# Hamiltonian Additional Damping Control for Suppressing Power Oscillation Induced by Draft Tube Pressure Fluctuation

^{1}

^{2}

^{*}

## Abstract

**:**

_{25}= 1.3, the minimum power oscillation amplitude is 0.5466, which is equivalent to an increase in D by 20. The maximum oscillation amplitude decreases by 4.6%, and the operation limited zone is reduced by 10.1%. The proposed additional damping control can effectively suppress the power oscillation and expand the regulation range.

## 1. Introduction

## 2. Problem Description

_{t}defined by the IEEE Working Group [28] is expressed as

_{t}only applies to the unconstrained draft tube with zero inlet pressure, and h

_{t}only refers to the pressure at the hydro turbine admission. However, the strict definition of the water head is the pressure difference between the upstream of the hydro turbine and the downstream.

_{t0}is h

_{t}in Equation (1).

_{t0}is p

_{t}in Equation (1) and Δp = A

_{t}h

_{p}(q − q

_{nl})sin(ω

_{p}t).

_{t0}in Equation (3), we have:

_{p}= A

_{t}h

_{p}(q − q

_{nl}).

## 3. Additional Damping Control for the Hamiltonian System

**x**∈

**R**

^{n}and control input

**u**∈

**R**

^{m}, m < n, H(

**x**) is the energy function,

**g**(

**x**) is the input matrix, the structure matrix

**J**(

**x**) is antisymmetric, and the damping matrix

**R**(

**x**) is a semi-positive definite symmetric matrix.

- (i)
- Assume that
**x**_{*}is the equilibrium, the constant control**v**_{*}is the solution of$$\mathbf{0}=[J({x}_{\ast})-R({x}_{\ast})]\frac{\partial H}{\partial x}({x}_{\ast})+g({x}_{\ast}){v}_{\ast}$$

**v**

_{*}is the input of stabilization control in system (7).

- (ii)
- Given the constant structural modification J
_{a}and R_{a}, the desired Hamiltonian structure matrix is**J**_{d}(**x**) =**J**(**x**) +**J**_{a},**R**_{d}(**x**) =**R**(**x**) +**R**_{a}, which satisfies**J**_{d}(**x**) = −**J**_{d}(**x**)^{T},**R**_{d}(**x**) =**R**_{d}(**x**)^{T}. So, the system (7) can be rewritten as$$\begin{array}{cc}\hfill \dot{x}& =\left[\left({J}_{d}(x)-{J}_{a}\right)-\left({R}_{d}(x)-{R}_{a}\right)\right]\frac{\partial H}{\partial x}(x)+g(x)u(x)\hfill \\ & =[{J}_{d}(x)-{R}_{d}(x)]\frac{\partial H}{\partial x}(x)-[{J}_{a}-{R}_{a}]\frac{\partial H}{\partial x}(x)+g(x)u(x)\hfill \\ & =[{J}_{d}(x)-{R}_{d}(x)]\frac{\partial H}{\partial x}(x)+g(x)\beta (x)\hfill \end{array}$$$$g(x)\beta (x)=-[{J}_{a}-{R}_{a}]\frac{\partial H}{\partial x}(x)+g(x)u(x)$$

**β**(

**x**) is the equivalent control after structure modification.

**x**

_{*}, the modified Hamiltonian system (9) also satisfies:

**β**(

**x**) in Equation (10) replacing

**β**(

**x**

_{*}) in a neighbourhood around the equilibrium point is valid. Assume that

**g**(

**x**) is a constant matrix, that is,

**g**(

**x**) =

**g**(

**x**

_{*}) =

**g**. Combining with Equations (10) and (11), there are:

- (iii)
- Let
**α**(**x**) =**u**(**x**)-**v**_{*}. If**g**(**x**) is full of rank, (**g**^{T}**g**) is reversible. In addition,$$\alpha (x)={({g}^{T}g)}^{-1}{g}^{T}({J}_{a}-{R}_{a})[\frac{\partial H}{\partial x}(x)-\frac{\partial H}{\partial x}({x}_{\ast})]$$

**α**(

**x**) is the additional damping control of the Hamiltonian system (7) with (

**J**

_{a},

**R**

_{a}), and

**α**(

**x**

_{*}) = 0. Notice that

**α**(

**x**) only plays a role in the transient process.

**x**

_{*,}which is its limitation. The control design method is selected by the load state of the system in the actual operation. The additional damping control is appropriate for this research, which aims to suppress the power oscillation induced by pressure fluctuation.

