# Dynamic Modeling and Simulation of Deep Geothermal Electric Submersible Pumping Systems

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## Abstract

**:**

## 1. Introduction

- Loose cable connections on the motor side (e.g., due to vibrations), leading to an increased electric resistance (possibly differing per phase) and lowering the motor output power.
- Motor insulation faults, resulting in currents among the windings or between the windings and ground.
- Solid parts (scalings) entering the pump, reducing the flow rate and causing fluctuations in the pump pressure and load torque.
- Bearing wear, resulting in higher mechanical friction and overheating of components.
- Shaft fracture, due to abrupt changes in the mechanical load.

## 2. State-Space Model of Deep Geothermal ESP Systems

**Voltage-source inverter (VSI)**(producing variable frequency and amplitude output voltages),**Sine filter**(converting the pulsed VSI output voltages into almost sinusoidal voltages),**Cable**(transmitting the electrical power to the downhole motor),**Motor**(driving the pump by converting electrical into mechanical power),**Protector (Seal)**(serving as axial bearing and oil reservoir placed between motor and pump),**Shaft**(transmitting the mechanical power from the motor to the pump),**Pump**(generating pressure by converting mechanical into hydraulic power), and**Pipe system and geothermal reservoir**(representing the hydraulic load).

#### 2.1. Electrical Subsystem

#### 2.1.1. Inverter

**Assumption**

**1**(Ideal switches).

- no current may flow if the switch is open,
- bidirectional current may flow without voltage drop, if the switch is closed and
- the switching takes place instantaneously (no switching delay).

**Assumption**

**2**(VSI capacitance).

#### 2.1.2. Filter

#### 2.1.3. Cable

**Assumption**

**3**(Cable shunt conductance).

#### 2.1.4. Electrical Machine

**Assumption**

**4**(Motor modeling).

- the motor is star-connected, i.e. the secondary ends of the phase windings are interconnected at the motor star point ${\mathtt{Y}}_{M}$,
- the multi-rotor configuration can be considered a single rotor with combined electromagnetic properties, i.e., no torsional effects among individual rotors are considered, and
- iron losses can be neglected.

**Assumption**

**5**(Magnetic linearity).

#### 2.2. Hydraulic Subsystem

#### 2.2.1. Pump

**Assumption**

**6**(Incompressible flow).

**Assumption**

**7**(Average streamline).

**Assumption**

**8**(Multi-stage characteristics).

**Assumption**

**9**(Leakage flow).

#### 2.2.2. Pipe System and Geothermal Reservoir

**Assumption**

**10.**

#### 2.3. Mechanical Subsystem

#### 2.3.1. Shaft (Spring-Damper-System)

**Assumption**

**11**(Flow dynamics).

#### 2.3.2. Decoupling of the Hydraulic and Mechanical System Dynamics

#### 2.4. Overall System Dynamics

## 3. Simulation Results and Discussion

`ode4`solver with a fixed step time of 100 $\mathrm{n}\mathrm{s}$ for the duration of 100 $\mathrm{s}$. The displayed data was sampled at the end of each PWM cycle, since at this point the voltage over time integral of the inverter output voltage equals the voltage over time integral of the sampled reference voltage.

#### 3.1. Test Scenario

#### 3.2. Results and Discussion

#### 3.2.1. Overall System (See Figure 13)

#### 3.2.2. Power and Efficiency (See Figure 14)

#### 3.2.3. Detailed Views on Electrical and Mechanical Subsystems (See Figure 15)

## 4. Conclusions

- Identification of primary system components of geothermal ESP systems,
- Simplification and abstraction of the physics based on feasible assumptions,
- Consistent and detailed state-space modeling of the system components,
- Provision of a set of realistic system parameters, and
- Simulative validation of the overall system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

$\mathbb{N},\mathbb{R}$ | Natural, real numbers. |

$x\in \mathbb{R}$ | Real scalar. |

$x:=y$ | x “defined as” y. |

$x\stackrel{!}{=}y$ | x “forced to be equal to” y. |

$\mathit{x}:={({x}_{1},\dots ,{x}_{n})}^{\top}\in {\mathbb{R}}^{n}$ | Column vector of magnitude $\widehat{x}:=\sqrt{{x}_{1}^{2}+\dots +{x}_{n}^{2}}$. |

${\mathit{x}}^{\top}$ | Transpose of vector $\mathit{x}$. |

$\mathit{X}\in {\mathbb{R}}^{m\times n}$ | Matrix with m rows and n columns. |

$diag(\mathit{x})\in {\mathbb{R}}^{n\times n}$ | Square matrix with diagonal elements $\mathit{x}$ and off-diagonal elements 0. |

${\mathbf{0}}_{m\times n}$ | Zero matrix. |

${\mathit{I}}_{n}\in {\mathbb{R}}^{n\times n}$ | Identity matrix. |

${\mathbf{0}}_{n}:={(0,\dots ,0)}^{\top}$ | Zero (column) vector. |

${\mathbf{1}}_{n}:={(1,\dots ,1)}^{\top}$ | Unit (column) vector. |

∧, ∨ | Logical “and” and “or”. |

x | Signal (e.g., current i and voltage u). |

z | Location or assigned component (e.g., $\mathrm{c}$ = cable and $\mathrm{f}$ = filter). |

$p\in {\{}^{\prime},\ast \}$ | Signal variants (i.e., per-unit-length, reference). |

$n\in \{1,2\}$ | Input and output. |

y | Assigned reference frame, (i) $\mathit{a}-\mathit{b}-\mathit{c}=(ab,bc,ca)$ for line-to-line signals, (ii) $\mathit{abc}=(a,b,c)$ for phase signals (three-phase) and (iii) $\mathit{\alpha}\mathit{\beta}=(\alpha ,\beta )$ for the two-phase representation. |

## Abbreviations

ESP | Electric submersible pump |

VSI | Voltage source inverter |

PWM | Pulse-width modulation |

SVM | Space-vector modulation |