 Journal of Power and Energy Engineering, 2014, 2, 130-138 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jpee http://dx.doi.org/10.4236/jpee.2014.24019 How to cite this paper: Dong, Z. (2014) PD Power-Level Control Design for MHTGRs. Journal of Power and Energy Engineering, 2, 130-138. http://dx.doi.org/10.4236/jpee.2014.24019 PD Power-Level Control Design for MHTGRs Zhe Dong Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing, China Email: dongzhe@mail.tsinghua.edu.cn Received December 2013 Abstract Due to its inherent safety feature, the modular high temperature gas-cooled reactor (MHTGR) has been seen as one of the best candidates in building next generation nuclear plants (NGNPs). Since the MHTGR dynamics has high nonlinearity, it is necessary to develop nonlinear power-level con- troller which is not only beneficial to the safe, stable, efficient and autonomous operati o n of the MHTGR but also easy to be implemented practically. In this paper, based on the concept of shifted- ectropy and the physically-based control design approach, it is proved theoretically that the sim- ple proportional-differential (PD) output-feedback power-level control can provide globally asymptotic closed-loop stability. Numerical simulation results verify the theoretical results and show the influence of the controller parameters to the dynamic response. Keywords Modular High Temperature Gas-Cooled Reactor (MHTGR); Power-Level Control; Closed-Loop Stability 1. Introduction Due to its inherent safety performance, the modular high temperature gas-cooled reactor (MHTGR) has been seen as one of the best candidates for building next generation nuclear plants. The MHTGR uses helium as coo- lant and graphite as moderator and structural material, and its inher ent safety is given by the low pow er density, strong negative temperature feedback effect and slim reactor shape [1]. China began to study the MHTGR at the end of 1970 s, a nd a 10 MWth pebble-bed high temperature gas-cooled test reactor HTR-10 designed by institute of nuclear and new energy technology (INET) of Tsinghua University ach ieved its criticality in December 2000 and full power in January 2003 [2]. Then, six safety demonstration tests were done on HTR-10, which manifest- ed its in heren t safety and self-stabilizin g featu res [3]. Based on the experience of the HTR-10 project, a high tem- perature gas-cooled reactor pebble-bed module (HTR-PM) project was then proposed [4]. As shown in Figure 1, the HTR-PM plant cons ists of two one-zone MHTGRs with combined thermal power of 2 × 250 MW th, and has the structure of two nuclear steam supplying sys tems (NSSSs) driving one steam turbine [4]. Here, the NSSS is composed of an MHTGR, a helical coiled once-through steam generator (OTSG) and some connecting pipes. Since a MHTGR is essentially a nonlinear dynamical system, it is necessary to develop nonlinear power-level control laws of the MHTGR for safe , stable and efficient operation. Actually, nonline ar power-level control de- sign is a hot field in nuclear engineering, and there have been some promising nonlinear reactor control design methods. Shtessel gave a nonlinear power-level regulator based on sliding mode control and observation tech-  Z. Dong niques for space reactor TOPAZ II [5]. Dong designed a dynamic output feedback dissipation power-level con- trol for the pressurized water reactors (PWRs) [6] by the use of the backstepping technique [7] and dissipa- tion-based high gain filter (DHGF) [8,9]. Etchepareborda and Eliasi proposed the nonlinear MPC (NMPC) me- thod for PWR power-level control design [10-12]. However, the forms of the above nonlinear power-level con- trol laws are too complicated to be implemented practically. Control desig n by fully using the good natural sys- tem d ynamics, i.e. the physically-based control design method can lead to simple and effective controllers, and is a promising trend of advanced control theory [13-15]. Very recently, based on the physically-based design approach, Dong proposed a nonlinear dynamic output-feedback power-level control for the PWRs [16], and also proved theoretically that the simple proportional-differential (PD) power-level control could guarantee globally asymptotic closed-loop stability for the PWRs [17]. Since the dynamic features of the MHTGR is different from that