## 4. Hamiltonian Model of Hydropower Units

**x**= [x

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}]

^{T}= [q, y, δ, ω, E′

_{q}]

^{T},

**F**(

**x**) is the external excitation, and the Hamilton function is:

_{p}(

**x**) is the control realized by a feedback dissipative, X

_{d∑}= X

_{d}+ X

_{T}+ X

_{L}, X′

_{d∑}= X′

_{d}+ X

_{T}+ X

_{L}and X

_{q∑}= X

_{q}+ X

_{T}+ X

_{L}.

_{1}(

**x**) in C

_{T}(

**x**) of structure matrix

**J**(

**x**) in Equation (14). It is only a simple hydraulic condition with a penstock and a non-elastic water column. If the elastic water column is adopted, the model becomes a five-order model, and the derived Hamiltonian model has a more complex expression, while HU is the eighth-order model. It is bound to make additional damping control (13) more difficult.

## 5. Control Design

#### 5.1. Damping Injection

**R**(

**x**) reflects the port dissipation characteristics. We want to add the corresponding Hamiltonian damping factor

**R**

_{a}to

**R**(

**x**) to increase the system damping. In HU, the active power belongs to the electrical systems, and the pressure fluctuation of DL belongs to the hydraulic systems. So,

**R**

_{a}is the correlation item between hydraulic and electrical systems. The desired structure matrix is selected as follows:

**u**(

**x**) =

**α**(

**x**) +

**v**

_{*}to obtain:

**u**(

**x**) is a state feedback control, which builds the coupling correlation between the different systems.

_{25}in (16) should be calculated by the positive definiteness of the Hessian matrix. Only in this way is the PCH system with the desired structure matrix (

**J**

_{d},

**R**

_{d}) asymptotically stable. However, for a high-order system, it is too complicated to confirm the positive definite condition of the Hessian matrix, and the obtained range of r

_{25}at the given equilibrium is not the expected optimal value. Therefore, this paper adopts the numerical example simulation to determine its value range.

#### 5.2. Analysis of Control Law

_{p}(

**x**) in the PCH system (14) can be rewritten as:

_{t}and flow q are both affected by Δh. From the above discussion, it seems feasible to calculate Δh in the elastic water hammer model and add it to the additional damping control of the PCH system. This problem needs more research by simulation.

#### 5.3. Control Structure

_{f}. We can regard the additional damping control (17) as a part of coordinated control consisting of the traditional speed governor and excitation controller. The structure of the Hamiltonian additional damping control is shown in Figure 1.

_{p}= 5.0, K

_{I}= 1.7, K

_{D}= 1.3, and the excitation controller adopts thyristor excitation PID: K

_{P1}= 1.0, K

_{I1}= 1.5, K

_{D1}= 0.0001. The main parameters of the system are A

_{t}= 1.127, T

_{w}= 2.242, T

_{y}= 0.5, T

_{j}= 8.999, T

_{d0′}= 5.4, X

_{d}= 1.07, X

_{d}′ = 0.34, X

_{q}= 0.66, X

_{f}= 1.29, X

_{ad}= 0.97, D = 5.

