Consistency control of grid-connected substation voltage regulation based on photovoltaic inverter power coordination

Abstract

Owing to the different access locations and capacities of distributed photovoltaics, there is inconsistency in the voltage regulation between the grid-connected areas, posing the difficulties for the power grid operation and control. To address this, a consistency control method for the voltage regulation in the grid-connected substations is proposed, based on the photovoltaic-inverter power coordination. By analyzing the impact of exceeding voltage limits after the photovoltaic grid connection, this method ensures effective voltage regulation in the grid-connected substation area. The approach coordinates the photovoltaic-inverter power and uses the linear calculations to determine the active and reactive power adjustments. A consistency algorithm constructs a voltage-regulation control model, achieving the uniform voltage regulation in the grid-connected substations. The experimental results show that this method can effectively reduce the voltage fluctuations caused by the distributed photovoltaic access, control the maximum instantaneous overvoltage of transformers within the set threshold range, improve the accuracy of voltage regulation consistency control in the grid-connected substations, and ensure the stable operation of the power grid.

Introduction

The grid-connected substation areas specifically refer to the transformer-powered areas connected to the main power grid, which can receive and distribute the electricity from the main grid and may also integrate the distributed energy sources^1,2,3. Due to the global emphasis on environmental protection and sustainable development, the distributed photovoltaic power generation has been widely adopted in the urban and rural areas owing to its flexibility, efficiency, and environmental benefits. However, the integrating large-scale distributed photovoltaics into the grid introduces several challenges, particularly the voltage regulation^4,5,6. The photovoltaic power generation exhibits the fluctuations and intermittency, complicating the grid load forecasting and voltage regulation. To ensure the safe and stable grid operation, the voltage in the grid-connected substation area must be accurately regulated.

The scholars in the related fields have conducted research on the consistency control of voltage regulation in the grid-connected substations. Liao D et al.⁷ discussed in detail the relationship between DDA and other diffusion algorithms, and illustrated the superiority of DDA by comparing with diffusion and consensus algorithms. The z-domain model of the whole DC mi-crogrid is established by considering the discrete nature and different sampling times of the digital controller and communication network. The effects of communication and secondary control parameters on the stability of the system are investigated and the tolerable communication rates are obtained based on the established model. The real-time simulations performed on the OPAL-RT platform verify the effectiveness of the proposed scheme and demonstrate its advantages in terms of convergence speed and stability. However, the controller and network are linear discrete systems, and controller saturation, network delay and nonlinear characteristics are not fully considered in practice, which may lead to model mismatch and affect the accuracy of stability analysis. Beltrán CA et al.⁸ designed a passivity-based control scheme for the output voltage regulation of a fuel cell/boost converter system. The proposed control scheme is designed as a current-mode control scheme with an outer loop for the voltage regulation and an inner loop for the current reference tracking. The inner loop is designed by considering the Euler-Lagrange formulation to realize a standard PBC, while the outer loop is realized by a standard PI controller. An adaptive law based on the immersion and invariance theory is also designed to enhance the performance of the closed-loop system by the asymptotic approximation of uncertain parameters such as load and inductor parasitic resistance. However, the outer-loop PI control based on the Euler-Lagrange model relies on the accurate load and inductor parasitic resistance parameters, and parameter drift in practice can lead to control failure. Saleem O et al.⁹ proposed a novel fuzzy incremental model-referenced adaptive voltage regulation strategy for the DC-DC buck converters to enhance its adaptability to random input variations and load transients. A universal proportional-integral-derivative (PID) controller is employed, whose gain is adjusted offline by a pre-calibrated linear-quadratic optimization scheme. A model-referenced adaptive controller is employed, which uses the Lyapunov gain adaptive law to modify the PID gain online. The adaptive controller is also augmented with an auxiliary fuzzy self-regulation system that acts as a high-level regulator that dynamically updates the adaptation rate of the Lyapunov gain adaptation law as a nonlinear function of the system’s classical error and its normalized acceleration. The proposed fuzzy system utilizes the knowledge of the relative rate of the system for the better self-regulation of the adaptation rate, in turn flexibly steering the adaptability and responsiveness of the controller when the error conditions change. But the rule base of the fuzzy system needs to be designed manually, which may lead to incomplete coverage of the rules and affect the adaptive capability if the sufficient operating condition data is lacking. Li Z et al.¹⁰ proposed a novel distributed event-triggered secondary control method to overcome the drawbacks of the primary control of DC microgrids. Through the event-triggered distributed communication, the proposed control method can realize system-wide control of parallel distributed generators. A simple event-triggered condition is also proposed that does not require an additional state estimator, so that only when the event-triggered condition is satisfied, the limited communication between neighboring devices is required, greatly reducing the communication burden at the network layer. But the stateless estimator design relies on the real-time communication of neighboring nodes, which may lead to local control failure and spread to the whole network if the communication link fails or packet loss. Li C et al.¹¹ proposed a node voltage sequencing regulation algorithm to reduce the number of battery charging and discharging, improve the service life of the battery, and set the charging and discharging thresholds of the energy storage system according to the node voltage value, so that the nodes that have not exceeded the limit can participate in the voltage regulation. And detect the state of charge of the battery, select the appropriate charging and discharging combinations according to the SOC to avoid overcharging and overdischarging problems. But the fixed thresholds cannot cope with the rapidly fluctuating distributed loads, which may lead to regulation triggered only after the voltage crosses the limit, with lagging response. Cheng C et al.¹² investigated a battery energy storage system based on the modular multilevel converter (MMC) topology for the integrated regulation of wind farm power and voltage. Due to the fluctuation of wind farm output power caused by the randomness of wind speed, it is proposed to input the active power of the wind farm into a first-order low-pass filter. The difference between the filter output and the input is used as the MMC-BESS active reference signal to smooth the grid-connected power of the wind farm. It is proposed to make the difference between the d-axis component of the PCC voltage and the rated value generate the reactive reference current through the PI regulation to realize the PCC voltage regulation. But the PI controller has poor adaptability to the nonlinear loads, which may lead to the insufficient reactive power compensation or voltage oscillation. Ou Y et al.¹³ proposed a reactive self-responsive voltage control method by considering the reactive regulation characteristics of the converter, and the reactive action threshold of the method was optimally adjusted. The reactive regulation potentials of PV converters and energy storage converters are utilized to effectively deal with the voltage overrun problem in the distribution networks with a high percentage of distributed PV. The voltage operating characteristics of the distributed PV distribution network with the high percentage of PV are analyzed, and the reactive voltage regulation characteristics of the converter are analyzed to study the constraints affecting the reactive power regulation capability of the converter. However, in weak grids or regions with the low short-circuit capacity, the reactive power regulation may trigger the voltage inverse overrun. The advantages and disadvantages of the existing methods and the voltage regulation consistency control method proposed in this paper are shown in Table 1.

