Open AccessArticle

Physics-Informed Neural Network-Based VSC Back-to-Back HVDC Impedance Model and Grid Stability Estimation

School of Electrical Engineering, Korea University, Anam-ro, Sungbuk-gu, Seoul 02841, Republic of Korea

Author to whom correspondence should be addressed.

Electronics 2024, 13(13), 2590; https://doi.org/10.3390/electronics13132590

Submission received: 20 May 2024 / Revised: 24 June 2024 / Accepted: 29 June 2024 / Published: 1 July 2024

(This article belongs to the Special Issue Advances in Enhancing Energy and Power System Stability and Control)

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Abstract

With the increase in the number of power electronic devices in power systems, various techniques for assessing their stability have emerged. Among these techniques, impedance model-based stability analysis techniques have been widely used. However, conducting such analyses across multiple operating points requires abundant impedance measurement data from power electronic devices. In this paper, we propose a method for constructing impedance models of equipment with fewer impedance measurement data in voltage-source converter (VSC) back-to-back high-voltage direct current (HVDC) systems using physics-informed neural networks. Furthermore, given the power system states, we present a neural network approach to estimate grid stability at different operating points. Validation via PSCAD/EMTDC simulations and a PyTorch neural network confirmed the adequacy of these models.

Keywords:

back-to-back HVDC; impedance modeling; physics-informed neural network; stability estimation; VSC HVDC

1. Introduction

Power systems are undergoing a significant transformation to achieve carbon neutrality, marked by the integration of renewable energy sources, advancements in battery energy storage systems, and the adoption of high-voltage direct current (HVDC) technologies. Among these changes, renewable energy sources present challenges to the grid owing to their unpredictability, varying output responses, and different connection conditions compared with traditional sources. Conversely, energy storage systems and HVDCs play crucial roles in supporting grids. A key characteristic shared by these facilities is the utilization of DC power. For example, photovoltaic sources produce DC power, wind sources require AC-DC-AC conversion for frequency conversion, and battery energy storage systems utilize DC batteries for energy storage and release. Consequently, the increasing use of grid-connected AC-DC converter facilities, particularly voltage-source converters (VSCs), is driving this transformation [1,2]. VSCs offer a high capacity, proven performance, sophisticated protection, and controllability. In particular, VSC back-to-back HVDC systems with both converters installed at grid vulnerability points enhance grid stability by adjusting the power flow and reducing the fault current in neighboring regions.

As VSC back-to-back HVDCs are essential for grid stability at vulnerable points, any operational error can severely impact grid operations [3]. Consequently, grid operators must conduct various stability evaluations to ensure the reliable operation of these systems. Although converter-driven grid stability is not unique to VSC back-to-back HVDCs, the IEEE recently introduced a converter-driven stability criterion to underscore its importance and ensure stable operations [4]. However, owing to the converters’ faster response time and more vulnerable performance against transient disturbances than traditional generators, verifying their transient stability requires conducting electromagnetic transient (EMT) dynamic simulations with shorter time steps in the microsecond range. Through EMT calculations, these simulations can validate the ability of converters to remain operational and aid in grid recovery during various grid faults [5]. However, obtaining accurate control modeling and the parameters required for EMT dynamic simulations, often proprietary to equipment manufacturers, remains challenging, thus making it difficult for grid operators to execute the simulations [6]. Even with precise data, conducting comprehensive EMT dynamic simulations for an entire grid remains computationally intensive [7].

To address the challenges in the EMT dynamic stability determination, grid operators commonly adopt a method that involves representing each operating point of the VSCs in an impedance form [8,9,10,11,12,13,14,15]. This method utilizes basic Circuit Theory and the General Nyquist Criterion, which facilitate the efficient determination of grid stability by comparing the grid and converter impedances at every operating point [16]. If the combined impedance at a particular operating point approaches zero or becomes negative at any frequency, resonance, and stability violations at the operating point are highly likely. Although numerous studies have been conducted to analytically calculate impedance from the past to the present [15,17,18,19], there is a significant challenge in performing these analyses without knowledge of the control structure of the equipment. However, impedance information can also be obtained through a perturb-and-observe method by injecting arbitrary currents and measuring the voltage responses [20]. This method makes it possible to conduct stability assessments without requiring assistance from facility manufacturers.

However, understanding the intricate behavior of power electronic devices such as VSC back-to-back HVDC systems remains challenging. Unlike passive components, their impedance varies non-linearly with frequency and operating point [21]. Therefore, a multi-operating point (MOP) impedance model is required to monitor grid stability. Creating MOP impedance models for converters requires considerable impedance data obtained through the perturb-and-observe method. Such perturbations are time-consuming, highlighting the need to minimize usage.