## 6. Simulation Research

#### 6.1. Pressure Fluctuation Analysis

_{p}as the pressure fluctuation frequency and f

_{p}= ω

_{p}/2π. In the following simulation, we try to find the possible resonance phenomenon with f

_{p}= 0.9 (Hz), h

_{p}= 0.05 (p.u) and f

_{p}= 0.7 (Hz), h

_{p}= 0.05 (p.u). The pressure fluctuation occurs at t = 1.0 (s). The mechanical power p

_{t}of the hydro turbine and the active power p

_{e}of the generator is shown in Figure 2.

_{t}and p

_{e}have the same oscillation frequency after entering the steady state. When f

_{p}= 0.9 (Hz) is close to the natural frequency (1.0 (Hz)), there is a larger difference in the oscillation amplitude between p

_{e}and p

_{t}than that of f

_{p}= 0.6 (Hz). Here is a resonance zone of the pressure fluctuation and power oscillation. The article [18] also found that the frequency of pressure fluctuation has a great influence on the oscillation amplitude of active power, and there is an amplification point of the maximum amplitude. This result is consistent with the classical vibration mechanics theory.

#### 6.2. Damping Injection Simulation

_{p}= 0.05 (p.u), f

_{p}= 0.9 (Hz), and the pressure fluctuation occurs at t = 1.0 (s). The responses of p

_{e}under the control laws (17) with different r

_{25}s are shown in Figure 3.

_{25}= 0 is equivalent to not adding the additional damping in control (17), and r

_{25}= 1 is added to the damping. Compared to the result of no control, the Hamiltonian additional damping control can obviously narrow the amplitude of power oscillation.

_{25}needs to be determined by simulation. r

_{25}has a reasonable range to guarantee the convergence of the system to the desired neighbourhood. The oscillation amplitude changes with r

_{25}are as follows in Figure 4.

_{25}= 0, the oscillation amplitude of p

_{e}is the reference value marked by a red dotted line.

**α**(

**x**) with r

_{25}< 0 supplies negative damping. In addition,

**α**(

**x**) with r

_{25}> 0 shows a positive damping characteristic. This variation of damping characteristics is consistent with the expectation of the damping injection method. When r

_{25}exceeds the range limited to the X-axis in Figure 4, the response curve of p

_{e}will gradually diverge and lose its stability, and when r

_{25}= 1.3, the oscillation amplitude is the smallest, which can function as the optimal value.

_{25}s, the oscillation amplitude change with f

_{p}is shown in Figure 5.

_{25}= 0 is also used as the reference value marked by the blue line. The intersection point of any two lines tells that the damping characteristic changes dynamically with f

_{p}. When f

_{p}is below a specific value located at the intersection, the positive damping induced by r

_{25}= 1 adjusts to the negative damping. This situation also applies to r

_{25}= −1.

_{25}. The additional damping control based on the desired structure matrix (

**J**

_{d},

**R**

_{d}) can modify the self-vibration characteristics and natural frequency. This phenomenon reveals a new way to improve the oscillation characteristic of the system, which needs further study.

#### 6.3. Damping Injection Quantization

_{25}. We try to obtain the value of r

_{25}indirectly by calculating D.

_{25}= 0, D = 25 and r

_{25}= 1.3, D = 5 for example, we select the maximum amplitude of power oscillation in steady state as a reference. The responses under two conditions are shown in Figure 6.

_{25}= 1.3 is equivalent to an increase in D by 20.

#### 6.4. Expand the Stable Operation Zone

_{25}s cannot completely restrain power oscillation induced by pressure fluctuation. The expansion of the oscillation amplitude is the main cause of the loss of stability. We want to use the Hamiltonian additional damping control to expand the load regulation range, which is analysed as follows with an example simulation.

#### 6.5. Simulation of Vibration Zone Crossing

_{25}= 0, only the PID control of speed regulation and excitation.

_{25}= −0.5. In the transient process of load increase, the target value in control law is obtained by the difference between the initial and final active values of p

_{e}.

_{25}= −0.5. The target value only corresponds to load p

_{e}= 1, ignoring its variation in the load increase.

