Power-level regulation and simulation of nonlinear pressurized water reactor core with xenon oscillation using H-infinity loop shaping control

This investigation is to solve the power-level control issue of a nonlinear pressurized water reactor core with xenon oscillations. A nonlinear pressurized water reactor core is modeled using the lumped parameter method, and a linear model of the core is then obtained through the small perturbation linearization way. The H∞loop shapingcontrolis utilized to design a robust controller of the linearized core model.The calculated H∞loop shaping controller is applied to the nonlinear core model. The nonlinear core model and the H∞ loop shaping controller build the nonlinear core power-level H∞loop shaping control system.Finally, the nonlinear core power-level H∞loop shaping control system is simulatedconsidering two typical load processes that are a step load maneuver and a ramp load maneuver, and simulation results show that the nonlinear control system is effective.


Introduction
Nuclear energy as a clean energy can be sustainable and renewable.Hence, developing technologies of nuclear power plants (NPPs) for electricity generation is inevitable.Of these technologies, it is necessary to investigate the power control technology of nonlinear nuclear reactor cores with xenon oscillations.
Several researchers have made great efforts for the improvement of reactor core power control systems.The state-feedback basedcontrol with classical control part was proposed by Edwards et al. [1][2] to devise power control systems of pressurized water reactor (PWR) cores.The linear quadratic Gaussian with loop transfer recovery (LQG/LTR) control wasadopted to contrivePWR core controllers for power regulations [3][4].However, the controllers in references [1][2][3][4] are designed based on a linear core model without xenon oscillations, and not always optimal or even ineffective for nonlinear cores.
Based on this consideration, the H∞loop shaping control [5] with stronger robustness than that of the statefeedback or LQG/LTR control is adopted to design a nonlinear power control system ofa PWR core with xenon oscillations.Finally, the nonlinear PWR core power H∞loop shaping control system is simulated and conclusions are drawn.

Modeling for PWR core
The According to the point reactor core modeling [6][7][8], the nonlinear PWR core is modeled adopting the point kinetics equations with six groups of delayed neutrons and reactivity feedbacks due to control rod movement and variations in fuel temperature, coolant temperature and xenon concentration.Main model parameters are given in Table 1.The nonlinear core model is showed as Eq.( 1).One group delayed neutron model is utilized and the coolant inlet temperature is treated as a constant [2][3][4].The small perturbation linearization methodology is utilized to linearize the nonlinear core model (1).The transfer function and the state equation of the core are separately calculated and expressed by where u=δrod-the input; y=δP r -the output; a i (i=0,1,2,5)numerator coefficients; b i (i=0,1,2,3,6)-denominator coefficients; s-laplace operator; x=[x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ] T =[δP r , δc r , δI r , δX f , δT f , δT e ] T -the state matrix; A-the R 6×6 system matrix; B-the R 6×1 input matrix; C-the R 1×6 output matrix; D-the zero matrix.
The transfer function G is calculated by using parameters from Ref. [9], in which the total primary heat output is 2200 MW, the primary coolant inlet temperature is 285 o C, the primary coolant outlet temperature is 317 o C, the primary coolantaverage pressure is 15.5 MPa, and the primary coolant mass flow is 12861.1 kg/s.

H ∞ loop shaping control
The H ∞ loop shaping design approach was proposed by McFarlane et al. [5] to implement tradeoffs between performance and robust stability of a stable H ∞ loop shaping control system for a plant, where a normalized coprime factor H ∞ robust stabilization issue needs to be solved to guarantee closed-loop stability and a certain level of robust stability of the system at all frequencies.
The loop shaping design procedure for the core.This H ∞ loop shaping design approach consists of two key techniques, namely loop shape and robust stabilization.The loop shaping design procedure is shown in Fig. 1, where W 1 and W 2 are weight matrices or compensators; G a is a shaped plant based on G; K s is the H ∞ controller of G a ; K is the controller to be devised; δP rd denotes the reference input, namely the desired change of power level.(2) Robust stabilization Step one: Calculation of max where M s is a normalized coprime factor of G a subjecting to G a = M s -1 N s .
Step two: Select (3) The eventual feedback controller: The stability margin H is both a measurement for robustness of a loop shape control system and the indicator of the success of the loop shaping.Generally speaking, K s is solved so that ICMIT 2016 0 can guarantee that the control system possess satisfied robustness.A small value of max H indicates incompatibility among a desired loop shape, a controlled plant and closed-loop robust stability.In the presence of this case, W 1 and W 2 need to be regulated again to obtain a decent loop shape.