Researchers are exploring methods for minimizing the perturbations required to create MOP impedance models, with artificial neural networks (ANNs) considered a promising option [22,23]. Typically, ANNs learn solely from impedance data, independent of internal control specifics. Early research proposed recurrent-based ANNs to predict impedance at specific operating points [24]. Fortunately, while detailed control methods for VSC back-to-back HVDC systems may vary, overarching control strategies are generally standardized. By leveraging this characteristic, researchers can derive impedance-like equations based on basic VSC control principles, thereby accelerating the ANN training process [25,26]. This approach, known as a physics-informed neural network (PINN), enables accurate MOP impedance modeling with fewer data points and, thus, fewer perturbations.

To construct a PINN, there are several approaches: (1) Physics-Informed Design of Architecture, where neurons in the ANN are specifically interconnected to mimic the form of the relationship equation when only the form of the relationship equation between the input and output is known; (2) Physics-Informed Loss Function, which utilizes derivative of the actual relationship equation when the relationship equation between the input and output is almost certain to reflect a more definite physical state; and so on [26]. A recent study successfully utilized PINN to model the MOP impedance of a single VSC by utilizing the Physics-Informed Design of Architecture method [6]. However, as this method does not utilize the complete equation, there is room for improvement.

This study expands upon this work by proposing a Physics-Informed Loss Function-based method to model the MOP impedance of VSC back-to-back HVDC systems. We calculated impedance-like equations tailored to VSC back-to-back HVDC systems to achieve a more accurate representation of the physical state through detailed impedance and differential equations. Additionally, our comprehensive model utilizes the PINN-based VSC back-to-back HVDC MOP impedance model to estimate the stability of the entire grid system. This maximizes the utility of the PINN impedance model.

The remainder of this paper is organized as follows. Section 2 describes a general VSC back-to-back HVDC system schematic and control model. Section 3 delves into the equations describing small-signal impedance-like matrix variations while controlling general VSC back-to-back HVDC systems concerning operating points and frequencies. Section 4 examines the impedance acquisition using the perturb-and-observe method. Section 5 proposes a PINN-based MOP impedance and an overall grid stability estimation model using the equations in Section 3. Finally, the proposed structure is validated using the simulation described in Section 6.

2. VSC Back-to-Back HVDC Model Configuration

Figure 1 depicts the basic schematic diagram of a VSC back-to-back HVDC system [27]. The two VSCs are connected to their respective AC systems via transformers and reactor impedances. The AC grid is characterized by an ideal voltage source and the grid’s characteristic impedance, known as the Short Circuit Ratio (SCR) [28]. The connection between the VSCs is represented as a direct link without line impedance, as they are installed at the exact location. In such VSC back-to-back HVDC systems, active power is received from one AC system and transmitted to the other AC system, while both converters supply or absorb appropriate reactive power to control voltage levels. When both AC systems are interconnected, VSC back-to-back HVDC can change the neighboring region’s active power flow and thus stabilize the grid.

In common practice, three-phase converters employ synchronous reference frame transformation to convert three-phase currents into dq-axes, where the d-axis current,

i_{d}

, conventionally relates to the active power and the q-axis current,

i_{q}

, to the reactive power [29]. This method allows for intuitive and straightforward control, with the active power and reactive power determined by a

|v_{s}|

−

i_{d}

product and a negative of a

|v_{s}|

−

i_{q}

product, respectively. In VSC back-to-back HVDC systems, each converter independently controls its d- and q-axis components, referencing the voltage at the connection point

v_{s}

with each AC grid. D-axis control typically involves maintaining a constant active power output or DC voltage,

v_{d c}

. The constant active power control directly adjusts the

i_{d}

to achieve the desired active power. In contrast, the constant DC voltage control modulates

i_{d}

to maintain a constant

v_{d c}

, which fluctuates due to an active power imbalance. Each converter adopts a distinct control strategy to prevent conflicts when targeting different active power values or DC voltage levels. Meanwhile, q-axis control primarily focuses on generating constant reactive power. The overall control strategy also incorporates a compensation logic to mitigate the coupling effect between the d- and q-axes induced by inductances

L_{1}

and

L_{2}

3. VSC Back-to-Back HVDC Small-Signal Impedance-like Equation

3.1. Control-Design-Based Small-Signal Impedance-like Equation

The control diagram provides a method for estimating the impedance of the control equipment at various operational points. In VSC back-to-back HVDC systems, this estimation relies on the current and voltage control loops, omitting detailed control, including VSC IGBT control. Although not entirely precise, this approach provides values similar to the actual impedance in a steady-state scenario. Here, considering that both the calculation and utilization of admittance are better than those of impedance, this paper describes the admittance matrix instead of the impedance matrix. The impedance matrix can be obtained by inversing the admittance matrix.