Table 1 Comparison of limitations between existing methods and the proposed method.

The traditional voltage regulation methods, such as reactive power compensation, on-load voltage regulating switches and static reactive power compensators, have significant changes in the grid characteristics after the distributed PV access, and these traditional methods expose the limitations in the precisely regulating voltage. Although the reactive power compensation can regulate the voltage to a certain extent, its response speed is slow in the face of rapid fluctuations in the distributed PV power, making it difficult to track the voltage changes in real time. When the on-load voltage regulator switch regulates the voltage, the regulation process is relatively lagging, and frequent regulation may affect the equipment life. The static reactive power compensator has limited ability to suppress the harmonics in the complex distributed PV grid environment, which may lead to the poor voltage regulation. In contrast, the voltage regulation consistency control method based on the PV inverter power coordination proposed in this paper deeply analyzes the overvoltage situation after PV grid-connection and fully considers the multiple PV grid-connected scenarios and different voltage control stages of grid-connected substations. Through an innovative linear calculation method, the active and reactive power regulation of PV inverters is accurately determined to realize the power balance and the coordinated operation between the PV inverters and the grid. And the consistency algorithm is used to construct a control model to realize the exchange and synchronization of multi-node voltage data to respond to the voltage fluctuations caused by the distributed PV systems more quickly and accurately, effectively avoiding the equipment damages and grid failures due to the voltage abnormalities, and improving the reliability and stability of grid operation.

The innovations of this paper are as follows:

• 1. A new consistency control method is proposed: for the problem of inconsistent voltage regulation in the grid-connected area caused by the distributed photovoltaic (PV) access, a consistency control of voltage regulation in the grid-connected substation based on the power coordination of PV inverters is proposed. By coordinating the power of PV inverters, the effective regulation of voltage in the grid-connected substation area is ensured, providing the new ideas and methods for solving the voltage regulation problems caused by the distributed PV grid connection.

• 2. In-depth analysis of the impact of voltage overrun: the voltage overrun after the grid-connection of PV is analyzed in detail, and the relationship between the PV inverter power and the local voltage and its impact mechanism on the voltage rise and overrun are elucidated.

• 3. Innovative PV inverter power coordination strategy: taking full account of the multi-PV grid-connected situation and combining the different voltage control stages of the grid-connected substation, a linear calculation method is adopted to obtain the active and reactive the power regulation quantities of the PV inverters. The power balance and coordinated operation between the PV inverters and the grid are realized, and the stability and reliability of the system are improved.

The specific contributions of this paper are as follows:

• 1. Solving the voltage regulation problem of distributed PV grid-connection: a consistent control method of grid-connected substation voltage regulation based on the PV inverter power coordination is proposed, effectively dealing with the inconsistent voltage regulation in the grid-connected area caused by the different locations and capacities of distributed PV accesses and contribute to the stable operation and control of the power grid.

• 2. In-depth analysis of the mechanism of voltage influence after grid-connection of PV: through the analysis of voltage overrun after grid-connection of PV, it clarifies the relationship between the PV inverter power and the local voltage as well as the influence on voltage rise and overrun, providing a theoretical basis for the subsequent control strategy.

• 3. Constructing an effective voltage regulation control model: use the consistency algorithm to construct a voltage regulation control model for the grid-connected substations, realize the exchange and synchronization of multi-node voltage data, and improve the reliability of data. Construct the multi-objective control function, set constraints, optimize the control effect, and realize accurate voltage regulation consistency control.

The main purpose of this paper is to solve the problem of inconsistent voltage regulation in the grid-connected area caused by the distributed PV grid-connection and to ensure the safe and stable operation of the power grid. In order to achieve the consistency control of voltage regulation in the area of the grid-connected substation, to ensure the reliability of the power supply, and ultimately to realize the efficient use of energy and sustainable development, the impact of PV grid-connected voltage overrun is thoroughly analyzed. Based on the PV inverter power coordination, the linear calculation method is used to determine the inverter active and reactive power regulation amount. The consistency algorithm is then applied to construct the voltage regulation control model.

Consistency control method for voltage regulation in grid-connected substation area

Analysis of voltage exceeding limits after photovoltaic grid connection

After the photovoltaic grid connection, the PV inverters primarily operate in maximum power point tracking mode under the unit power factor. This can cause the voltage rise and overvoltage, directly compromising the safety and stability of the grid-connected substation^11,12,13. To ensure the effective voltage regulation in the grid-connected substation area, this study analyzes the impact of overvoltage following the photovoltaic grid connection.

