# Dynamic Equivalent Modeling for Small and Medium Hydropower Generator Group Based on Measurements

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## Abstract

**:**

## 1. Introduction

## 2. Equivalent Model for the Hydropower Generator Group

#### 2.1. Equivalent Generator Model

_{j}is the inertia time constant; P

_{m}is the mechanical power; E

^{′}is the electromotive force after x

^{′}

_{d}and instead of q-axis transient electromotive force; V is the generator voltage; x

^{′}

_{d}is the d-axis transient reactance and the value is equal to q-axis synchronous reactance; δ is the power angle; D is the damping coefficient; t is time; ω is the rotor speed of generator; ω

_{f}is the rotor speed of reference; T

^{′}

_{d0}is the d-axis transient open circuit time constant; E

_{f0}is the initial excitation voltage; K

_{v}is the feedback coefficient of excitation voltage; V

_{0}is the initial generator voltage; x

_{d}is the d-axis synchronous reactance.

#### 2.2. Equivalent Load Model

_{s}is the load active power; P

_{s0}is the initial load active power; N

_{p}is the index of voltage characteristics on active power; Q

_{s}is the load reactive power; Q

_{s0}is the initial load reactive power; N

_{q}is the index of voltage characteristics on reactive power.

_{l}is the active power in tie line; Q

_{l}is the reactive power in tie line; P

_{e}is the generator active power; Q

_{e}is the generator reactive power.

## 3. Methodology of the Proposed Equivalence

#### 3.1. Frame of the Equivalence Method

_{j}, x

_{d}, x

^{’}

_{d}, T

^{’}

_{d0}, K

_{v}, D, P

_{s0}, Q

_{s0}, N

_{p}, N

_{q}]. The overall framework is shown in Figure 2.

_{l}is the measured value of active power; P

^{′}

_{l}is the calculated value of active power; Q

_{l}is the measured value of reactive power; Q

^{′}

_{l}is the calculated value of reactive power.

^{′}from different times have to be calculated. The two most classic numerical methods are the improved Euler method and Runge-Kutta method [24]. The 4th order Runge-Kutta method is used widely when solving ordinary differential equations because of its briefness and accuracy [25,26,27]. In this paper, the 4th order Runge-Kutta method is used to solve generator dynamic equations and the state variables of time k + 1 moment such as ω, δ, E

^{′}can be calculated by time k moment. The initial value of state variables can be determined by the steady-state phasor diagram. The power outputs in the tie line are obtained with the certain equivalent model parameters based on Equations (1)–(4). The frame is shown in Figure 3.

#### 3.2. Dynamic Multi-Swarm Particle Swarm Optimizer Algorithm

_{n}is the velocity vector of the particle on n-th iteration; w is the weight factor; c

_{1,2}is the learning factor; rand is a random number between 0 and 1.

^{′}

_{nj}is the mutated x

_{nj}; l is the random number of 0 or 1; x

_{nj}is the j-dimensional component of the n-th particle; U

_{j}is the upper limit of x

_{ij}; L

_{j}is the lower limit of x

_{ij}; b is the random number of 0–1.

_{max}is the upper limit of inertia weight which is set to 0.9; w

_{min}is the lower limit of inertia weight which is set to 0.4; n is the current iteration number; n

_{max}is the maximum number of iterations.

_{f}, P

_{l}and Q

_{l}in the tie line; (2) Initialize the parameters of each particle, swarm and iteration number; (3) Work out the outputs based on Equations (1)–(4); (4) Calculate the objective function and confirm the initial value of pbest and gbest; (5) Update iterations and velocity and position of the particle; (6) Update iterations; (7) Work out the outputs based on Equations (1)–(4); (8) Calculate the objective function and then update pbest and gbest; (9) Determine whether n

_{max}/R is integer or not. If the answer is Yes, regroup the population randomly, otherwise go on; (10) Determine whether the terminal condition is satisfied or not. If the answer is Yes, stop calculating and output gbest, otherwise determine whether mutate or not. If the answer is Yes, generate velocity and position randomly and then return to (6), otherwise return to (5).

