# Dynamic Output Feedback Power-Level Control for the MHTGR Based On Iterative Damping Assignment

## Abstract

**:**

_{2}disturbance attenuation performance under modeling uncertainty or exterior disturbance, and can also guarantee the globally asymptotic closed-loop stability without uncertainty and disturbance. This newly built control strategy is then applied to the power-level regulation of the HTR-PM plant, and numerical simulation results show both the feasibility and high performance of this newly-built control strategy. Furthermore, the relationship between the values of the parameters and the performance of this controller is not only illustrated numerically but also analyzed theoretically.

## 1. Introduction

_{th}pebble-bed high temperature gas-cooled test reactor (HTR-10), which was built at Institute of Nuclear and New Energy Technology (INET) of Tsinghua University [4], achieved its criticality in December 2000 and full power level in January 2003. Six safety demonstration tests have also been done on the HTR-10, which have manifested both the inherent safety feature and the self-stabilizing features [5].

_{th}power, and adopts the operation scheme of two modules connected to one steam turbine/generator set. Here, the module is a nuclear steam supplying system (NSSS) composed of an MHTGR, a helical coiled once-through steam generator (OTSG) and some connecting pipes. The MHTGR and OTSG of one NSSS are arranged side by side and housed in independent steel pressure vessels, and the schematic view of the HTR-PM plant is illustrated in Figure 1.

_{2}disturbance attenuator when there exist exterior disturbances or modeling uncertainties, and it guarantees the globally asymptotic closed-loop stability if there is no disturbance and uncertainty. The IDA-PLC is then applied to the power-level control of the NSSS of the HTR-PM plant, and numerical simulation results show not only the feasibility of this newly-built MHTGR power-level control strategy but also the relationship between its performance and its parameters.

## 2. Nonlinear State-Space Model and Problem Formulation

#### 2.1. Nonlinear State-Space Model

_{r}is the relative nuclear power; c

_{r}is the relative concentration of delayed neutron precursor; β is the fraction of delayed fission neutrons; Λ is the effective prompt neutron life time; ρ

_{r}is the reactivity provided by the control rods; λ is the effective radioactive decay constant of the delayed neutron precursor; α

_{c}and α

_{r}are respectively the reactivity coefficients of the fuel and reflector temperatures; P

_{0}is the rated reactor thermal power; T

_{c}is the average fuel temperature; T

_{d}is the average temperature of the helium inside the pebble-bed; T

_{d}is the temperature of the helium entering into the pebble-bed; T

_{c,m}and T

_{d,m}are initial equilibrium values of T

_{c}and T

_{d}respectively; T

_{r}is the reflector temperature; Ω

_{cd}and Ω

_{cr}are respectively the heat transfer coefficient between the fuel and helium in the pebble-bed and that between the fuel and reflector inside the riser; M

_{p}is the mass flowrate times the heat capacity of the helium inside the primary loop; μ

_{c}and μ

_{d}are respectively the total heat capacities of the fuel or helium inside the pebble-bed; G

_{r}is the differential reactivity worth of the control rod, and z

_{r}is the rod speed signal generated by the corresponding power-level control strategy.

_{r}, c

_{r}, T

_{c}, T

_{d}, T

_{din}, T

_{r}and ρ

_{r}from their equilibrium values, i.e., n

_{r0}, c

_{r0}, T

_{c0}, T

_{d0}, T

_{din0}, T

_{r0}and ρ

_{r0}are respectively defined as:

_{r}changes very slowly and its amplitude is also very small. Moreover, δT

_{din}reflects the influence of the other parts of the MHTGR to the reactor core dynamics. Therefore, it is quite reasonable to view

**w**defined in Equation (4) as the disturbance.

#### 2.2. Problem Formulation

_{2}disturbance attenuator is introduced as follows:

_{2}disturbance attenuator if there is a semi-positive smooth function V(x) such that following γ-dissipation inequality is satisfied:

**x**) is a semi-positive function; ||·||

_{2}is the Euclidean norm, and here γ is a positive scalar called the L

_{2}gain from disturbance w to evaluation signal ζ.