If the number of load nodes in the grid-connected substation area is N, and the photovoltaic power source is connected at the k node, the voltage at the power access point k is ${U_k}$, and the active and reactive power of the power source are ${P_{DG,k}}$ and ${Q_{DG,k}}$, respectively. The load power is ${P_{L,i}}+j{Q_{L,i}}$, and the voltage loss at the power access point k after photovoltaic grid connection is $\Delta {U_k}$. The calculation formula for $\Delta {U_k}$ is:

$$\Delta {U_k}=\frac{{{P_{DG,k}}\sum\limits_{{i=0}}^{k} {{R_i}+{Q_{DG,k}}\sum\limits_{{i=0}}^{k} {{X_i}} } }}{{U_{N}^{2}}} \times 100\%$$

(1)

In formula (1), ${R_i}$ and ${X_i}$ respectively represent the equivalent resistance and reactance of the i-section line, and R represents the local voltage.

The voltage rise rates caused by ${U_N}$ on ${P_{DG,k}}$ and ${Q_{DG,k}}$ are ${\mu _{P,k}}$ and ${\mu _{Q,k}}$, respectively, and their calculation formulas are:

$${\mu _{P,k}}=\frac{{\partial \left( {0 - \Delta {U_k}\% } \right)}}{{\partial {P_{DG,k}}}}$$

(2)

$${\mu _{Q,k}}=\frac{{\partial \left( {0 - \Delta {U_k}\% } \right)}}{{\partial {Q_{DG,k}}}}$$

(3)

.

In formulas (2) and (3), $\Delta {U_k}\%$ represents the percentage of voltage loss.

According to the above formula, during the grid-connected PV operation, the output power of the PV inverter directly influences the local voltage, contributing to the voltage rise and potentially causing the voltage limit violations.

Power coordination of photovoltaic inverters

Based on the analysis of the voltage limit violations following the PV grid connection, we conclude that the PV inverter power coordination can effectively mitigate these violations and ensure the grid-connected substation voltage stability. Therefore, this study first achieves the PV inverter power coordination to determine the required power adjustments. The methodology fully considers the multiple PV grid-connection scenarios, specifically addressing the multi-inverter power coordination. Furthermore, by considering different voltage control stages in the grid-connected substations, a linear calculation method¹⁴ calculates the required active and reactive power adjustments for the PV inverters, establishing the foundation for consistent voltage regulation control in the grid-connected substations.

(1) Reactive power compensation stage of photovoltaic inverter:

If the photovoltaic power station is operating normally, the inverter is in the maximum power factor point operating state. At this time, the voltage ${U_k}$ at the photovoltaic power supply connection point needs to satisfy formula (4):

$${U_o}={U_k}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_j}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_j}} } \right){X_i}} \right] - {P_{DG,k}}{R_k}} }}{{{U_k}}}$$

(4)

.

In formula (4), ${U_o}$ represents the bus voltage of the power grid, ${P_j}$ and ${Q_j}$ represent the active and reactive power of the photovoltaic inverter during the compensation stage, and ${R_k}$ represents the resistance at k.

In this stage, the ${Q_{DG,k}}$ result output by the photovoltaic inverter can be adjusted to the voltage at the photovoltaic power source connection point, so that it approaches the target voltage ${U_{k,\lim }}$. At this time, ${U_{k,\lim }}$ needs to satisfy formula (5):

$${U_o}={U_{k,\lim }}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_j}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_j}} } \right){X_i}} \right] - {P_{DG,k}}{R_k}+{Q_{PV,b}}{X_k}} }}{{{U_{k,\lim }}}}$$

(5)

.

In formula (5), ${Q_{PV,b}}$ represents the inductive reactive power output by the photovoltaic inverter, and this power occurs when ${U_k}$ drops to ${U_{k,\lim }}$. ${X_k}$ represents the reactance at k.

Combining the above two formulas, it can be concluded that:

$$\left\{ \begin{gathered} {Q_{PV,b}}=\frac{A}{{{X_k}}} \hfill \\ A={U_o}\left( {{U_k} - {U_{k,\lim }}} \right)+U_{{k,\lim }}^{2} - U_{k}^{2} \hfill \\ \end{gathered} \right.$$

(6)

.

If the power S of the photovoltaic inverter reaches the maximum value ${S_{\hbox{max} }}$ and the voltage at k still exceeds the limit, then it enters the next control phase.

In this control system, in the reactive power compensation stage, based on Eqs. (4)-(6), the reactive power output is precisely calculated according to the voltage at the PV power connection point to ensure that the reactive power injection is minimized under the premise of meeting the voltage regulation requirements. When the voltage is close to the target value, the reactive power output is gradually reduced. In addition, during the entire control process, the system voltage and power changes are monitored in real time by the consistency algorithm, and the reactive power injection is dynamically adjusted to avoid the problems of increased losses and oversized components caused by the improper reactive power injection.

(2) Maximum power adjustment stage of photovoltaic inverter:

When the photovoltaic inverter is at its maximum capacity, its power factor angle is represented by ${\theta _{\lim }}$, and ${U_k}$ needs to satisfy formula (7):

$${U_o}={U_k}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_{PV,i}}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_{PV,i}}} } \right){X_i}} \right]} }}{{{U_k}}} - \frac{{{S_{\hbox{max} }}\left( {{R_k}\cos {\theta _{\lim }}+{X_k}\sin {\theta _{\lim }}} \right)}}{{{U_k}}}$$

(7)

.