## 4. Case Study

#### 4.1. Sensitivity Analysis of Equivalent Model Parameters

_{j}is the j-th parameter; R

_{sensitivity_j}is the sensitivity of the j-th parameter.

_{j}, k) is the active power with changed X

_{j}at k moment; P(X

_{0}, k) is the active power with basic X

_{0}at k moment; P(X

_{0}, 1) is the active power with basic X

_{0}parameter at the initial time; Q(X

_{j}, k) is the reactive power with changed X

_{j}at k moment; Q(X

_{0}, k) is the reactive power with basic X

_{0}at k moment; Q(X

_{0}, 1) is the reactive power with basic X

_{0}at initial time.

_{s0}, inertia time constant T

_{j}and d-axis transient reactance x

^{′}

_{d}are the three most sensitive parameters. Initial load reactive power Q

_{s0}, index of voltage characteristics on active power N

_{p}, and index of voltage characteristics on reactive power N

_{q}are the three lowest sensitivity parameters.

#### 4.2. Identifiability Verification

_{j}, x

^{′}

_{d}, P

_{s0}are very close. Certain parameters deviate from the actual values because these parameters have less influence on the dynamic characteristic and the simplified excitation model is introduced. Besides, the accuracy of the dynamic process is not affected because this method focuses on global optimization. As shown in Figure 6, the identified and the actual response curves have high fitting degree and the error is very small.

_{j}, x′

_{d}, P

_{s0}are still relatively accurate in Table 2, and the trends of response curves are consistent in Figure 7. Therefore, it can be said that the DMS-PSO algorithm used for parameter identification is completely feasible when the noise of input measurements is not big. It can filter out the noise at first and then identify the parameters based on the proposed method when the noise is big.

#### 4.3. Equivalence with Simulation Data

#### 4.4. Equivalence with PMU Data

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Flow chart of parameter identification based on dynamic multi-swarm particle swarm optimizer (DMS-PSO) algorithm.

**Figure 6.**(

**a**) Active power curves based on identified parameters and actual parameters; (

**b**) reactive power curves based on identified parameters and actual parameters.

**Figure 7.**(

**a**) Active power curves based on identified parameters and actual parameters; (

**b**) reactive power curves based on identified parameters and actual parameters.

**Figure 8.**(

**a**) Active power curves about different model; (

**b**) reactive power curves about different model.

**Figure 10.**(

**a**) Active power curves between before and after equivalence; (

**b**) reactive power curves between before and after equivalence.

**Figure 11.**(

**a**) Active power curves between before and after equivalence; (

**b**) reactive power curves between before and after equivalence.

**Figure 12.**(

**a**) Active power curves between before and after equivalence; (

**b**) reactive power curves between before and after equivalence.

Parameter | Identified Value | Actual Value | Parameter | Identified Value | Actual Value |
---|---|---|---|---|---|

T_{j} | 50.96 | 49.50 | D | 0.165 | 0.2 |

x_{d} | 0.1038 | 0.1460 | P_{s} | 0.987 | 1 |

x′_{d} | 0.0512 | 0.0523 | Q_{s} | 0.298 | 0.5 |

T^{″}_{d0} | 8.682 | 10.430 | N_{p} | 1.143 | 1.5 |

K_{v} | 0.3672 | - | N_{q} | 1.767 | 1.2 |

**Table 2.**Comparison of the identified parameters and the actual parameters of the model for 50 dB signal-to-noise ratio (SNR) inputs (p.u).