Design of nonlinear PWR core power control system
According to Section 3, solving K is the following.
(1) Loop shaping G a is exploited and defined as a shaped plant of G.Of selected weight matrices for G, W 1 is I 1×1 ; W 2 is expressed by Eq.( 6), and parameters of W 2 are shown in Table 2, where e i =10 i .6 0 2 6 (2) Solve controllers The controller K s is solved, and then the final controller K of G is calculated by K=W 1 K s W 2 .K is expressed by Eq.( 7), and parameters of K of G are shown in Table 3.The solved stability margin H of K is shown in Table 3.  (3) Effect of local controllers When the reference input δP rd is taken as a 0.1 step, responses of the output δP r in G with K are shown in Fig. 2, where the curve is step response of δP r in G with K.According to the curve, the output δP r in G with K in response to a step possesses good performance, such as the zero overshoot, the short adjustment time and the zero static error.Hence, the controller Kis desirable.
The solved controller K is applied to the nonlinear core model.The combination of the nonlinear core model and K is the nonlinear core power H ∞ loop shapingcontrol system.The schematic of the nonlinear core power H ∞ loop shaping control system is shown in Fig. 3.

Simulations
The nonlinear core power H ∞ loop shaping control systemis simulated considering two typical load variations that are the -10% step process and the ramp process with ±0.0556%•min -1 variations.The direction from the bottom to the top of core is treated as the positive direction of y-axis for simulation work, the core bottom is the zero point of y-axis and the height of core is 366cm.When the core power level is at 100%, the initial rod position is at the top of the core, namely at 366cm.
When the reference input δP rd is taken as a -10% step load in 24 hours, change trajectories of core parameters during the core following the step load (100%→90%) are shown in Fig. 4, where S1 is the desired load change trajectory for the start of reactorfull power level operation, the middle of reactor shut-down operation with a -10% step variation, and the end of 90% power level operation; S2 is a change trajectory of the core power level; S3 is a change trajectory of the control rod position in the core.
When the reference input δP rd is taken as a ramp load with ±0.0556%•min -1 variations, change trajectories of core parameters during the core following the ramp load (100%→90%→100%) are shown in Fig. 5, where R1 is the desired load change trajectory for the start of reactor full power level operation, the middle ofreactor shutdown operation with a -6%•min -1 ramp variation and 90% power level operation, and the end of reactor start-up operation with a 6%•min -1 ramp variation; R2 is a change

Conclusions
The power-level control for PWR cores with xenon ocsillationsis difficult to be implemented, which is also a challenge reaesrch direction.In this paper, the power-level control problem for a PWR core is handled applying the H ∞ loop shapingcontrol strategy to this nonlinear core.
The nonlinear PWR core and its linearized model are separately modeled using the lumped parameter method and the small perturbation linerization method that are capable of be reference modeling approaches to study other nonlinear plants.
In order to implement applications of the controller designed on a linearizedcore model to a nonlinearcore model, the solved H ∞ loop shaping controller has strong robustness of accommodating model errors between the nonlinear model and the linear model.
The devised nonlinear PWR core power H ∞ loop shaping control systemcan control the core powerlevelsatisfactorily in response to different load changes such as a step or ramp.Hence, the nonlinear PWR core power H ∞ loop shaping control system is effective.
(1) Loop shaping Singular values of a controlled plant G are regulated to obtain a desired open loop shape adopting an additional pre-compensator W 1 and/or an additional postcompensator W 2 .The plant G is shaped to be G a introducing W 1 and/or W 2 , G a =W 2 GW 1 .Suppose W 1 and W 2 are such that G a contains no hidden modes with unstability.

Table 1 .
Main model parameters.

Table 3 .
Parameters of K.