In formula (7), ${P_{PV,i}}$ and ${Q_{PV,i}}$ respectively represent the active and reactive power that the i photovoltaic power source can output. In this stage, the photovoltaic inverter always maintains ${S_{\hbox{max} }}$, and after adjusting the power factor, meeting the standard of ${U_{k,\lim }}$ is the voltage adjustment target at k. At this time, formula (8) needs to be satisfied:

$${U_o}={U_{k,\lim }}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_{PV,i}}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_{PV,i}}} } \right){X_i}} \right]} }}{{{U_{k,\lim }}}} - \frac{{\left( {{R_k}{P_{PV,t}}+{X_k}\sqrt {S_{{\hbox{max} }}^{2} - P_{{PV,t}}^{2}} } \right)}}{{{U_{k,\lim }}}}$$

(8)

By combining formulas (7) and (8) above, the active power ${P_{PV,t}}$ output by the photovoltaic inverter at time t after adjustment in this stage can be obtained. The calculation formula is:

$${P_{PV,t}}=\frac{{\left( {A+B} \right){R_k}+C{X_k}}}{{R_{k}^{2}+X_{k}^{2}}}$$

(9)

$$\left\{ \begin{gathered} B={S_{\hbox{max} }}\left( {{R_k}\cos {\theta _{\lim }}+{X_k}\sin {\theta _{\lim }}} \right) \hfill \\ C={\left[ {{{\left( {{S_{\hbox{max} }}{R_k}} \right)}^2}+{{\left( {{S_{\hbox{max} }}{X_k}} \right)}^2} - {{\left( {A+B} \right)}^2}} \right]^{\frac{1}{2}}} \hfill \\ \end{gathered} \right.$$

(10)

(3) Power reduction stage of photovoltaic inverter:

If the voltage at k still exceeds the limit after the above two stages of adjustment, the active power of the photovoltaic inverter should be reduced^15,16,17.

The maximum value of ${\theta _{\lim }}$ is ${\theta _{\hbox{max} }}$. If the photovoltaic inverter is in this situation, ${U_k}$ needs to satisfy formula (11):

$${U_o}={U_k}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_{PV,i}}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_{PV,i}}} } \right){X_i}} \right]} }}{{{U_k}}} - \frac{{{S_{\hbox{max} }}\cos {\theta _{\hbox{max} }}\left( {{R_k}+{X_k}\tan {\theta _{\hbox{max} }}} \right)}}{{{U_k}}}$$

(11)

.

In this stage, the inverter always keeps ${\theta _{\hbox{max} }}$, and after active power reduction, adjust the voltage at k to make it meet the standard of ${U_{k,\lim }}$. The formula is:

$${U_o}={U_{k,\lim }}+\frac{{\sum\limits_{{i=0}}^{k} {\left[ {\left( {\sum\limits_{{j=i}}^{n} {{P_{PV,i}}} } \right){R_i}+\left( {\sum\limits_{{j=i}}^{n} {{Q_{PV,i}}} } \right){X_i}} \right]} }}{{{U_{k,\lim }}}} - \frac{{{P_{PV,s}}\left( {{R_k}+{X_k}\tan {\theta _{\hbox{max} }}} \right)}}{{{U_{k,\lim }}}}$$

(12)

Combining the above two formulas, the active power ${P_{PV,s}}$ of the photovoltaic inverter after processing in this stage is obtained as follows:

$${P_{PV,s}}={S_{\hbox{max} }}\cos {\theta _{\hbox{max} }}+\frac{A}{{{R_k}+{X_k}\tan {\theta _{\hbox{max} }}}}$$

(13)

.

(4) Calculation of power adjustment for photovoltaic inverters:

Based on the control results of three stages, the PV inverter power adjustment is calculated by using the following formula:

$$\Delta {P_{PV,s}}=\frac{{\left( {{U_{k,\lim }} - {U_{k,s}}} \right)\left( {{P_{PV,t}}{ - _{\hbox{max} }}\cos {\theta _{\hbox{min} }}} \right)}}{{{U_{k,t}} - {U_{k,1}}}}$$

(14)

$$\Delta {Q_{PV,s}}=\frac{{{Q_{PV,s}}\left( {{U_{k,\lim }} - {U_{k,s}}} \right)}}{{{U_{k,s}} - {U_{k,m}}}}$$

(15)

.

In formulas (14) and (15), $\Delta {P_{PV,s}}$ and $\Delta {Q_{PV,s}}$ respectively represent the adjustment amounts of active and reactive power of the photovoltaic inverter, ${U_{k,s}}$ represents the input voltage at k when formula (13) is satisfied, and ${U_{k,m}}$ and ${U_{k,1}}$ respectively represent the input voltage at k of the photovoltaic inverter when ${\theta _{\lim }}$ and ${\theta _{\hbox{max} }}$ are satisfied.

Modeling of voltage regulation consistency control in grid-connected substation areas

After determining the active and reactive power adjustment values for the PV inverters, we employ a consistency algorithm to develop a voltage regulation control model for the grid-connected substation area. These algorithms operate on the distributed clusters, and utilize a dedicated protocol to coordinate the operations and maintain the result consistency, ensuring data uniformity across the system^18,19,20. The consistency algorithm is based on the theory of distributed system synergy, and its core idea is to make the state of each node consistent through the information interaction between the nodes. In this study, each node continuously updates its own voltage data and exchanges and synchronizes data with the neighboring nodes according to Eq. (16). Taking a node in the power grid as an example, it will gradually adjust its state according to the consistency matrix, combined with the voltage state values of itself and its neighboring nodes. In order to effectively monitor and regulate the overall system voltage, all node voltage data must reach stability and consistency²¹. The proposed consistency algorithm, when applied to the grid-connected substation voltage regulation control models, facilitates the multi-node voltage data exchange and synchronization, enhancing the system-wide data reliability. In this study’s framework, when the instantaneous overvoltage occurs in a grid-connected substation area, the algorithm propagates this information throughout the PV grid-connected system, enabling the rapid corrective actions and achieving the coordinated voltage regulation across all substations. The data synchronization process of the consistency algorithm operates as follows:

$$\left\{ \begin{gathered} {z_l}=\left[ {\frac{1}{{{\alpha _q}\left( {{T_c}+{t_0} - t} \right)}}+1} \right]{a_{lk}}\left( {{x_l} - {x_k}} \right) \hfill \\ {y_l}={z_l} \times x\left( t \right) \hfill \\ \end{gathered} \right.$$

(16)

.