Parameter | Identified Value | Actual Value | Parameter | Identified Value | Actual Value |
---|---|---|---|---|---|

T_{j} | 52.34 | 49.50 | D | 0.1297 | 0.2 |

x_{d} | 0.2156 | 0.1460 | P_{s} | 1.1681 | 1 |

x′_{d} | 0.0586 | 0.0523 | Q_{s} | 0.2743 | 0.5 |

T^{″}_{d0} | 17.35 | 10.430 | N_{p} | 2.292 | 1.5 |

K_{v} | 2.2693 | - | N_{q} | 1.987 | 1.2 |

Parameter | T_{j} | x_{d} | x′_{d} | T′_{d0} | K_{v} |

Value | 68.3137 | 0.5811 | 0.0455 | 15.3374 | 13.8422 |

Parameter | D | P_{s0} | Q_{s0} | N_{p} | N_{q} |

Value | 0.3093 | 0.7765 | 0.5897 | 0.5982 | 1.8078 |

Position | Type | Before Equivalence | After Equivalence | Relative Errors |
---|---|---|---|---|

Jiulong500–Shimian500 | Active power | 14.5465 | 14.55 | 0.024% |

Reactive power | −0.34845 | −0.34935 | 0.25% | |

Jiulong500 | Voltage amplitude | 1.00641 | 1.00637 | −0.0039% |

Jianshan500–Pengzhu500 | Active power | 4.41634 | 4.41727 | 0.0047% |

Reactive power | 1.41196 | 1.41087 | 0.079% | |

Jianshan500 | Voltage amplitude | 0.97591 | 0.97589 | −0.0020% |

Huangyan500–Wanxian500 | Active power | 15.90473 | 15.90617 | 0.0091% |

Reactive power | −1.177 | −1.17682 | 0.015% | |

Huangyan500 | Voltage amplitude | 0.98683 | 0.98682 | −0.001% |

Meiguhe220–Puti220 | Active power | 0.53659 | 0.53659 | 0% |

Reactive power | 0.49844 | 0.49845 | 0.0020% | |

Puti220 | Voltage amplitude | 0.9913 | 0.9913 | 0% |

Caoba220–Mingshan220 | Active power | 0.29271 | 0.29271 | 0% |

Reactive power | −0.16314 | −0.16313 | 0.0061% | |

Caoba220 | Voltage amplitude | 0.98993 | 0.98991 | −0.0020% |

grid | Frequency | 1 | 1 | 0% |

Number | Fault Descriptions | Relative Errors |
---|---|---|

1 | Three phase short circuit occurred in 1 s in a 500 kV Yuecheng-Puti transmission line and broke this line in 1.1 s | 40.49 |

2 | Three phase short circuit occurred in 1 s in a 500 kV Nantian-Dongpo transmission line and broke this line in 1.1 s | 39.65 |

3 | Three phase short circuit occurred in 1 s in a 500 kV Shuzhou-Danjing transmission line and broke this line in 1.1 s | 34.17 |

4 | Three phase short circuit occurred in 1 s in a 500 kV Ya’an-Jianshan transmission line and broke this line in 1.1 s | 37.48 |

5 | Three phase short circuit occurred in 1 s in a 500 kV Tanjiawan-Nanchong transmission line and broke this line in 1.1 s | 28.56 |

Parameter | T_{j} | x_{d} | x′_{d} | T′_{d0} | K_{v} |

Value | 6.0684 | 0.5548 | 0.1475 | 1.6488 | 4.2420 |

Parameter | D | P_{s0} | Q_{s0} | N_{p} | N_{q} |

Value | 0.1982 | 1.1065 | 3.1636 | 0.8321 | 0.6005 |

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## Share and Cite

**MDPI and ACS Style**

Hu, B.; Sun, J.; Ding, L.; Liu, X.; Wang, X.
Dynamic Equivalent Modeling for Small and Medium Hydropower Generator Group Based on Measurements. *Energies* **2016**, *9*, 362.
https://doi.org/10.3390/en9050362

**AMA Style**

Hu B, Sun J, Ding L, Liu X, Wang X.
Dynamic Equivalent Modeling for Small and Medium Hydropower Generator Group Based on Measurements. *Energies*. 2016; 9(5):362.
https://doi.org/10.3390/en9050362

**Chicago/Turabian Style**

Hu, Bowei, Jingtao Sun, Lijie Ding, Xinyu Liu, and Xiaoru Wang.
2016. "Dynamic Equivalent Modeling for Small and Medium Hydropower Generator Group Based on Measurements" *Energies* 9, no. 5: 362.
https://doi.org/10.3390/en9050362