_{2}-disturbance attenuator, the problem to be solved in this paper is given as follows:

_{2}-disturbance attenuator for System (6) with evaluation Signal (11)? That is to say, how to design a power-level controller for the MHTGR with L

_{2}-disturbance attenuation performance?

## 3. Power-Level Control Based on Iterative Damping Assignment

_{2}-disturbance attenuation performance of the IDA-PLC are both verified.

#### 3.1. Introduction to the Concept of Feedback Dissipation Control

^{n}and ω(O) = O. If there exists a smooth function Γ(·): R

^{n}→R

^{+}= [0, ∞) called Hamiltonian function so that inequality is satisfied:

^{n}such that System (15) can be expressed as:

^{n×n}is called the structure matrix. Moreover, if structure matrix T can be written as:

^{n}is the system state vector; v ∈ R

^{p}is the control input; θ ∈ R

^{m}is the system output, and ω(O) = O. For a given Hamiltonian function Γ(χ), feedback control υ is called a feedback dissipation control if Inequality (20) is satisfied:

#### 3.2. Design of the Power-Level Control Based on Iterative Damping Assignment

_{2}disturbance attenuator corresponding to a given evaluation signal.

_{2}disturbance attenuator corresponding to evaluation signal Equation (49). Moreover, the L

_{2}gain can be adjusted by feedback gain K. Moreover, if there is no disturbance, i.e.:

_{2}attenuator of system Equation (6) corresponding to evaluation signal Equation (49). Moreover, from Inequality (55), the L

_{2}gain from disturbance

**w**to evaluation signal ζ is:

_{1}, it is quite clear that:

_{1}and x

_{3}, i.e., δn

_{r}and δT

_{d}can be obtained through measurement, it is quite necessary to design a convergent state-observer for the implementation of this newly-built power-level control strategy.

## 4. Dynamic Output-Feedback Power-Level Control Strategy

_{2}disturbance attenuation.

#### 4.1. Observation Strategy

^{n}is the state-vector; η ∈ R

^{m}is the system output; μ ∈ R

^{p}is the control input and vector-valued functions ψ, φ and χ are all smooth.

_{o}∈ R

^{n×m}is the gain matrix of the observer,

#### 4.2. Design and Analysis of the Dynamic Output-Feedback Power-Level Control Strategy

_{2}disturbance attenuator.

_{2}disturbance attenuator for System (6) if the gain matrix K

_{O}satisfies:

_{O}(·) is only minimal at e

_{y}≡ O. It is also easily to derive that:

_{y}≡ O and w ≡ O to observation error dynamics Equation (80), and we have:

_{3}can be arbitrarily given, it is easily to see from Equation (89) that:

_{y}. Since state-feedback control Equation (48) is a globally asymptotic stabilizer if Condition (51) is satisfied, it can be seen from Lemma 1, Equations (86,87,90) that control strategy Equation (79) is still a globally asymptotical stabilizer and Observer (78) is convergent under Condition (51).

_{2}disturbance attenuator even if w ≠ O. Define the extended Hamiltonian function for the closed-loop system composed of Equations (6,79) as:

_{O}(·) are determined by Equations (40,86), respectively. Differentiating Equation (91) along the trajectory given by Equations (6,79), we can derive that:

**L**and

**M**are respectively defined by Equations (53,85).

_{1}and θ

_{2}are both small positive scalars.

_{1}+ θ

_{2}. Based upon Inequality (95) and Definition 1, it is now quite clear that dynamic output feedback control strategy Equation (79) with observer gain Equation (84) and a small enough ε is still a L

_{2}disturbance attenuator in case of w ≠ O. This is completes the proof of Proposition 2.