In formula (16), ${z_l}$ represents the consistency matrix, ${\alpha _q}$ represents the node state value of the grid-connected substation, ${T_c}$ represents the convergence time of the consistency algorithm, ${t_0}$ represents the initial time, and t represents the current convergence process time, ${a_{lk}}$ represents the control parameter value of the node voltage in the grid-connected substation area, ${x_l}$ represents the instantaneous voltage value of the transformer at time l, ${x_k}$ represents the instantaneous voltage value of the transformer at time k, ${y_l}$ represents the transformer voltage data processed by using consistency algorithm, and $x\left( t \right)$ represents the original transformer voltage data of different nodes.

By applying the consistency matrix from the preceding formulation, we process the voltage data from the grid-connected substation nodes to enhance the control model’s real-time performance and derive the corresponding objective function. The multi-objective function incorporates three key control criteria: (1) minimization of transformer instantaneous overvoltage amplitude; (2) minimization of control action rate-of-change; (3) minimization of control process energy consumption. After the distributed photovoltaic system is connected to the power grid, the output power of 15 sets of photovoltaic panels will fluctuate due to factors such as lighting and temperature. By coordinating the power of photovoltaic inverters and adjusting their active and reactive power outputs at different voltage control stages, the voltage changes caused by photovoltaic power fluctuations can be suppressed, and the instantaneous overvoltage amplitude of transformers can be controlled within a reasonable range. This yields the following constructed objective function:

$${J_k}=\int\limits_{{{t_0}}}^{{{t_f}}} {\left[ {{\alpha _1}{{\left( {{V_t} - {V_r}} \right)}^2}+{\alpha _2}{{\left( {\frac{{dV}}{{tdt}}} \right)}^2}+{\alpha _3}{P_t}} \right]dt}$$

(17)

.

In formula (17), ${J_k}$ represents the multi-objective control objective function for the instantaneous overvoltage in the constructed grid-connected substation area, ${V_t}$ represents the instantaneous overvoltage amplitude of the grid-connected substation area at time t, and ${V_r}$ represents the expected voltage value of the grid-connected substation area, V represents the voltage output value of the grid-connected substation, t represents the control time of the grid-connected substation, ${P_t}$ represents the energy consumption of the grid-connected substation during the control process, ${t_0}$ represents the start time of the grid-connected substation control process, ${t_0}$ represents the end time of the grid-connected substation control process, and ${\alpha _1}$, ${\alpha _2}$, and ${\alpha _3}$ represent the weight coefficients corresponding to the three control objectives mentioned above. The determination of the weight coefficients ${\alpha _1}$, ${\alpha _2}$, and ${\alpha _3}$ is based on the principle of multi-objective optimization. ${\alpha _1}$ is mainly used to minimize the instantaneous over-voltage amplitude of the grid-connected transformer. When the value of ${\alpha _1}$ is taken between 0.4 and 0.6, it can effectively suppress the over-voltage while ensuring that the other two objectives are not overly affected, thus ${\alpha _1}$ is finally taken as 0.5. ${\alpha _2}$ is used to minimize the rate of change of the control action, a too large value may lead to too slow voltage regulation, and too small value may not be able to effectively smooth the control process. If ${\alpha _2}$ = 0.3, ${\alpha _3}$ is used to minimize the energy consumption of the control process, combined with the actual operating costs and system efficiency requirements, and the value is 0.2. These weighting coefficients are configured to enable the multi-objective control function to optimally balance all objectives, achieving the system-wide performance optimization.

The key indicator of transformer instantaneous overvoltage amplitude in the objective function is directly related to the instantaneous overvoltage that results from the changes in the transformer voltage caused by the variations in the distributed photovoltaic system’s output power after it is connected to the power grid. For example, when the output power of the photovoltaic panel increases, it may cause the local voltage to rise, resulting in an increase in the instantaneous overvoltage amplitude of the transformer²². Conversely, it decreases. This article indirectly controls the instantaneous overvoltage amplitude of the transformer in the objective function by coordinating the power of the photovoltaic inverter and adjusting its active and reactive power output according to different voltage control stages, ensuring the stable operation of the power grid.

Based on the objective function constructed above, the corresponding constraint conditions are set to ensure the authenticity and reliability of the control objectives. The specific constraint conditions are as follows:

$$\left\{ \begin{gathered} {V_{\hbox{min} }} \leqslant {V_t} \leqslant {V_{\hbox{max} }} \hfill \\ {V_c}^{{\hbox{min} }} \leqslant {V_c} \leqslant {V_c}^{{\hbox{min} }} \hfill \\ {\eta _z}^{{\hbox{min} }} \leqslant {\eta _z} \leqslant {\eta _z}^{{\hbox{max} }} \hfill \\ {P_t} \leqslant {P_{\hbox{max} }} \hfill \\ \end{gathered} \right.$$

(18)

.

In formula (18), ${V_{\hbox{min} }}$ represents the minimum instantaneous overvoltage that occurs in the grid-connected substation during the operation, and ${V_{\hbox{max} }}$ represents the maximum instantaneous overvoltage that occurs in the grid-connected substation during the operation, ${V_c}^{{\hbox{min} }}$ represents the minimum output voltage of the grid-connected substation, ${V_c}^{{\hbox{max} }}$ represents the maximum output voltage of the grid-connected substation, ${V_c}$ represents the output voltage of the grid-connected substation, ${\eta _z}^{{\hbox{min} }}$ represents the minimum rate of change of control actions in the grid-connected substation area, ${\eta _z}$ represents the rate of change of control actions in the grid-connected substation area, ${\eta _z}^{{\hbox{max} }}$ represents the maximum rate of change of control actions in the grid-connected substation area, and ${P_{\hbox{max} }}$ represents the maximum energy consumption allowed in the control process of the grid-connected substation area.

