# A Speed-Governing System Model with Over-Frequency Protection for Nuclear Power Generating Units

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Nuclear Power Unit Overspeed Protection and Super-Acceleration Protection

_{g}and ω

_{REF}are the generator frequency and the frequency reference value, respectively. P

_{e}and P

_{REF}are the generator output power and power reference value, respectively. P

_{m}is the turbine output power. P

_{CV}is the high-pressure regulating valve opening. P

_{IV}is the opening degree of the medium-pressure regulating valve. P

_{VR}is the digital proportional–integral–differential (PID) governor’s output value. U

_{L}and f

_{L}represent the generator’s terminal voltage and frequency. V

_{REF}, V

_{t}, and E

_{fd}are the terminal reference voltage, terminal voltage, and excitation voltage, respectively.

#### 2.1. Overspeed Protection of the Nuclear Power Unit

#### 2.2. Super-Acceleration Protection of Nuclear Power Unit

_{CVR}which represents the main steam flow) and the speed acceleration of the unit is segmented as shown in Equation (1).

_{1}and α

_{2}are threshold values with the unit of pu/s (i.e., per unit/second). P

_{VR0}is the digital PID governor’s given initial value.

_{1}pu/s, the main steam flow is controlled by the governor. When the speed acceleration exceeds α

_{1}pu/s, the main steam flow is directly given by the super-acceleration protection.

## 3. Speed-Governing System and Core System Models of a Nuclear Power Unit

#### 3.1. Power–Frequency Electro-Hydraulic Speed-Governing Model

_{R1}is the inertia time constant of the frequency measurement. K

_{P}, K

_{D}, and K

_{i}are the integral, differential, and integral link coefficients of the PID controller.

_{P1}, K

_{D1}, and K

_{I1}are the integral, differential, and integral link coefficients of the sub-loop PID controller, respectively. V

_{O}and V

_{C}are the normal opening and closing rate limits of the oil motive, respectively.

_{IV}is the action time constant of the oil motor. T

_{1}, T

_{CV1}, and T

_{IV1}are the time constants of the valve opening displacement sensor. P

_{IVR}is the given opening of the medium-pressure regulator with a value of 1 pu during normal operation. It is assumed that the medium-pressure regulator is fully open during normal operation. P

_{CV,MAX}, P

_{IV,MAX}, P

_{CV,MIN}, and P

_{IV,MIN}are the maximum or minimum valve opening degree. In Figure 5a, Switch 1 is selected to be in the second position if the super-acceleration protection is in action. Switch 2 is selected to be in the first or second position corresponding to the closing/opening of the valve, respectively.

#### 3.2. Steam Bypass Control System Model

_{b1}, τ

_{b2}, τ

_{b3}, and τ

_{b4}are the compensator and filter time constants. T

_{ref}is the coolant’s average temperature reference value, and P

_{bp}is the bypass valve opening degree. T

_{avg}is the measured average temperature in the primary circuit.

_{ref}and T

_{avg}.

#### 3.3. Steam Turbine Model

_{s}and P

_{sn}are the rated and actual main steam pressure, respectively.

_{CH}, T

_{RH}, and T

_{CO}are the volume time constants of high-pressure steam, intermediate reheat steam, and low-pressure steam, respectively. F

_{HP}, F

_{IP}, and F

_{LP}are the percentage of steady-state power output of the high-pressure cylinder, medium-pressure cylinder, and low-pressure cylinder in the total power output, respectively. λ

_{h}is the natural power overshooting co-efficient of the high-pressure cylinder.

#### 3.4. Reactor Core System Model

_{r}is the neutron flux density in the core with the unit of neutron number/cm

^{3}, indicating the reactor’s thermal power. T

_{cav}is the coolant’s average temperature in the core. T

_{c1}and T

_{c2}are the core coolant’s inlet and outlet temperatures, respectively. T

_{p}is the average temperature of the primary coolant. h

_{s}is the specific enthalpy of steam at the outlet of the second circuit of the steam generator. D

_{sp}is the main pump speed by per-unit value, which is related to the main coolant pump flow.