## 5. Simulation Results with Discussion

#### 5.1. Description of the Numerical Simulation

#### 5.2. Simulation Results

_{1}and C

_{2}are so small that the influence of the value of K to the control performance is much weaker than that of σ. Therefore, in this study, we only check the influence of parameters σ and ε to the control performance.

**Figure 3.**Dynamic responses in Case A of (

**a**) relative nuclear power; (

**b**) average fuel temperature; (

**c**) outlet helium temperature and (

**d**) control rod speed signal with different σ and constant ε.

**Figure 4.**Dynamic responses in Case A of (

**a**) relative nuclear power; (

**b**) average fuel temperature; (

**c**) outlet helium temperature and (

**d**) control rod speed signal with constant σ and different ε.

#### 5.3. Discussion

_{r}and δT

_{d}to be positive and increasing, which drives the power-level controller to give a negative control rod speed signal. Inserting the control rods leads to the decreases of both the nuclear power and fuel temperature. The closed-loop system comes into an equilibrium state if the positive reactivity caused by the decrease of the fuel temperature nearly cancels the negative reactivity caused by the rod insertion. The generation of the control rod speed signal is driven by the variations of both the nuclear power and average coolant temperature obtained from measurement. These two variation signals lead state-observer Equation (78) to give a convergent observation of the state-variations which then cause state-feedback power-level control Equation (63) to generate the speed signal of control rods. Also from Figure 3, we can see that the steady regulation error is smaller if controller parameter σ is larger. Actually, from Equation (56), it is clear that a larger σ can result in a larger L

_{2}gain from the disturbance to the evaluation signal, which means that the closed-loop system is more robust to both the modeling uncertainty or exterior disturbances. Thus, a smaller steady control error reflects that the closed-loop system has a stronger ability to sustain the disturbances. However, from Proposition 2, to maintain the L

_{2}disturbance attenuation property of state-feedback law Equation (56), it is necessary to set parameter ε to be a small enough positive scalar. This can be clearly seen from Figure 4 that scalar ε is larger, the L

_{2}disturbance attenuation performance of the closed-loop is weaker.

**Figure 5.**Dynamic responses in Case B of (

**a**) relative nuclear power; (

**b**) average fuel temperature; (

**c**) outlet helium temperature and (

**d**) control rod speed signal with different σ and constant ε.

_{r}and δT

_{d}, which results in the generation of a positive speed signal of control rods. The withdrawal of the control rods then causes increases of both the nuclear power and the fuel temperature. Similarly with the case of load drop, the closed-system come to the steady state if the negative reactivity caused by the increase of the fuel temperature nearly cancels the positive reactivity induced by withdrawing the control rods. Moreover, it is clear from Figure 5 and Figure 6 that the control performance is also deeply influenced by the values ε and σ. The reason is the same as that given in the above paragraph.

_{2}disturbance attenuation performance of this newly-built control law is guaranteed by choosing the values of both parameters ε and σ to be small enough. With comparison to the power-level controller presented in [29], the transition periods of both the fuel and coolant temperatures caused by the newly-built controller in this paper is much smaller than those corresponding to the controller in [29]. This is just an instance for the fact that dynamic output feedback control is stronger than static output feedback control, and this is also the key improvement of the work in this paper relative to that in [29].

**Figure 6.**Dynamic responses in Case B of (

**a**) relative nuclear power; (

**b**) average fuel temperature; (

**c**) outlet helium temperature and (

**d**) control rod speed signal with constant σ and different ε.

## 6. Conclusions

_{2}disturbance attenuation performance, and has analytic expressions with clear physical meaning. Moreover, there is a clear relationship between the controller parameters and the regulation performance. Both simulation results and theoretical analysis have shown that the performance of this newly developed power-level controller can be satisfactory high with large enough σ and ε. The results given here have shown the theoretic feasibility of this newly-developed power-level control law. For engineering implementation of this control strategy for the MHTGR such as the reactors of HTR-10 or HTR-PM, the future study lies in verifying the performance of this control strategy on the hardware-in-loop simulation platform proposed in [30].