Using formulas (17) and (18), we construct a voltage regulation consistency control model for the grid-connected substations. By solving this model with the prescribed constraints, we achieve the coordinated voltage regulation across the substation network. This control system is a distributed architecture, with the control modules integrated into each photovoltaic inverter. Each inverter control module interacts with other nodes through a communication network to receive and process the voltage and power information from other nodes²³. Based on the consistency algorithms and power coordination strategies, it independently calculates and adjusts its own active and reactive power output, achieving the consistent control of the voltage in the entire grid-connected substation area. The solution process is illustrated in Fig. 1.

Fig. 1

Solution process of voltage regulation consistency control model for grid-connected substation area.

The control process shown in Fig. 1 begins with initializing the model output. We then calculate both the current objective function value and its approximate gradient. Using this gradient information, we iteratively adjust the output to reduce the objective function value. At each iteration, we verify whether the solution satisfies all constraints. If the constraints are violated, we make corrections until they are met. This process repeats until the convergence criteria are satisfied, at which point we output the final optimized objective function value as the control result. The detailed computational procedure is described below.

$$\left\{ \begin{gathered} {T_j}=\left\| {{J_k}} \right\| \times \frac{{{J_0}}}{{{J_k}}} \times {e_k} \hfill \\ {S_k}=\sum\limits_{{j=1}}^{m} {{T_j} \times \kappa } \times {p_c} \hfill \\ \end{gathered} \right.$$

(19)

.

In formula (19), ${T_j}$ represents the approximate gradient value of the objective function, ${J_0}$ represents the initial value of the objective function, ${e_k}$ represents the discretization parameter of the objective function, ${S_k}$ represents the output result of the grid-connected substation voltage regulation consistency control model, $\kappa$ represents the number of convergence times, and ${p_c}$ represents the convergence step size of the control model. Based on the above formula, the consistency control model for the voltage regulation in the grid-connected substation area is solved. Based on the solution results, the consistency control of voltage regulation in the grid-connected substation area is achieved to ensure its stable operation.

At night, due to the reduction of light intensity and photovoltaic power generation, the main source of voltage fluctuation may change from intermittent photovoltaic power generation to load change. Although the consistency control method proposed in this paper mainly aims at the voltage fluctuation caused by photovoltaic grid connection, its core idea is to achieve voltage consistent regulation by coordinating the power of photovoltaic inverter, which is also applicable to the voltage fluctuation caused by night load change. At night, when voltage fluctuation is caused by load change, voltage stability can be maintained by adjusting the energy storage system or coordinating control with other distributed generators. Specifically, the energy storage system can release electric energy when the voltage is too low, and absorb electric energy when the voltage is too high, so as to realize the stable regulation of voltage.

Experimental analysis

To verify the effectiveness of the photovoltaic-inverter power coordination-based voltage regulation consistency control method proposed in this study, we selected an 11 kV high-power grid-connected substation as the research object. The substation comprises 15 photovoltaic panels, 10 DC transformers, and 2 energy storage devices. We simulate the grid-connected system by using MATLAB/Simulink, and the simulation results are used for the experiment. The 11 kV grid connected substation model built in Matlab/Simulink is built by Simulink’s power system tools. The data acquisition module records the DC transformer voltage, power and other parameters at 10ms interval. According to formula (4)–(15), realize three-stage power regulation (reactive power compensation, maximum power point adjustment, power reduction), and output the active/reactive power reference value of the inverter. The specific configuration is shown in Fig. 2.

Fig. 2

Distribution of a high-power grid-connected substation area.

As shown in Fig. 2, the simulation model built in MATLAB Simulink includes 15 photovoltaic panels, 10 DC transformers and 2 energy storage devices. Photovoltaic panels generate electricity by absorbing solar energy and input electric energy into the grid. DC transformers and energy storage equipment are used to regulate and store electric energy. In the model, scenarios of different light intensity, load change and grid voltage fluctuation are set to simulate the actual operating conditions. The data collector records the voltage, power and other parameters under different nodes to evaluate the performance of the consistency control method. The experimental testing allows for an in-depth evaluation of the PV inverter-based voltage regulation consistency control method in the grid-connected substations. The specific experimental parameters are listed in Table 2.

Table 2 Experimental parameters.

As shown in Table 2, the DC transformer’s data collector records the voltage, power, and other parameters under the varying operating conditions over time. These data fully reflect the operational characteristics of the grid-connected substation area with the distributed PV integration, providing the comprehensive and practical support for evaluating the voltage regulation consistency control method. During the experiments, we test the aforementioned parameters and adjust them in real time.

In this paper’s methodology, we conduct a sensitivity analysis to evaluate the impact of parameter variations on the system performance in a grid-connected substation. The results are presented in Table 3.

Table 3 Experimental parameters.

The sensitivity analysis results in Table 3 reveal the distinct parameter-VSI relationships. When the transformer capacity increases by 10% from 100 MVA to 110 MVA, the VSI decreases slightly (ΔVSI = −0.05, sensitivity = −0.005). Conversely, a 5% line resistance increase from 0.1 Ω/km to 0.105 Ω/km elevates VSI values (ΔVSI = + 0.02, sensitivity = + 0.004). Reduce the load factor by 10% (70% → 63%) lowers VSI (ΔVSI = −0.03, sensitivity = −0.003), while a 20% faster control system response (50ms → 40ms) causes a marginal VSI reduction (ΔVSI = −0.01, sensitivity = −0.001). Most notably, a 30% increase in the solar power penetration (20% → 26%) significantly boosts VSI (ΔVSI = + 0.10, sensitivity = + 0.010), demonstrating its strong positive correlation with the system performance.

