to the external disturbance and parameters uncertainties. Besides, it was observed that the sliding mode observer was satisfactory in the presence of the parameters uncertainties and disturbance. The comparison between S.M.C. and the conventional PID controller showed a significant improvement in the desired core power tracking with S.M.C. system and an increased ability in disturbance rejection.
Reactor Relative Power
0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
0
100
200
300
400
500 600 Time (sec)
700
800
900
1000
Fig. 11. Reactor relative power using S.M.C. with sign(s).
A.2. Derivation of Eq. (23) According to the desirable sliding surface (21) and considering control rod speed zr as the control input, to derive the sliding mode controller the attractive Eq. (11) using Eq. (12) should be satisfied. Therefore, at the first, we achieve the first-time derivative of sliding surface (21) as follows:
Appendix A. s˙ = me˙ + e¨ , In this section, derivation of Eqs. (20) and (23) are given as follows: A.1. Derivation of Eq. (20)
−ˇ l
+ tanh
enr +
e nr
−
ϕ
ˇ ec + ecr − k2 ecr − k1 enr l r
1 enr
−
2 ecr
− ec2r
(28)
Now, to ensure negative-definiteness of Eq. (28) we can choose observer gains as:
− ˇ
k1
l
k2 = +
1
1,
0≺
(29)
ˇ l 2
10−2
Therefore, based on the above observer gains, Eq. (20) is achieved. Considering equilibrium vector of system in the state-space as:
Xe =
enr ecr
1
0,
ϕ
= me˙ +
−ˇ l
n˙ r +
˙ ˇ nr + c˙ r − n¨ rd l l
(33)
A.3. Comparison between S.M.C. with sign function and S.M.C. with tanh function Also, in order to further analysis, performance of the S.M.C. with sign function has been compared with that of the S.M.C. with tanh function. performance of the S.M.C. with sign function is presented for 80% → 50% → 80% demand power level change with the rate of 10%/min as Fig. 11. Also, the performance of the S.M.C. with tanh function is presented for 80% → 50% → 80% demand power level change as Fig. 12. Smooth response and better tracking of the desired power can be observed for the S.M.C. with tanh function rather than S.M.C. with sign function. Also, computation time is very larger for S.M.C. with sign function.
0.9
=0
(30)
It can be seen that every term of Eq. (20) is negative for Xe = / 0 because: k1 0,
s
Now, using Eqs. (4) and (5) and estimated values c r from observer, control input is chosen as Eq. (23).
en2r 0 ⇒ −k1 en2r ≺ 0 enr · tanh(
enr ) 0⇒− ϕ
1 enr
tanh(
enr )≺0 ϕ
(31)
0 ⇒ −ec2r ≺ 0 Therefore, Eq. (20) is negative for Xe = / 0 and is zero for Xe = 0. Hence, negative-definiteness of Eq. (20) is ensured.
Reactor Relative Power
V˙ = enr
(32)
According to the point kinetics equations and using Eqs. (12) and (32) can be written as follows: − tanh
Considering Eq. (19), by calculating e˙ nr & e˙ cr and substituting into (19), yields the following:
e = nr − nrd ⇒ s˙ = me˙ + n¨ r − n¨ rd
0.8 0.7 0.6 0.5 0.4
0
200
400
600
800
1000
Time(sec)
Fig. 12. Reactor relative power using S.M.C. with tanh s/ .
8
G.R. Ansarifar et al. / Nuclear Engineering and Design 296 (2016) 1–8
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