## Acknowledgement

## References

- Reutler, H.; Lohnert, G.H. The modular high-temperature reactor. Nucl. Technol.
**1983**, 62, 22–30. [Google Scholar] - Reutler, H.; Lohnert, G.H. Advantages of going modular in HTRs. Nucl. Eng. Des.
**1984**, 78, 129–136. [Google Scholar] [CrossRef] - Lohnert, G.H. Technical design features and essential safety-related properties of the HTR-Module. Nucl. Eng. Des.
**1990**, 121, 259–275. [Google Scholar] [CrossRef] - Wu, Z.; Lin, D.; Zhong, D. The design features of the HTR-10. Nucl. Eng. Des.
**2002**, 218, 25–32. [Google Scholar] [CrossRef] - Hu, S.; Liang, X.; Wei, L. Commissioning and operation experience and safety experiment on HTR-10. In Proceedings of 3rd International Topical Meeting on High Temperature Reactor Technology, Johnanneshurg, South Africa, 1–4 October 2006.
- Zhang, Z.; Wu, Z.; Wang, D.; Xu, Y.; Sun, Y.; Li, F.; Dong, Y. Current status and technical description of Chinese 2 × 250MW
_{th}HTR-PM demonstration plant. Nucl. Eng. Des.**2009**, 239, 1212–1219. [Google Scholar] [CrossRef] - Zhang, Z.; Sun, Y. Economic potential of modular reactor nuclear power plants based on the Chinese HTR-PM project. Nucl. Eng. Des.
**2007**, 237, 2265–2274. [Google Scholar] [CrossRef] - Edwards, R.M.; Lee, K.Y.; Schultz, M.A. State-feedback assisted classical control: An incremental approach to control modernization of existing and future nuclear reactors and power plants. Nucl. Technol.
**1990**, 92, 167–185. [Google Scholar] - Ben-Abdennour, A.; Edwards, R.M.; Lee, K.Y. LQR/LTR robust control of nuclear reactors with improved temperature performance. IEEE Trans. Nucl. Sci.
**1992**, 39, 2286–2294. [Google Scholar] [CrossRef] - Arab-Alibeik, H.; Setayeshi, S. Improved temperature control of a PWR nuclear reactor using LQG/LTR based controller. IEEE Trans. Nucl. Sci.
**2003**, 50, 211–218. [Google Scholar] [CrossRef] - Shtessel, Y.B. Sliding mode control of the space nuclear reactor system. IEEE Trans. Aerosp. Electron. Syst.
**1998**, 34, 579–589. [Google Scholar] [CrossRef] - Ku, C.C.; Lee, K.Y.; Edwards, R.M. Improved nuclear reactor temperature control using diagonal recurrent neural networks. IEEE Trans. Nucl. Sci.
**1992**, 39, 2298–2308. [Google Scholar] [CrossRef] - Na, M.G.; Hwang, I.J.; Lee, Y.J. Design of a fuzzy model predictive power controller for pressurized water reactors. IEEE Trans. Nucl. Sci.
**2006**, 53, 1504–1514. [Google Scholar] [CrossRef] - Huang, Z.; Edwards, R.M.; Lee, K.Y. Fuzzy-adapted recursive sliding-mode controller design for power plant control. IEEE Trans. Nucl. Sci.
**2004**, 51, 1504–1514. [Google Scholar] [CrossRef] - Van der Schaft, A.J. L
_{2}-Gain and Passivity Techniques in Nonlinear Control; Springer: Berlin, Germany, 1999. [Google Scholar] - Maschke, B.M.; Ortega, R.; van der Schaft, A.J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans. Autom. Control