Select the high-voltage side bus of the DC transformer at node 15, and use the data collector to collect and record the changes of the DC transformer in the grid connected system over time, as shown in Fig. 3.

Fig. 3

Voltage variation of DC transformer.

As shown in Fig. 3, the DC transformer experiences the significant voltage fluctuations, including instances where the transient overvoltage exceeds the threshold. While the power output varies with these voltage changes, the fluctuation amplitude remains relatively small. To address these voltage variations, three control methods are compared: the approach from reference⁷ with the parameters set to proportional gain Kp = 2.0, integral gain Ki = 0.5, and derivative gain Kd = 0.1, using an overvoltage threshold of 120% rated voltage. The method from reference⁸ features a 5 ms response time and 100 Hz low-pass filter for the noise suppression and the proposed control method. The comparative results demonstrate the effectiveness of the proposed approach in managing these voltage transients. The voltage fluctuation amplitude is small, which indicates that the proposed method can quickly respond to the voltage change, adjust the PV inverter power in time, and maintain the voltage stability. However, it should also be noted that the voltage still fluctuates briefly at the beginning of the rapid load change, indicating that there is still room for improvement in the response speed of the model.

In order to more comprehensively evaluate the performance of the consistency control method under the scenarios of different light intensity, load change and grid voltage fluctuation, nodes 1–10 are selected to access photovoltaic at the same time, and the relevant voltage fluctuation data are recorded, as shown in Table 4.

Table 4 Voltage fluctuation data under different scenarios for nodes 1–10.

By analyzing Table 4, it can be seen that under the comprehensive influence of different light intensity, load change and grid voltage fluctuation, the voltage fluctuation of each node shows significant differences. When the light intensity is constant (1000 w/m ²) and there is no fluctuation between the load and the grid voltage (node 1), the voltage fluctuation range is the smallest and the recovery time is the shortest (50ms). As the light intensity decreases (node 3600 w/m ²) or the load decreases (nodes 3, 7, 9, −10% to −20%), the voltage fluctuation amplitude increases and the recovery time prolongs. Especially, under the joint action of the light intensity, load reduction and grid voltage drop, the voltage fluctuation of node 7 is the most severe and the recovery time is the longest (90ms). On the contrary, the increase of light intensity (node 81000 w/m ² to + 20%) or grid voltage fluctuation (+ 5% to + 10%) also led to the increase of voltage fluctuation, but the recovery time was slightly shorter than the load change scenario.

In the experiment, three methods are used to control the instantaneous overvoltage of the DC transformer in the grid-connected substation area, and the control results of these three methods are statistically analyzed. Set the solar irradiance to vary sinusoidally within the range of 100–1000 W/m ², with a variation period of 300 s, to simulate the fluctuation of actual light intensity. The initial load power is set to 500 kW. During the experiment, 50 kW is randomly increased every 60 s to simulate the load changes in the actual power grid. During the experiment, the data acquisition equipment is used to monitor the voltage, power, and other data of the DC transformer in real-time with a sampling interval of 10ms. The control results are shown in Table 5.

Table 5 Control results of three methods.

Table 5 demonstrates the maximum instantaneous overvoltage control results of the three methods for 10 transformers. Overall, this paper’s method performs better in controlling the maximum instantaneous overvoltage, and the maximum instantaneous overvoltage values of each transformer under its control are generally lower than those of the other two methods. Specifically, the maximum instantaneous overvoltage value of some transformers under the method of literature⁷ exceeds 1300 V, and some under the method of literature⁸ exceeds 1400 V, which are beyond the set threshold. While the maximum instantaneous overvoltage value under the control of this paper’s method is 1023 V, which is within the range of the set threshold, indicating that this paper’s method can control the transformer’s instantaneous overvoltage more effectively, guarantee the stability of the voltage of grid-connected station areas, and reduce the risk of faults caused by overvoltage. The risk of faults caused by the overvoltage. The maximum instantaneous overvoltages recorded for the methods from references^7,8 are 1,325 V and 1,422 V respectively, both exceeding the predetermined threshold. In contrast, the application of the proposed method results in a transformer maximum instantaneous overvoltage of 1,023 V, well within the acceptable threshold range. These results demonstrate the method’s effectiveness in controlling the transformer transient overvoltages. In contrast, this article proposes a consistency control method based on the photovoltaic inverter power coordination, which comprehensively considers the differences in the distributed photovoltaic access locations and capacities. Through the innovative power coordination strategies, the inverter power is accurately adjusted at different voltage control stages. Simultaneously utilizing consistency algorithms to construct control models, real-time synchronization and exchange of multi node voltage data are achieved, effectively solving the problem of inconsistent voltage regulation caused by the distributed photovoltaic grid connection. The proposed method exhibits the high control accuracy in practical applications. This superior performance stems from its multi-objective optimization approach, which considers both the objective function and corresponding constraints across multiple dimensions. By simultaneously minimizing the transient overvoltage amplitude while accounting for the rate of control action change and energy consumption during the operation, the method achieves smoother and more efficient voltage regulation. Consequently, it significantly enhances the transformer’s transient overvoltage control capability and ensures the voltage stability in the grid-connected substation area. However, the proposed model also has some limitations. In a complex grid environment, the voltage regulation accuracy of the model may be affected when the multiple distributed PV access points simultaneously experience the power mutations with the harmonic disturbances. This is because the accuracy and timeliness of the data synchronization of the consistency algorithm will be challenged when dealing with a large amount of mutation data and harmonic disturbances, leading to deviations in the regulation of the PV inverter power.

To further verify the effectiveness of the proposed method in practical applications, we use the control process overshoot as the evaluation index to compare the performance of the three methods. The corresponding statistical results are presented in Fig. 4.

Fig. 4

Overshoot of three methods.

