A. P. Tiwari

Spatial control of a large pressurized heavy water reactor

The paper presents tile design of a near optimal linear regulator for controlling xenon-induced spatial oscillations in a large, pressurized heavy water reactor. The nonlinear mathematical model of the reactor including xenon iodine dynamics is characterized by 56 state variables land 14 inputs. This nonlinear model is linearized over rated power of the reactor and then the singularly perturbed structure of the linear model is exploited to decompose it into a fast subsystem of 14th order and a slow subsystem of 42nd order. The slow subsystem regulator problem is formulated as a cheap control problem that entails the solving of regulator problems of a 28th-order submodel and a 14th-order submodel. The fast subsystem regulator problem is also solved, Separately designed regulators are finally combined to obtain the near-optimal composite control for the original 56th-order model.

Multirate Output Feedback Based Sliding Mode Spatial Control for a Large PHWR

The paper presents a novel method to design a spatial control system based on multirate output feedback robust sliding mode control for a large pressurized heavy water reactor (PHWR). The non-linear model of PHWR including xenon and iodine dynamics is characterized by 70 state variables and 14 inputs and outputs each. Linear nodal model is obtained by linearizing the non-linear dynamic equations of the reactor about the full power operating point. A multirate output feedback based discrete sliding mode spatial controller is designed for the linearized model of the reactor. The proposed control method does not require state information of the system for feedback purposes and hence may be easier to implement. From simulation of the non-linear model of the reactor in representative transients, the proposed control scheme is found to be superior to other methods.

Spatial control of a large PHWR by piecewise constant periodic output feedback

The paper presents the design of piecewise constant periodic output feedback control for a discrete-time singularly perturbed system resulting from the discretization of a continuous-time standard singularly perturbed system. By a suitable linear transformation of state variables, the given continuous-time singularly perturbed model is converted into a block triangular form in which the fast subsystem is decoupled. The discrete-time model corresponding to the transformed model also exhibits a two time scale property if sampling period is larger than the parameter E. Now an output injection matrix is found that stabilizes the slow subsystem. The periodic output feedback gain is then calculated only for the slow subsystem and the same for the fast subsystem is set equal to zero. Finally the periodic output feedback gain for the composite system is obtained using the periodic output feedback gains computed separately for the slow and fast subsystems. An approach has been suggested whereby the determination of periodic output feedback gain for the slow subsystem can be converted into an optimization problem. By minimization of the suggested performance index the closed loop system behavior is improved. The method has been applied to a large pressurized heavy water reactor (PHWR) for control of xenon-induced spatial oscillations. A particular grouping of state variables has been suggested for obtaining the model in standard singularly perturbed form. The periodic output feedback gain is then calculated. The efficacy of control has been demonstrated by simulation of transient behavior of the nonlinear model of the PHWR.

Spatial control of a large pressurized heavy water reactor by fast output sampling technique

In this paper a method is presented to design a controller for discrete two time scale system based on fast output sampling technique. Using similarity transformation the two time scale system is converted into a block diagonal form which is then partitioned into two subsystems, namely, a fast subsystem and a slow subsystem. Now state feedback controls are designed separately for the slow subsystem and the fast subsystem. Then a composite state feedback control is obtained from the state feedback controls designed for subsystems to assign the eigenvalues of the entire system at arbitrary locations. This composite state feedback gain is realized by using fast output sampling feedback gain. Thus the states of the system are not needed for feedback. But in practice there are two problems in realizing the state feedback gain exactly viz. poor error dynamics and noise sensitivity. So a linear matrix inequality formulation is used to overcome these undesired effects. The method has been applied to a large pressurized heavy water reactor (PHWR) for control of xenon induced spatial oscillations. A particular grouping of the state variables has been considered to decompose the system into the slow subsystem and the fast subsystem. Then fast output sampling fedback gains have been calculated. The efficacy of the control has been demonstrated by simulation of transient behavior of the nonlinear model of the PHWR.

Spatial Control of a Large PHWR by Decentralized Periodic Output Feedback and Model Reduction Techniques

Nuclear reactors of small and medium size are generally described by point kinetics model, however, this model is not valid in case of a large reactor, because in that flux shape undergoes appreciable variation with time. The behaviour of large reactor core can be explained with reasonable accuracy by spatial model like nodal model, which considers the reactor to be divided into number of regions or nodes. The thermal feedbacks which introduce nonlinearity into the problem, should be considered for realistic modeling. The spatial model of 540-MWe PHWR developed in is augmented with the dynamics of coolant and fuel temperatures and a 72nd-order model is obtained. As working with such a large model is difficult from the point of view of numerical computations, the new model having 14 inputs and 14 outputs, is suitably reduced by aggregation technique to obtain a 26th-order reduced model, which is more suitable to handle. The design of spatial controller by a state feedback based on reduced model needs the availability of all the states of the system for feedback purpose. As all the states of the reactor are not accessible for measurement, one has to resort to output feedback. Also, as the stability is not guaranteed by static output feedback, here the spatial controller is designed by periodic output feedback which is static in nature and at the same time guarantees complete closed-loop pole assignability. The various zones in a large reactor are coupled and a change in the control input to any zone causes respective change in the neutron flux of the other neighbouring zones, which may not be desirable. Therefore, a decentralized controller would serve as a better option, as it ensures that the input to any zone affects corresponding zone only and other zones are not affected by it. The above idea of periodic output feedback controller design yields a gain matrix with large magnitude, which amplifies measurement noise and is difficult to implement practically. Hence, it is desirable to keep the gain low. This objective is suitably expressed as LMI problem and putting appropriate design constraints, a better gain is obtained. The nonlinear model of the 540-MWe PHWR with the controller as above is tested for the reactivity transient simulation and the results of such a simulation are presented