of the PWR, the power-level control de- signed for the PWR cannot directly applied to the MHTGR. It is necessary to develop nonlinear power-level controller for the MHTGRs. Dong designed a nonlinear state-feedback power-level control strategy to the MHTGR based on the technique of iterative damping assignment (IDA) [18]. Although this IDA-based control can provide globally asymptotic closed-loop stability, its mathematical form is too complex to be implemented practically. Based upon the physic ally-based control design approach, Dong also gave a nonlinear dynamic out- put feedback power-level controller for the MHTGR [19]. However, th is control is still complicated in its form. Therefore, it is necessary to design simple power-level control laws for the MHTGR with stro ng load following capability. In this paper, based on the concept of shifted-ectropy and physically-based control design method, it is proved theoretically that the static output -feedback control with simple PD structure can globally asymptotically stabil- ize the MHTGR. Numerical simulation results not only verify the theoretical results but also illustrate the rela- tionshi p be t ween performanc e an d c ontroller parameters. 2. Problem Formulation 2.1. Nonlinear State-Space Model As shown in Figure 1, the MHTGR and OTSG of the NSSS is arranged side by side, and is connected to each other by a horizontal coaxial gas duct. The cold helium enters the main blower mounted on top of the OTSG, and is pressurized before flowing into the c old gas duct. It enters the channels inside the reflector of the core, and then passes through the pebble-bed from top to bottom where it is heated to a high temperature. The hot he- lium leaves the hot gas chamber at the bottom reflector, and flows into the primary side of the OTSG through the hot gas duct. The primary loop can be nodalized as the elements given in Figure 2. By adop ting the point kinetics with one equivale nt delayed neutron group and w ith the temperature reactivity feedback effect of the pebble-bed/reflector community, the dynamical model for control design can be written as ( ) ( ) ( ) ( ) ( ) rR rrrr RR,m r rr p0 RRH r RR ps H RHHS HH r rr , , , , , nncnT T c nc P TTT n T TTTT Gz ρβ α β λ µµ µµ ρ − = ++− Λ ΛΛ = − Ω =− −+ ΩΩ =−− − = (1) where nr is the relative neutron power, cr is the r elative concentration of delayed neutron precursor, β is the frac- tion of delayed fission neutrons, Λ is the effective prompt neutron life time, ρr is the reactivity provided by the control rods, λ is the effective radioactive decay constant of the precursor, TR and αR is the temperature and reac- tivity feedback co efficient of the community constituted by both the pebble-bed and reflector respectively, TR,m is the initial equilibriu m value of TR, P0 is the rated reactor thermal power, TH is th e av erag e helium temperature of the primary side, TS is the average coolant temperature of the secondary side of the OTSG, Ωp is the heat transfer coefficient between the helium and pebble- bed/reflector community, Ωs is the heat transfer coefficient  Z. Dong Figure 1. Composition of the HTR-PM plant. Figure 2. Nodalization of the primary loop. between the two sides of OTSG, μR and μH is respectively the total heat capacities of the pebble-bed/reflector community and helium inside the primary loop, Gr is the total differential reactivity worth of the contro l rod, and zr is the rod s p e e d signal. Here, note that αR is guaranteed t o be negative by physical design of the MHTGR. Define the deviations of the actual values of nr, cr, TR, TH, TS and ρr from their equilibrium values, i. e. nr0, cr0, TR0, TH0 and ρr0 as δnr = nr − nr0, δcr = cr − cr0, δTR = TR − TR0, δTH = TH − TH0, δTS = TS − TS0, and δρr = ρr −ρr0. Here, δTS reflects the influence of the secondary to primary loop, and can be well suppressed by adjusting the feedwater flow-rate of the OTSG. Therefore, in this paper, the influence of δTS is omitted. Let , (2) , (3) and . (4) Here, x is the reactor state-vector of the MHTGR. Then, the nonlinear state-space model for control design can be written as (5) where  Z. Dong ( ) ( ) ( ) ( ) ( ) ( ) R 12r01 3 12 p 01 34 RR ps 344 HH x xnxx xx Px xx xx x α β λ µµ µµ − −++ ΛΛ − Ω = −− ΩΩ −− fx , (6) ( ) T r0 113 nx × + = Λ gx O , (7) and . (8) 2.2. Theoretic Problem Based on the above