_{e}of different control strategies receives different degrees of influence from pressure pulsation when the unit runs in the part-load vibration zone during the start-up (0~30 s). After 30 s, the unit leaves the vibration zone and gradually enters a steady state, and the influence of pressure pulsation disappears. The power oscillation may occur when it crosses the vibration zone too slowly or when the pressure fluctuation amplitude is high. Compared to the control effect of Case 1, the control law in Case 2 can greatly reduce the amplitude of power oscillation, and Case 3 can let the unit pass through the vibration zone rapidly.

## 7. Conclusions

- The simulation results show that adding the Hamiltonian damping factor is mathematically equivalent to increasing the oscillation damping, and it is effective to use additional damping control based on the damping injection method.
- The resonance point of pressure fluctuation and power oscillation shifts with the values of the Hamiltonian damping factor. The damping characteristic of the same factor is a variation of positive-negative near the resonance point. In application, the values of these damping factors should be selected by the load condition.
- PID + Hamiltonian additional damping control can expand the stable operation zone. During the start-up, HU applying the Hamiltonian additional damping control can faster pass through the vibration zone and have a smaller power oscillation than the PID control.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

DF | Draft tube |

HU | Hydropower unit |

PCH | Port-controlled Hamiltonian |

A_{t} | Constant proportionality factor |

h_{t} | Water head of the hydro turbine (p.u) |

q | Flow of the hydro turbine (p.u) |

q_{nl} | No-load flow of the hydro turbine (p.u) |

h_{p} | Amplitude of Pressure fluctuation in DL (p.u) |

Δp | Mechanical power oscillation (p.u) |

T_{j} | Inertia time constant (s) |

p_{e} | Active power of the generator (p.u) |

D | The equivalent damping coefficient |

Fp | Amplitude of power oscillation in the hydro turbine (p.u) |

u | Speed governor output (p.u) |

E_{f} | Excitation controller output (p.u) |

y | Guide vane opening (p.u) |

y_{0} | Initial guide vane opening (p.u) |

T_{w} | Water inertia time (s) |

f_{p} | Water head loss coefficient |

T_{y} | Time constant of the main servomotor (s) |

E_{q}_{′} | Internal transient voltage (p.u) |

U_{s} | Infinite-bus voltage (p.u) |

X_{d} | The d-axis synchronous reactance |

X_{d}′ | Transient reactance of the generator (p.u) |

X_{T} | Reactance of the transformer (p.u) |

X_{L} | Reactance of the transmission line (p.u) |

X_{q} | The q-axis synchronous reactance (p.u) |

X_{f} | Reactance of the excitation winding (p.u) |

X_{ad} | The d-axis armature reaction reactance (p.u) |

T′_{d}_{0} | Time constant (s) |

ω_{p} | Angular frequency of pressure fluctuation (rad/s) |

ω | Angular velocity of the generator (p.u) |

ω_{B} | Basic angular velocity, ωB = 314 rad/s |

δ | Power angle (rad) |

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**MDPI and ACS Style**

Zeng, Y.; Yu, S.; Dao, F.; Li, X.; Xu, Y.; Qian, J.
Hamiltonian Additional Damping Control for Suppressing Power Oscillation Induced by Draft Tube Pressure Fluctuation. *Water* **2023**, *15*, 1479.
https://doi.org/10.3390/w15081479

**AMA Style**

Zeng Y, Yu S, Dao F, Li X, Xu Y, Qian J.
Hamiltonian Additional Damping Control for Suppressing Power Oscillation Induced by Draft Tube Pressure Fluctuation. *Water*. 2023; 15(8):1479.
https://doi.org/10.3390/w15081479

**Chicago/Turabian Style**

Zeng, Yun, Shige Yu, Fang Dao, Xiang Li, Yiting Xu, and Jing Qian.
2023. "Hamiltonian Additional Damping Control for Suppressing Power Oscillation Induced by Draft Tube Pressure Fluctuation" *Water* 15, no. 8: 1479.
https://doi.org/10.3390/w15081479