- Core neutron dynamic module:$$\begin{array}{l}\frac{\mathrm{d}{N}_{\mathrm{r}}}{\mathrm{d}t}=\frac{{R}_{\mathrm{ext}}-\beta}{l}{N}_{\mathrm{r}}+\frac{\beta}{l}{C}_{\mathrm{r}}+[{\alpha}_{\mathrm{f}}({T}_{\mathrm{F}}-{T}_{\mathrm{F}0})+{\alpha}_{\mathrm{c}}({T}_{\mathrm{cav}}-{T}_{\mathrm{cav}0})]{N}_{\mathrm{r}}\\ \frac{\mathrm{d}{C}_{\mathrm{r}}}{\mathrm{d}t}=\lambda {N}_{\mathrm{r}}-\lambda {C}_{\mathrm{r}}\end{array}$$
- Core fuel and coolant temperature module:$$\begin{array}{l}\frac{\mathrm{d}{T}_{\mathrm{F}}}{\mathrm{d}t}=\frac{{F}_{\mathrm{f}}{P}_{0}}{{\mu}_{\mathrm{f}}}{N}_{\mathrm{r}}-\frac{\Omega}{{\mu}_{\mathrm{f}}}{T}_{\mathrm{F}}+\frac{\Omega}{2{\mu}_{\mathrm{f}}}{T}_{\mathrm{cav}}+\frac{\Omega}{4{\mu}_{\mathrm{f}}}{T}_{\mathrm{HL}}+\frac{\Omega}{4{\mu}_{\mathrm{f}}}{T}_{\mathrm{CL}}\\ \frac{\mathrm{d}{T}_{\mathrm{cav}}}{\mathrm{d}t}=\frac{(1-{F}_{\mathrm{f}}){P}_{0}}{{\mu}_{\mathrm{C}}}{N}_{\mathrm{r}}+\frac{\Omega}{{\mu}_{\mathrm{C}}}{T}_{\mathrm{F}}-\frac{4M+\Omega}{2{\mu}_{\mathrm{C}}}{T}_{\mathrm{cav}}+\frac{4M-\Omega}{2{\mu}_{\mathrm{C}}}{T}_{\mathrm{c}1}\\ \frac{\mathrm{d}{T}_{\mathrm{c}2}}{\mathrm{d}t}=\frac{(1-{F}_{\mathrm{f}}){P}_{0}}{{\mu}_{\mathrm{C}}}{N}_{\mathrm{r}}+\frac{\Omega}{{\mu}_{\mathrm{C}}}{T}_{\mathrm{F}}+\frac{4M-\Omega}{2{\mu}_{\mathrm{C}}}{T}_{\mathrm{cav}}-\frac{4M+\Omega}{2{\mu}_{\mathrm{C}}}{T}_{c2}\end{array}$$
- Hot line and cold line temperature module:$$\begin{array}{l}\frac{\mathrm{d}{T}_{\mathrm{HL}}}{\mathrm{d}t}=\frac{1}{{\tau}_{\mathrm{HL}}}({T}_{\mathrm{c}2}-{T}_{\mathrm{HL}})\\ \frac{\mathrm{d}{T}_{\mathrm{c}1}}{\mathrm{d}t}=\frac{1}{{\tau}_{\mathrm{CL}}}({T}_{\mathrm{CL}}-{T}_{\mathrm{c}1})\end{array}$$
- Primary loop average temperature module:$${T}_{\mathrm{avg}}=\frac{1}{{\tau}_{\mathrm{c}}s+1}\left(\frac{{T}_{\mathrm{HL}}+{T}_{\mathrm{CL}}}{2}\right)$$
- Steam generator module:

- f.
- Main coolant pump module:

^{−1}. α

_{F}and α

_{C}are the reactivity co-efficients of the fuel temperature and coolant temperature with unit of pcm/°C, respectively. T

_{F0}and T

_{cav0}are the initial temperature values. P

_{0}is the rated core thermal power. F

_{f}is the heating fuel share. Ω is the heat transfer coefficient between the fuel and coolant in the core. μ

_{f}and μ

_{C}are the heat capacity of the fuel and core coolant, respectively. M = D

_{sp}× C

_{pc}× m

_{Cn}, in which C

_{pc}is the coolant heat capacity and m

_{Cn}is the rated coolant mass flow. Ω

_{p}is the heat transfer coefficient between the coolant in the steam generator and the U-shaped heat pipe. Ω

_{S}is the heat transfer coefficient between the U-shaped heat pipe and the secondary loop steam. μ

_{p}and μ

_{m}are the heat capacity of the coolant in the steam generator and the U-shaped heat pipe, respectively. h

_{fw}is the inlet temperature specific enthalpy of the secondary loop feed water. K

_{Ps}is the steam pressure time constant. K

_{Ps_Ts}(P

_{s}) is the conversion relation between the main steam pressure and the temperature of the secondary circuit. τ

_{HL}and τ

_{CL}are the coolant hot line and cold line time constants, respectively. τ

_{c}is the measuring time constant of the coolant temperature.