modeling, the theoretic problem to be solved in this paper is summarized as follows. Problem 1. How to design an output-feedback PD c o ntrol law of n onlinear s y s tem (5) taking the form as , (9) so that x→O as t→∞? □ 3. PD Power-Level Control Design Following Theorem 1, i.e. the main resu lt of this paper, shows that simple output-feedback PD power-level con- trol law can guarantee asymptotic closed-loop stability of reactor state-variables . Theorem 1. There exists a PD power-level control law of nonlinear system (5) that provides globally asymptotic closed-loop stability fo r th e reac tor state of the MHTGR, i.e. x→O as t→∞. Proof: Based upon the idea of backstepping, a virtual control input ξr is firstly designed for subsystem ( )( ) ( ) r , . ξ = + = xf xgx y hx (10) From [17], the shifted-ectropy of neutron kinetics is ( ) 11 22 N1 2r0r0 r0r0 r0 ,1ln 11ln 1. xx xx xx nnn nn β ζλ =Λ+−+ ++−+ (11) Based on (11), let the Lyapunov function for the neutron kinetics be ( )( )( ) 2 I N 12N 121 0 ,, d 2 t k Vxxxxxss ζ =+ ∫ . (12) Here, the objective of adding the second term of VN is to minimize the steady error of nr by feedback control. Then, dif fe renti a t e VN along the trajectory given by neutron kinetics, and we have ( )( ) ( )() ( ) 2 12 N121rR3I1 r0 1 r020 ,d t xx Vxxxxk xss nxnx βξα − =−+ ++ ++ ∫ . (13) Moreover, it is clear that the shifte d-ectrop y of reactor thermal-hydraulics can be written as ( ) ( ) 22 T3 4R3H4 1 ,2 xxx x ζ µµ = + . (14) Then, based on (14), let the Lyapunov function of reactor thermal-hydraulics be  Z. Dong ( ) () ()( ) T 34T 34T 34 ,1 ,,V xxxxxx γ ζγς =−+ , (15) where γ is a positive given constant satisfying 0 < γ < 1, and ( ) 2 TR3H4S 4 R0 1d 2 t x xxss ς µµ µ =+ +Ω ∫ (16) denotes the energy variation of the thermal-hydraulic loo ps. Differentiate (15) along the trajectory g iven by the reactor thermal-hydraulics, and we have ( )()() ( ) ( ) ( ) 22 s H T0130144p 34s4 RR 0 2 pps2 p s433 ps ps d11 1, t VPxxPxxx ssxxx xx x µ γ γη µµ γη Ω = ++−−−Ω−+Ω Ω ΩΩ −−Ω +Ω−+ Ω +ΩΩ +Ω ∫ (17) where η is given positive constant satisfying 0 < η < 1. Choose the Lyapunov function for subsystem (10) as ( )() ( ) R 1N 12T 34 0 ,, q VVxxVxx P = +x , (18) where qR is a given positive constant, VN and VT is given by (12) and (15) respectively. Differentiate (18) along the trajector y given by subsystem dyna mics (10), and we can derive that ( ) ( )() ( )( ) ( ) ( ) ( )( ) 22 12 2 R 1p 34s4 r0 1 r020 22 ps p2 RR R433 3R1 0ps RR 2 s RR RH 1r1 I1R44 R RR 00 11 11 2 1d d 2 t xx q Vxx x nxnx P q q xx xxx Pq q xxkxssqxxss Pq βγη α γη αµ ξγ µµ − =−−−− Ω−+Ω ++ Ω +ΩΩ − −−−+−Σ+ Ω +ΩΣ Ω Σ +++ +++ ∫ t ∫ (19) where ( ) ( ) p s0 Rps 1 P γη Ω +Ω Σ= − ΩΩ . (20) From equation (19), if we design virtual control ξr as ( )( ) s H rND 1I1R44 RR 00 d d, tt kx kxssqxxss µ ξγ µµ Ω =−− − + ∫∫ (21) where ( ) ( ) 2 Rps R ND R 0p s 11 2 q kq P γη α −ΩΩ >+ Ω +Ω , (22) then the closed-loop subsystem constituted by (10) and (21) is globally asymptotically stable. Now, we design the control law for entire system (5). Choose the Lyapunov function of the entire system as ( ) ( ) 2 21 ,2 e VeV k ξ ξ ξ = +xx , (23) where kξ is a given po s itiv e consta nt, and , (24)  Z. Dong Differentiate (23) along the trajectory given by entire system dynamics (5), and we have ( ) ( )() ( )( ) ( ) ( ) ( ) 22 12 2 R 2p 34s4 r0 1 r020 22 ps p2 RR R433 3R1 0ps RR 2 ND 11r 11 11 2 1, xx q Vxx x nxnxP q q xx xxx Pq kxxee u k ξξ ξ βγη α γη ξ − =−−−− Ω−+Ω ++ Ω +ΩΩ − −−−+−Σ+ Ω +ΩΣ − ++− (25) where ( ) ( ) 2 Rps R ND NDR 0p s 11 2 q kk q P γη α −ΩΩ =−+ Ω +Ω . (26) From (25), if we choose feedback control u as ( ) 1rNP1ND 1TP4TD4 u kxkxkxkxkx ξ ξ =− +=−+++ , (27) where , (28) , (29) and , (30) then we ha ve ( ) ( )() ( )( ) ( ) ( ) 22 12 22 R 2ND1p34s4 r0 1 r020 22 ps p2 RR R433 3R1 0ps RR 11 1 1. 