With the increasing integration of solar power into the electrical grid, the stability of power systems has become a critical concern due to the intermittent nature of solar energy. Solar inverters play a pivotal role in converting direct current (DC) from photovoltaic (PV) panels into alternating current (AC) for grid integration. Understanding the dynamic behavior of solar power systems under various conditions is essential for assessing their impact on distribution networks. In this study, we focus on developing a comprehensive model for solar inverters and simulating their dynamic characteristics using electromagnetic transient analysis software. We explore how factors like three-phase short-circuit faults, changes in irradiance, and temperature variations affect system performance, including output power, grid active power, and DC-side voltage. By analyzing these aspects, we aim to provide insights into the disturbances caused by solar power sources in distribution networks.
The modeling of solar inverters is foundational to this research. Solar inverters are complex devices that facilitate energy conversion in PV systems. A typical solar inverter consists of a DC source, an output circuit, and a three-phase inversion unit. To accurately represent the behavior of solar inverters, we begin by establishing a mathematical model based on theoretical principles. This model serves as the basis for designing control strategies that ensure stable operation under dynamic conditions.
In the following sections, we delve into the mathematical formulation of solar inverters, discuss control strategies such as voltage outer-loop and current inner-loop controls, and explain the application of Space Vector Pulse Width Modulation (SVPWM) technology. Subsequently, we present a detailed simulation setup and analyze the results under various disturbance scenarios. The findings highlight the resilience and vulnerabilities of solar power systems, emphasizing the importance of robust inverter design for grid integration.
Mathematical Modeling of Solar Inverters
The mathematical model of a solar inverter is derived from the fundamental equations governing its operation in a three-phase stationary coordinate system. In this system, the voltage vector of the solar inverter can be expressed using the following equation:
$$ U_{abc} = E_{abc} + I_{abc} R + L \frac{dI_{abc}}{dt} $$
where \( U_{abc} \) represents the three-phase voltage on the AC output side of the solar inverter, with its vector form given as \( (u_a, u_b, u_c)^T \). Here, \( u_a \), \( u_b \), and \( u_c \) denote the phase voltages corresponding to phases a, b, and c in the three-phase stationary coordinate system, and \( T \) indicates the transpose operation. \( E_{abc} \) is the three-phase grid voltage, represented as \( (e_a, e_b, e_c)^T \), where \( e_a \), \( e_b \), and \( e_c \) are the grid voltages for phases a, b, and c, respectively. \( I_{abc} \) is the three-phase current on the output side of the solar inverter, expressed as \( (i_a, i_b, i_c)^T \), with \( i_a \), \( i_b \), and \( i_c \) being the currents for phases a, b, and c. \( L \) is the inductance of the solar inverter, and \( \frac{dI_{abc}}{dt} \) is the time derivative of the current.
To facilitate control, the AC quantities such as voltage and current are transformed into DC quantities using mathematical transformations. This involves converting the abc three-phase stationary coordinate system to the dq synchronous rotating coordinate system. In this transformed system, the active power \( p_1 \) and reactive power \( p_2 \) are calculated as follows:
$$ p_1 = \frac{3}{2} (e_d i_d + e_q i_q) $$
$$ p_2 = \frac{3}{2} (e_d i_q – e_q i_d) $$
where \( e_d \) and \( e_q \) are the d-axis and q-axis components of the grid voltage in the dq synchronous rotating coordinate system, and \( i_d \) and \( i_q \) are the d-axis and q-axis components of the current. This transformation simplifies the control process by allowing the use of linear controllers, such as PI regulators, to manage the DC quantities indirectly, thereby controlling the AC outputs effectively.
The dynamics of solar inverters are influenced by various parameters, including resistance and inductance, which are critical for stability analysis. The mathematical model provides a foundation for simulating the behavior of solar inverters under different operating conditions, enabling us to predict their response to disturbances in the grid.