As shown in Fig. 4, within the same time frame, the methods from references^7,8 exhibit the relatively large overshoot values. In contrast, the proposed method demonstrats significantly smaller overshoot, indicating its superior capability to rapidly adjust transformer voltage and achieve the consistent voltage regulation control in the grid-connected substation area. This improved performance stems from the method’s minimal over-regulation characteristics, which originate from its unique PV inverter power coordination design. During the voltage regulation, the method precisely analyzes post-connection PV voltage variations and optimally coordinates the PV inverter’s active and reactive power outputs across different operational stages, including the reactive power compensation, maximum power point adjustment, and power curtailment. This refined coordination strategy enables the rapid and accurate response to the voltage fluctuations, facilitating timely the transformer voltage adjustments. Consequently, the method achieves the consistent voltage regulation control in the substation area while ensuring the grid stability.

Under the conditions of 90% light intensity drop, 20% grid voltage surge, and 50% sudden load change, the proposed method’s performance is thoroughly assessed by comparing it to the Literature^7,8,9,10 methods in terms of maximum immediate overvoltage, adjustment time, and overshoot. The specific situation is shown in Table 6.

Table 6 Performance comparison of different methods in different situations.

According to Table 6 analysis, under the conditions of a sudden drop of 90% in the light intensity, a sudden increase of 20% in the grid voltage, and a sudden change of 50% in the load, the method proposed in this paper performs well in the three indicators of maximum instantaneous overvoltage, regulation time, and overshoot. In terms of maximum instantaneous overvoltage, the values of the proposed methods are 1023 V, 1050 V, and 1030 V, which are lower than other methods in references^7,8,9,10. Among them, the method in reference⁸ has the highest value, reaching 1400 V, 1350 V, and 1370 V, respectively. In terms of adjustment time, the proposed method has a faster adjustment speed, with 40ms, 35ms, and 45ms respectively, while the method in reference⁸ has the longest adjustment time, with 80ms, 75ms, and 85ms respectively. In terms of overshoot, the proposed method has a relatively low overshoot of 5%, 4%, and 6%, respectively. The method in reference⁸ has the highest overshoot of 18%, 17%, and 19%, respectively. Overall, under different extreme operating conditions, the method proposed in this article can more effectively control the voltage, quickly adjust the system, and reduce the overshoot, demonstrating better performance.

By comparing the power factor stability and equipment loss rate of different methods in the heavy-loaded industrial areas and light-loaded residential areas, it is possible to comprehensively assess the actual performance of each method and clarify its applicability in different application scenarios. The comparison is shown in Table 7.

Table 7 Comparison of power factor stability and equipment loss rate under different load Conditions.

Analyzing Table 7, it can be seen that the power factor stability of the method proposed in this paper reaches 0.97, and the equipment loss rate is 3.0%, which is closer to 1 and lower than that of the methods in the literature^7,8,9,10. During the light load period in the residential area, the method of this paper has a power factor stability of 0.98 and equipment loss rate of 2.8%, which is also the best performance in terms of power factor stability, and the equipment loss rate is lower than the other comparative methods. This shows that the method proposed in this paper can effectively improve the power factor stability and reduce the equipment loss rate under different load scenarios, which is more advantageous in practical applications and more applicable in different application scenarios.

The Definition of Variable Symbol is shown in Table 8.

Table 8 Definition of variable symbol.

Conclusion

To mitigate voltage fluctuations caused by the output variations or load changes in the distributed energy resources (e.g., PV and wind power systems), this study proposes a grid-connected substation voltage regulation control method based on the PV inverter power coordination. The method maintains the grid voltage within the permissible ranges and enhances the system stability. By analyzing the post-connection overvoltage conditions and implementing the coordinated active/reactive power control through the PV inverters, the proposed approach achieves the consistent voltage regulation across the grid-connected substation area. The experimental results show that, in comparison, the overshoot value of our method is relatively small. Under the conditions of 90% decrease in the light intensity, 20% sudden rise in the grid voltage, and 50% sudden load change, our method performs well on the indicators of maximum instantaneous overvoltage of 1023 V, 1050 V, 1030 V, regulation time of 40ms, 35ms, 45ms, and overshoot of 5%, 4%, and 6%, respectively, being better than the comparative method.

The advantage of the proposed model lies in the comprehensive consideration of multi-PV grid-connection and different voltage control stages of the substation, and the precise power coordination and voltage regulation are realized by the linear calculation method and consistency algorithm. Compared with the existing methods, the model can respond to the voltage changes more quickly, effectively avoid the equipment damage and grid faults caused by the voltage abnormalities, and improve the reliability and stability of grid operation. Meanwhile, the model’s multi-objective optimization strategy takes into account the voltage regulation effect, controls the action smoothness and energy consumption, and improves the overall system performance. The proposed model helps to ensure the stable operation of the power grid, reduce the power outages and other accidents caused by the voltage problems, and guarantee the normal use of electricity for the social production and life. It also promotes the healthy development of distributed photovoltaic, improves the utilization rate of renewable energy, and reduces the carbon emissions to contribute to the environmental protection.

In the practical power grid applications, this model also faces some challenges. In a complex power grid environment, when the multiple distributed photovoltaic access points simultaneously experience the power mutations accompanied by the harmonic interference, the accuracy and timeliness of data synchronization in consistency algorithms will be affected. A large amount of abrupt data and harmonic interference may lead to deviations in the algorithm processing of data, resulting in the inaccurate regulation of photovoltaic inverter power and subsequently affecting the voltage regulation accuracy. In addition, the distributed photovoltaic output in the actual power grid is uncertain due to the natural conditions such as weather, and the power grid load is also in a dynamic state. These factors increase the difficulty of accurate prediction and control, which has a certain impact on the effectiveness of this method in practical applications.