2 xx q Vkxx xx nxnx P q q xx xxx Pq βγη α γη − =− −−−−Ω−+Ω ++ Ω +ΩΩ − −−−+−Σ+ Ω +ΩΣ (31) Based on equation (31), it is clear that there always exists a PD power-level controller (27) so that reactor state x of MHTGR dynamics (5) are globally asymptotically stable. This completes the proof of this theorem. □ Remark 1. From equation (12) and (28), the steady error of relative nuclear power can be suppressed by enlarging the proportional feedback gain kNP corresponding to δnr. Also from (18), the dynamic performance of the thermal-hydraulic loop can be strengthened through enlarging qR, which certainly leads to larger values of feedback gains kND, kTP and kTD. □ Remark 2. Since positive constants γ and η can be arbitrarily chosen between 0 and 1, inequality (22) is easy to be satisfied by choosing γ to be close enough to 1 and η to be close enough to 0. However, larger γ also leads to larger kTP and kTD. □ 4. Numerical Simulation with Discussions 4.1. Description of the Numerical Simulation To verify the stabilization capability of PD control (27), it is applied to the power-level regulation of an MHTGR of the HTR-PM plant. Here, the dynamic model of the MHTGR used in this simulation adopts that one composed of both nodal neutron kinetics and nodal reactor thermal-hydraulics given in [20]. The OTSG adopts the moving boundary model presented in [21]. The model of the steam turbine and that of the electrical genera-  Z. Dong tor are also included in the simulation code [22]. The controller parameters are selected as kNP = kND = 0.5, γ = 0.5. Here, qR is set to be variable. 4.2. Simulation Results In this simulation, the case of large-range pow er-level maneuver of the MHTGR is studied to show the feasibil- ity of PD power-level control (27). As the power demand signal decreases linearly from 100% full power-level (FP) to 50% FP in 5 minutes, the error signals of the nuclear power and the helium temperature cause the power- level control to generate proper control rod speed to cope with the decrease of power demand. The responses of relative nuclear power, average fuel temperature and outlet helium temperature as well as the designed rod speed with different values of qR are all shown in Figure 3. 4.3. Discussions From Figure 3, we can see that the dynamic performance of reactor thermal-hydraulic loop is higher if qR is larger. Moreover, a larger qR results in the deterioration of the response of neutron kinetics. Actually, this phe- nomenon can be interpreted by the proof of Theorem 1. From Equation (18), it is clear that the ratio of VT in V1 is higher if qR is la rger. S ince VT denotes the Lyapunov function of the thermal-hydraulic loop, larger ratio of VT results in faster convergence of those thermal-hydraulic state-variables, which can be easily seen from Figures 3(b) and (c). On the other hand, from Equation (26), larger qR leads to smaller , which then we aken the convergence of neutron kinetic states. As we can see from Figure 3(a), the oscillation of the rel at i ve nuc l e a r power is toughe r i f qR is larger. Thus, from the a bove disc us s i on, Figure 3. Numerical simulation results: (a) Relative nuclear power, (b) Average fuel temperature, (c) Outlet he lium temper- ature, (d) Designed control rod speed signal. 800 1000 1200 1400 1600 1800 2000 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 time/s n r (a) 8001000 1200 1400 1600 1800 2000 600 605 610 615 620 625 630 635 640 time/s T f / ℃ (b) q R =0.1 q R =0.05 q R =0.03 q R =0.01 80010001200 14001600 1800 2000 715 720 725 730 735 740 745 750 755 time/s Tcout / ℃ (c) 800 1000 1200 1400 1600 1800 2000 -1.5 -1 -0.5 0 0.5 1 1.5 2 time/s Rod Speed v r/(cm/s) (d)  Z. Dong we can see that the numerical simulation results given in Section 4.2 are in accordance with the theoretical anal- ysis in Section 3. Moreover, the numerical simulation results also illustrate the relationship between dynamic performance and controller parameters. 5. Conclusion Due to its inherent safety feature and potential economic competitive power, the modular high temperature gas- cooled reactor (MHTGR) has already been seen as one of the best candidates in building SMR-based nuclear power plant. Since power-level control is meaningful in providing safe, stable and efficient reactor operation, and an MHTGR is essentially a nonlinear dynamic system, it is crucial to develop nonlinear pow er-level control which can be easily implemented. Based upon the shifted-ectropies of both neutron kinetics and reactor ther- mal-hydraulics, it is proved theoretically that the simple PD power-level control can provide globally asymptotic closed-loop stability for the MHTGR. Numerical simulation results are consistent with the theoretical analysis, and also showed the relationship between the regulat ing perf ormance an d c ontroller param e ters. 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