Control Strategies for Solar Inverters
Solar inverters employ sophisticated control strategies to maintain stability and efficiency. In this study, we adopt a double-loop control strategy, which consists of an outer voltage loop and an inner current loop. This approach is based on grid voltage vector orientation and ensures precise regulation of power output.
Voltage Outer-Loop Control Strategy
The voltage outer-loop control is designed to track the DC-side voltage and manage the output power of the solar inverter. It aims to eliminate reactive power, ensuring that it remains zero. The control involves two axes: d-axis and q-axis. For the d-axis outer-loop control, the input signal is the DC voltage reference, denoted as \( u_{dc\_ref} \), which is derived from the maximum power point tracking (MPPT) device. This signal is processed through a PI regulator, producing an output current reference \( i_{d\_ref} \). Similarly, for the q-axis outer-loop control, the input is the reactive power reference \( q_{ref} \) injected into the grid by the solar inverter. After PI regulation, it outputs a current reference \( i_{q\_ref} \). The PI regulators are tuned to achieve optimal performance, with proportional and integral coefficients selected based on system requirements.
The voltage outer-loop control ensures that the DC-side voltage remains stable, even under varying environmental conditions. This is crucial for the overall efficiency of the solar power system, as fluctuations in DC voltage can lead to power losses and reduced system reliability.
Current Inner-Loop Control Strategy
The current inner-loop control operates in the dq rotating coordinate system and is responsible for regulating the output current of the solar inverter. The voltages for the d-axis and q-axis are computed using the following equations:
$$ u_q = e_q + L \omega i_d + \left( K_P + \frac{K_I}{S} \right) (i_{q\_ref} – i_q) $$
$$ u_d = e_d – L \omega i_q + \left( K_P + \frac{K_I}{S} \right) (i_{d\_ref} – i_d) $$
where \( u_q \) and \( u_d \) are the q-axis and d-axis voltage components in the dq rotating coordinate system, \( \omega \) is the synchronous rotational angular velocity of the coordinate axis, \( K_P \) is the proportional coefficient of the PI regulator, \( K_I \) is the integral coefficient, \( i_{q\_ref} \) and \( i_{d\_ref} \) are the q-axis and d-axis current components output by the PI regulator, and \( S \) is the complex frequency domain variable in the transfer function of the PI regulator. The current inner-loop control ensures that the output current closely follows the grid voltage, thereby maintaining stable power output and enhancing the dynamic response of the solar inverter.
The combination of voltage outer-loop and current inner-loop controls provides a robust framework for managing the operation of solar inverters. This dual-loop strategy is widely used in practical applications due to its effectiveness in handling disturbances and maintaining grid compatibility.
Application of SVPWM Technology in Solar Inverter Control
Space Vector Pulse Width Modulation (SVPWM) is a advanced technique used in solar inverters to improve the utilization of DC voltage and achieve a circular flux trajectory. It focuses on the overall output effect of the three-phase voltage, enhancing the performance of solar inverters.
Switching States of Three-Phase Solar Inverters
A three-phase solar inverter typically consists of six switching transistors, with each bridge arm having an upper and lower switch that operate in complementary states. The switching function for each phase is denoted as \( S_x \), where \( x \) can be a, b, or c, representing the three bridge arms. When \( S_x = 1 \), the upper switch of the corresponding phase is conducting, and when \( S_x = 0 \), the lower switch is conducting. The conduction states of the switching transistors can be represented as \( (S_a, S_b, S_c) \), and since each \( S_x \) has two states, there are eight possible switching states, labeled as \( U_0 \) to \( U_7 \). The vector representations of these states are summarized in Table 1.
| State Classification | U0 | U1 | U2 | U3 | U4 | U5 | U6 | U7 |
|---|---|---|---|---|---|---|---|---|
| Vector Representation | (0,0,0) | (0,0,1) | (0,1,0) | (0,1,1) | (1,0,0) | (1,0,1) | (1,1,0) | (1,1,1) |
| Description | All Off | One On | One On | Two On | One On | Two On | Two On | All On |
Among these states, \( U_1 \) to \( U_6 \) are non-zero vectors, while \( U_0 \) and \( U_7 \) are zero vectors. SVPWM utilizes these non-zero vectors to control the output voltage of the solar inverter, aiming to approximate a circular trajectory in the space vector diagram.