**2000**, 45, 1498–1502. [Google Scholar] [CrossRef] - Ortega, R.; van der Schaft, A.J.; Maschke, B.M.; Escobar, G. Interconnections and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica
**2002**, 38, 585–596. [Google Scholar] [CrossRef] - Ortega, R.; van der Schaft, A.J.; Castaños, F.; Astolfi, A. Control by interconnection and standard passivity-based control of port-Hamiltonian systems. IEEE Trans. Autom. Control
**2008**, 53, 2527–2542. [Google Scholar] [CrossRef] - Ortega, R.; Spong, M.W.; Gómez-Estern, F.; Blankenstein, G. Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Autom. Control
**2002**, 47, 1218–1233. [Google Scholar] [CrossRef] - Fujimoto, K.; Sugie, T. Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations. Syst. Control Lett.
**2001**, 44, 309–319. [Google Scholar] [CrossRef] - Liu, Q.J.; Sun, Y.Z.; Shen, T.L.; Song, Y.H. Adaptive nonlinear coordinated excitation and STATCOM controller based on Hamiltonian structure for multimachine power-system stability enhancement. IET Proc. Control Theory Appl.
**2003**, 50, 285–294. [Google Scholar] [CrossRef] - Wang, Y.; Cheng, D.; Li, C.; Ge, Y. Dissipative Hamiltonian realization and energy-based L
_{2}-disturbance attenuation control of multi-machine power systems. IEEE Trans. Autom. Control**2003**, 48, 1428–1433. [Google Scholar] [CrossRef] - Galaz, M.; Ortega, R.; Bazanella, A.S.; Stankovic, A.M. An energy-shaping approach to the design of excitation control of synchronous generators. Automatica
**2003**, 39, 111–119. [Google Scholar] [CrossRef] - Dong, Z.; Feng, J.; Huang, X.; Zhang, L. Dissipation-based high gain filter for monitoring nuclear reactors. IEEE Trans. Nucl. Sci.
**2010**, 57, 328–339. [Google Scholar] [CrossRef] - Dong, Z.; Huang, X.; Zhang, L. Output feedback power-level control of nuclear reactors based on a dissipative high gain filter. Nucl. Eng. Des.
**2011**, 241, 4783–4793. [Google Scholar] [CrossRef] - Li, H.; Huang, X.; Zhang, L. A simplified mathematical dynamic model of the HTR-10 high temperature gas-cooled reactor with control system design purpose. Ann. Nucl. Energy
**2008**, 35, 1642–1651. [Google Scholar] [CrossRef] - Dong, Z.; Huang, X.; Zhang, L. A nodal dynamic model for control system design and simulation of an MHTGR core. Nucl. Eng. Des.
**2010**, 240, 1251–1261. [Google Scholar] [CrossRef] - Li, H.; Huang, X.; Zhang, L. A lumped parameter dynamic model of the helical coiled once-through steam generator with movable boundaries. Nucl. Eng. Des.
**2008**, 238, 1657–1663. [Google Scholar] [CrossRef] - Dong, Z. Output feedback dissipation control for the power-level of modular high-temperature gas-cooled reactors. Energies
**2011**, 4, 1858–1879. [Google Scholar] [CrossRef] - Dong, Z.; Huang, X. Real-time simulation platform for the design and verification of the operation strategy of the HTR-PM. In Proceedings of the 19th International Conference on Nuclear Engineering, Chiba, Japan, 16–19 May 2011.

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Dong, Z.
Dynamic Output Feedback Power-Level Control for the MHTGR Based On Iterative Damping Assignment. *Energies* **2012**, *5*, 1782-1815.
https://doi.org/10.3390/en5061782

**AMA Style**

Dong Z.
Dynamic Output Feedback Power-Level Control for the MHTGR Based On Iterative Damping Assignment. *Energies*. 2012; 5(6):1782-1815.
https://doi.org/10.3390/en5061782

**Chicago/Turabian Style**

Dong, Zhe.
2012. "Dynamic Output Feedback Power-Level Control for the MHTGR Based On Iterative Damping Assignment" *Energies* 5, no. 6: 1782-1815.
https://doi.org/10.3390/en5061782