_{r}is the precursor nuclear density of the equivalent single set of delayed neutrons, with unit of core number/cm

^{3}. R

_{ext}is the reactivity introduced by the control rod, with unit of pcm. T

_{m}denotes the U-shaped heat pipe temperature. T

_{pj}is the inertia time constant of the main coolant pump rotor. ω

_{p}

^{*}and ω

_{pr}

^{*}are the speed and rated speed per-unit values of the asynchronous motor, respectively. U

_{L}

^{*}and f

_{L}

^{*}are per-unit values.

_{m}or N

_{r}due to sudden load shedding, etc. The dead zone, hysteresis link, and nonlinear conversion were applied to maintain the average primary coolant temperature in the designed control band.

_{j}is the inertial time constant of the generator, D

_{ω}is the damping coefficient. T

_{m}and T

_{e}are the mechanical torque and electromagnetic torque, respectively. δ is the power angle. ω

_{0}is the rated speed equal to 1.

## 4. Results and Discussion

#### 4.1. Simulation Example

#### 4.2. Simulation Analysis of Overspeed Protection Characteristics of Load Rejection

- (1)
- There is a large difference between the mechanical power of the nuclear turbine and the electromagnetic power of the generator (see Figure 11a,b) after a large disturbance on the grid side. This can lead easily to the overspeed of the unit, due to the large volume time constants for the nuclear steam turbine and the high proportion of the low-pressure cylinder power. Nuclear turbines are more likely to overspeed during tripping.
- (2)
- The speed-governing system and overspeed protection acted correctly, which quickly reduced the turbine’s mechanical power, with the maximum speed of the nuclear power unit reaching 1.07 pu. When using OPC protection that only considers the quick closing of the high-pressure regulating valve (i.e., strategy two), the maximum unit speed exceeded 1.10 pu. This would lead to the action of the ETS. This would remove the steam turbine directly and close the core rapidly, resulting in a significant recovery time.
- (3)
- After the steam bypass valve acted, the rapid rise of steam pressure in the secondary loop was limited. As a result, the deviation between the reactor power and the power output of the steam generator was reduced. The rate of decrease of the core power was slowed due to the reactor control, thus the safety of the core and thermal system was ensured.
- (4)
- In the process of load rejection, the maximum and minimum terminal voltage is 1.15 and 0.93 pu, respectively. This did not cause a voltage stability problem. The maximum and minimum speed of the main cooling pump reached 1.073 and 0.990 pu, respectively. This may lead to the action of the main cooling pump overspeed protection as the coolant flow is mainly affected by the change of the unit speed.

#### 4.3. Simulation of Overspeed Protection and Super-Acceleration Protection Characteristics of a Nuclear Power Unit during Large Grid Disturbance

- (1)
- After a three-phase short-circuit failure lasted for 0.25 s on the power grid side, the unit speed increased significantly in a short time. The maximum unit speed reached 1.12 pu by adopting the overspeed protection strategy which quickly closed the high-pressure regulating valve only (i.e., strategy three that is similar to the OPC protection strategy for the simulation of thermal power units). The multiple actions of the OPC caused the high-pressure regulating valve to close repeatedly. The maximum unit speed was 1.095 pu when we adopted the overspeed protection strategy which simultaneously quickly closed the high-pressure and medium-pressure regulating valves (i.e., strategy two).
- (2)
- This led to the action of the super-acceleration protection as the rate of change of the speed exceeds the set α1. The peak unit speed was 1.026 pu when the super-acceleration protection strategy (i.e., strategy one) was adopted, which suppressed the further unit speed increase with the minimum mechanical power dropping to 0.40 pu. As a result, the overspeed protection did not operate. However, the overspeed protection both acted in strategy two and strategy three, and the mechanical power dropped to zero. It rose to the rated value after fault clearance.
- (3)
- Since the main cooling pump was more sensitive to the power supply frequency as the unit speed rose after the grid three-phase short-circuit fault, the maximum speed of the main cooling pump reached 1.026, 1.103, and 1.114 pu under strategy one, two, and three, respectively. The minimum speed of the main cooling pump was slightly less than 0.975 pu during the fault.
- (4)
- The impact of the grid disturbance on the reactor and thermal system was minimized after adopting the super-acceleration protection, which avoided an excessive rise in unit speed. Strategy three had the greatest impact on the reactor and its thermal system.