Principles of SVPWM Control
According to the circuit principles of solar inverters, when the switching state \( S_a = 1 \), the phase voltage for phase a equals the source voltage \( U_{dc} \), and when \( S_a = 0 \), the phase voltage is zero. The same logic applies to phases b and c. The relationship between switching states and phase/line voltages is detailed in Table 2.
| Switching State (S_a, S_b, S_c) | Phase Voltage a | Phase Voltage b | Phase Voltage c | Line Voltage a | Line Voltage b | Line Voltage c |
|---|---|---|---|---|---|---|
| U1 (0,0,1) | 2/3 U_dc | -1/3 U_dc | -1/3 U_dc | U_dc | 0 | -U_dc |
| U2 (0,1,0) | -1/3 U_dc | 2/3 U_dc | -1/3 U_dc | -U_dc | U_dc | 0 |
| U3 (0,1,1) | 1/3 U_dc | 1/3 U_dc | -2/3 U_dc | 0 | U_dc | -U_dc |
| U4 (1,0,0) | -1/3 U_dc | -1/3 U_dc | 2/3 U_dc | 0 | -U_dc | U_dc |
| U5 (1,0,1) | 1/3 U_dc | -2/3 U_dc | 1/3 U_dc | U_dc | -U_dc | 0 |
| U6 (1,1,0) | -2/3 U_dc | 1/3 U_dc | 1/3 U_dc | -U_dc | 0 | U_dc |
In this table, \( U_{dc} \) represents the DC source voltage of the solar inverter. SVPWM adjusts the duration of these non-zero vectors to synthesize the desired output voltage, minimizing harmonics and improving the efficiency of solar inverters. This technique is particularly beneficial for grid-connected solar power systems, as it enhances the quality of the injected power and reduces stress on the grid components.
Simulation of Dynamic Characteristics in Solar Power Systems
To evaluate the impact of solar power on distribution networks, we conducted dynamic simulations using electromagnetic transient analysis software. The simulation model represents a 35 kV distribution network integrated with solar power sources. The network includes series circuits 1-2, 2-3, and 3-4, each connected to a 60 kW AC load. Two distributed solar power sources are incorporated: the first is connected between lines 1-2 and 2-3, and the second between lines 2-3 and 3-4. The parameters of the simulation model are listed in Table 3.
| Component | Parameter | Value |
|---|---|---|
| AC Feeder | Line Voltage (kV) | 35 |
| Transformer | Step-Up Ratio | 0.27/38.5 |
| Line 1-2 | Impedance (Ω/mH) | 0.35/0.5 |
| Line 2-3 | Impedance (Ω/mH) | 0.245/0.35 |
| Line 3-4 | Impedance (Ω/mH) | 0.245/0.35 |
The simulation involves analyzing the system’s response to three types of disturbances: a three-phase short-circuit fault, a decrease in irradiance, and a drop in temperature. Each scenario is simulated over an 8-second period, with disturbances introduced at the 2-second mark. The results provide valuable insights into the dynamic behavior of solar power systems and their interaction with the grid.
Analysis of Three-Phase Short-Circuit Fault
In this scenario, a three-phase short-circuit fault is simulated on line 3-4, lasting for 0.1 seconds starting at t=2 s. The simulation results are analyzed in terms of solar output power, generator output voltage, and grid-side power.
Solar Output Power Analysis: The active power output of the solar system exhibits a slight decrease during the fault period (2.0–2.5 s), dropping from approximately 70 kW to a minimum of 53 kW, before stabilizing back to 70 kW. The reactive power remains near zero throughout, indicating effective control by the solar inverters.