A Three-Time-Scale Approach for Design of Linear State Regulator for Spatial Control of Advanced Heavy Water Reactor

This paper introduces a technique for simultaneous decomposition of a non-autonomous singularly perturbed system into three subsystems namely “slow”, “fast 1” and “fast 2” respectively. Design of a composite controller in terms of the individual subsystem controllers is derived and the decomposition of the optimal control problem of the original high order system into three smaller order optimal control problems, with separate quadratic performance indexes extracted from the quadratic performance index of the original system, is discussed.

Design of Single-Input Fuzzy Logic Controller for Spatial Control of Advanced Heavy Water Reactor

In large nuclear reactors, such as the Advanced Heavy Water Reactor (AHWR), the spatial oscillations in the neutron flux distribution due to xenon reactivity feedback need to be properly controlled; otherwise, power density and rate of change of power at some locations in the reactor core may exceed limits of fuel failure due to “flux tilting.” Further, situations, such as online refueling, might cause transient variations in the flux shape from the nominal flux shape. For analysis and control of spatial oscillations in AHWR, it is necessary to design a suitable control strategy, which will be able to stabilize these oscillations. In this paper, a simplified scheme to design a fuzzy logic controller for spatial control of AHWR, known as the single-input fuzzy logic controller (SIFLC) is proposed. The SIFLC reduces the conventional two-input fuzzy logic controller (CFLC) to a single-input FLC. The SIFLC offers a significant reduction in rule inferences and simplifies the tuning of control parameters. The SIFLC requires less execution time compared to CFLC for the control of spatial oscillations in AHWR. Through the dynamic simulations, it is observed that the designed SIFLC is able to suppress spatial oscillations developed in AHWR and the performance is found to be better compared to the recently proposed approach in the literature.

Design of Fast Output Sampling Controller for Three-Time-Scale Systems: Application to Spatial Control of Advanced Heavy Water Reactor

This paper introduces a formulation for design of Fast Output Sampling (FOS) controllers for three-time-scale systems. It is shown that the FOS control gain for a three-time-scale system can be obtained by combining the solutions of the three subsystem problems, obtained separately. Since three smaller order subsystem problems are to be solved in lieu of one high order problem, numerical ill-conditioning is completely avoided. Techniques for block-diagonalization and composite control of a three-time-scale system are also discussed. The proposed method is applied to the problem of spatial control of advanced heavy water reactor (AHWR). The model of AHWR is decomposed into three subsystems respectively named “slow,” “fast 1,” and “fast 2,” and separate subsystem control problems are cast from which an FOS controller for the original system is derived. Efficacy of the controller thus obtained is demonstrated through dynamic simulations. The controller thus designed employs only the output information to achieve arbitrary pole placement.

Coupled neutronics-thermal hydraulics model of Advanced Heavy Water Reactor for control system studies

A coupled neutronics-thermal hydraulics model of advanced heavy water reactor (AHWR) which can be used for control system studies, simulation etc. is presented in this paper. A nodal model is developed for core neutronics and a simple two-phase thermal hydraulics model is developed from first principles. Neutronics-thermal hydraulics coupling is achieved through the void reactivity feedback model. Selection of a suitable nodalization scheme through steady state analysis is discussed, and dynamic response of the model is also presented.

Technical communique: Discrete-time output feedback sliding mode control for spatial control of a large PHWR

This paper presents a novel approach to design a sliding mode control using a new computationally efficient formulation of Multirate Output Feedback (MROF), that applies to models in a special form in which some of the states of the system are directly available as outputs. The computation of the remaining states of the system requires matrices of lower dimension and thus the method is less susceptible to ill-conditioning. The proposed method has been successfully applied to the spatial control problem of a large Pressurized Heavy Water Reactor (PHWR), in which the outputs are the 14 zonal power levels. Also, the zonal power levels are selected as state variables along with other variables. Thereby, the PHWR model is perfectly suitable for the application of this new formulation. The non-linear model of PHWR including xenon and iodine dynamics is characterized by 70 state variables and 14 inputs and outputs each. Linear nodal model is obtained by linearizing the non-linear dynamic equations of the reactor about the full power operating point. The non-linear simulation results in representative transients produced by the proposed method are found to be superior to other methods.