#### 4.4. Model Validation with the PCTRAN Software

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

T_{CH} | High pressure steam volume time constant |

T_{RH} | Volume time constant of the intermediate reheat steam |

T_{CO} | Low pressure steam volume time constant |

T_{O} | Opening time constant of the high-pressure regulating valve oil motive |

T_{C} | Shutdown time of the high-pressure regulating valve oil motive constant |

V_{CCVQ} | Quick closing speed limit of the high-pressure regulating valve |

V_{CIVQ} | Quick closing speed limit of the medium-pressure regulating valve |

K_{CVQ} | Quick closing co-efficient of the high-pressure regulating valve |

K_{IVQ} | Quick closing co-efficient of the medium-pressure regulating valve |

ω_{g} | Generator speed |

${\omega}_{g}^{\prime}$ | Rotational speed acceleration of the unit |

Pm | Turbine output power |

P_{CV} | High-pressure regulating valve opening degree |

P_{IV} | Medium-pressure regulating valve opening degree |

P_{bp} | Bypass valve opening degree |

U_{L}, f_{L} | Generator’s terminal voltage and frequency |

Dsp | Main coolant pump flow |

N_{r} | Neutron flux density in the core |

R_{ext} | Reactivity introduced by the control rod |

T_{avg} | Measured average temperature in the primary circuit |

T_{HL}, T_{CL} | Hot line and cold line temperatures of the primary coolant |

T_{F} | Reactor core fuel temperature |

Ps | Main steam pressure |

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**Figure 5.**Electro-hydraulic servo system model. (

**a**) Electro-hydraulic servo system model of the high-pressure regulating valve; (

**b**) electro-hydraulic servo system model of the medium-pressure regulating valve.

**Figure 11.**Dynamic response curves of the generator during full load rejection under two different strategies. (

**a**) Mechanical power; (

**b**) electromagnetic power; (

**c**) high-pressure regulating valve; (

**d**) medium-pressure regulating valve; (

**e**) generator frequency; (

**f**) terminal voltage of the unit.

**Figure 12.**Dynamic response curves of the reactor and thermal system during full load rejection under two different strategies. (

**a**) Core neutron flux density; (

**b**) reactivity of the control rods; (

**c**) fuel mean temperature; (

**d**) coolant mean temperature; (

**e**) bypass control valve opening; (

**f**) main steam pressure; (

**g**) coolant main pump speed.

**Figure 13.**Dynamic response curves of the generator during large grid disturbance under three different strategies. (

**a**) Mechanical power; (

**b**) generator frequency; (

**c**) high-pressure regulating valve; (

**d**) medium-pressure regulating valve; (

**e**) OPC instructions.

**Figure 14.**Dynamic response curves of the reactor and thermal system during large grid disturbance under three different strategies. (

**a**) Core neutron flux density; (

**b**) reactivity of the control rods; (

**c**) fuel mean temperature; (

**d**) coolant mean temperature; (

**e**) bypass control valve opening; (

**f**) main steam pressure; (

**g**) coolant main pump speed.

**Figure 15.**Simulation comparison of load rejection with the personal computer transient analyzer (PCTRAN). (

**a**) Coolant main pump flow and unit speed; (

**b**) rod position; (

**c**) core neutron flux density; (

**d**) fuel mean temperature; (

**e**) main steam pressure.

Parameter Symbols | Recommended Values | Parameter Symbols | Recommended Values |
---|---|---|---|

T_{CH} | 0.4 | V_{CCVQ} | −5 |

T_{RH} | 10 | V_{CIVQ} | −5 |

T_{CO} | 1 | T_{CV1} | 0.01 |

F_{HP} | 0.33 | T_{IV1} | 0.01 |

F_{IP} | 0 | P_{CV,MAX} | 1.03 |

F_{LP} | 0.67 | P_{CV,MIN} | 0 |

λ_{h} | 0.8 | P_{IV,MAX} | 1 |

T_{O} | 5 | P_{IV,MIN} | 0 |

T_{C} | 0.26 | α_{1} | 0.05 |

T_{IV} | 0.5 | α_{2} | 0.06 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Sun, W.; Zhao, J.; Liu, D.
A Speed-Governing System Model with Over-Frequency Protection for Nuclear Power Generating Units. *Energies* **2020**, *13*, 173.
https://doi.org/10.3390/en13010173

**AMA Style**

Wang L, Sun W, Zhao J, Liu D.
A Speed-Governing System Model with Over-Frequency Protection for Nuclear Power Generating Units. *Energies*. 2020; 13(1):173.
https://doi.org/10.3390/en13010173

**Chicago/Turabian Style**

Wang, Li, Wentao Sun, Jie Zhao, and Dichen Liu.
2020. "A Speed-Governing System Model with Over-Frequency Protection for Nuclear Power Generating Units" *Energies* 13, no. 1: 173.
https://doi.org/10.3390/en13010173