Generator Output Voltage Analysis: The generator output voltage, initially stable at 37 kV, experiences a sharp decline to about 19 kV at the onset of the fault. It then rapidly recovers to 35 kV within 0.2 seconds, followed by oscillations until stabilizing around 6 seconds. This demonstrates the transient nature of voltage disturbances caused by faults in solar-integrated grids.
Grid-Side Power Analysis: The grid-side active power, steady at 200 kW before the fault, plunges to -400 kW momentarily due to reverse power flow, then surges to 480 kW before oscillating and stabilizing at 200 kW by 7 seconds. The reactive power spikes to a maximum of 260 kVar during the fault and gradually returns to zero, highlighting the dynamic power exchanges in the system.
Impact of Irradiance Reduction
To assess the effect of changing environmental conditions, the irradiance is reduced from 1000 W/m² to 600 W/m² at t=2 s. The simulation focuses on solar output power, grid-side power, and DC-side voltage of the solar inverters.
Solar Output Power Analysis: The output power of the solar system decreases from 70 kW to 53 kW after the irradiance drop, reflecting the direct correlation between solar irradiance and power generation. This underscores the sensitivity of solar inverters to variations in sunlight intensity.
Grid-Side Power Analysis: The grid-side active power shows a minor disturbance, decreasing slightly from 200 kW to 198 kW before stabilizing. The reactive power also experiences a small reduction, settling around 100 kVar. These changes indicate that irradiance variations have a limited but noticeable impact on grid power flow.
DC-Side Voltage Analysis: The DC-side voltage of the solar inverters, initially constant at 600 V, dips momentarily upon the irradiance change but quickly recovers to 600 V. This transient response emphasizes the role of control strategies in maintaining voltage stability in solar power systems.
Impact of Temperature Reduction
In this case, the temperature is lowered from 25°C to 15°C at t=2 s to evaluate its effect on system performance. Temperature changes influence the efficiency of PV panels, potentially altering power output.
Solar Output Power Analysis: The solar output power decreases marginally from 70 kW to 69 kW, with minor fluctuations between 2.0 and 2.2 seconds before stabilization. This slight reduction suggests that temperature variations have a less pronounced effect compared to irradiance changes, due to the relatively low temperature coefficient of typical solar cells.
Grid-Side Power Analysis: The grid-side active power remains virtually unchanged at 200 kW throughout the simulation, indicating that temperature-induced disturbances in solar power systems have negligible impact on the grid. This reinforces the importance of focusing on irradiance and fault conditions for grid stability assessments.
Overall System Behavior: The simulation results confirm that solar power systems can introduce disturbances to distribution networks, particularly under fault conditions and irradiance variations. However, with advanced control strategies like those implemented in modern solar inverters, these systems can quickly recover and maintain stable operation.
Conclusion
In this study, we have developed a comprehensive model for solar inverters and conducted detailed simulations to analyze the dynamic characteristics of solar power systems. The mathematical model, based on the three-phase stationary coordinate system, provides a solid foundation for understanding inverter behavior. The double-loop control strategy, comprising voltage outer-loop and current inner-loop controls, effectively regulates DC-side voltage and output current, ensuring stable operation of solar inverters. Additionally, the application of SVPWM technology enhances voltage utilization and improves the overall performance of solar power systems.
The simulation results demonstrate that solar power systems are susceptible to disturbances from three-phase short-circuit faults and changes in irradiance, leading to fluctuations in output power and voltage. However, temperature variations have a minimal impact. The ability of solar inverters to quickly stabilize after disturbances highlights their robustness, but also underscores the need for continuous improvement in control algorithms to mitigate grid integration challenges.
Future work could focus on optimizing the parameters of solar inverters for specific grid conditions and exploring hybrid control strategies to enhance resilience. Overall, this research contributes to a deeper understanding of solar power dynamics and supports the development of more reliable and efficient renewable energy systems.