Upon examining the control scheme in Figure 2, a key component is the setting

i_{r e f}

to minimize the errors in

i

, which is then used to calculate the

v_{r e f}

. This logic is succinctly expressed in Equation (1), the fundamental equation of the control system [9,30].

[\begin{matrix} v_{d, r e f} \\ v_{q, r e f} \end{matrix}] = - (k_{P, i} + \frac{k_{I, i}}{s}) ([\begin{matrix} i_{d, r e f} \\ i_{q, r e f} \end{matrix}] - [\begin{matrix} i_{d} \\ i_{q} \end{matrix}]) - j ω L [\begin{matrix} i_{d} \\ i_{q} \end{matrix}] + F i l t e r (s) [\begin{matrix} v_{d} \\ v_{q} \end{matrix}]

(1)

Here,

k_{P, i}

and

k_{I, i}

represent the PI gains of the current control loop, whereas

F i l t e r (s)

denotes a filter transfer function crucial for stable

v_{s}

measurement and synchronous reference frame transformation. The filter for

v_{s}

measurement typically adopts a second-order configuration with a transfer function as expressed in Equation (2), where

ζ

represents the damping factor and

ω_{0}

signifies the cutoff frequency. The following equations are obtained assuming the same second-order filter:

F i l t e r (s) = \frac{ω_{0}^{2}}{s^{2} + 2 ζ ω_{0} s + ω_{0}^{2}}

(2)

The converter admittance can be inferred from the small-signal relationship between

v

and

i

. Therefore, substituting the reference values in Equation (1) is necessary. First, rearranging Equation (1) to isolate the expressions for

i

on the left-hand side and substituting

j ω

with

s

leads to Equation (3).

[\begin{matrix} i_{d} \\ i_{q} \end{matrix}] = \frac{k_{P, i} s + k_{I, i}}{L s^{2} + k_{P, i} s + k_{I, i}} [\begin{matrix} i_{d, r e f} \\ i_{q, r e f} \end{matrix}] + \frac{s}{L s^{2} + k_{P, i} s + k_{I, i}} ([\begin{matrix} v_{d, r e f} \\ v_{q, r e f} \end{matrix}] - F i l t e r (s) [\begin{matrix} v_{d} \\ v_{q} \end{matrix}])

(3)

Finally, a small-signal variation assessment is required. With further assumptions, equation simplification becomes feasible. Given our prior disregard for PWM’s role of PWM, merely approximating the actual

v

is sufficient for deriving an impedance-like expression. Equation (4) can be derived by integrating as follows:

[\begin{matrix} {∆ i}_{d} \\ {∆ i}_{q} \end{matrix}] = g_{c} (s) [\begin{matrix} {∆ i}_{d, r e f} \\ {∆ i}_{q, r e f} \end{matrix}] + y_{i} (s) [\begin{matrix} {∆ v}_{d} \\ {∆ v}_{q} \end{matrix}]

(4)

where

\{\begin{matrix} g_{c} (s) = \frac{k_{P, i} s + k_{I, i}}{L s^{2} + k_{P, i} s + k_{I, i}}, \\ y_{i} (s) = \frac{s^{2} (s + 2 ζ ω_{0})}{(L s^{2} + k_{P, i} s + k_{I, i}) (s^{2} + 2 ζ ω_{0} s + ω_{0}^{2})} \end{matrix} .

(5)

As an admittance of any device is the amount of change in current when the unit changes in voltage, if the

∆ i_{r e f}

can be substituted by

∆ i

and

∆ v

in Equation (4), the equivalent admittance of the VSC can be derived.

3.2. Constant Active Power Control

In VSCs executing constant active power control, obtaining the relationship between

∆ i_{r e f}

and

∆ v

is straightforward because

i_{d, r e f}

directly controls the active power. Active power, expressed as the product of

|v|

and

i_{d}

, can derive

∆ i_{d, r e f}

by dividing a fixed

P_{r e f}

by the measured

|v|

. Measured voltages rather than actual voltages are utilized when calculating references, necessitating the inclusion of a second-order filter,

F i l t e r (s)

, expressed as Equation (6).

i_{d, r e f} = \frac{P_{r e f}}{F i l t e r (s) v_{d}}

(6)

Here,

v_{d}

is used instead of

|v|

because in dq-axes control, the reference phase angle used in the synchronous reference frame transformation is from the angle of

v

, aiming to set

v_{q}

to zero. Thus,

|v|

and

v_{d}

have almost the same magnitudes in the steady-state case. Calculating the small-signal component of Equation (6) yields Equation (7), an expression of

{∆ i}_{d, r e f}

in the form of

∆ v

, which can be substituted into Equation (4) to replace

{∆ i}_{d, r e f}

{∆ i}_{d, r e f} = - \frac{P_{r e f}}{v_{d}^{2}} F i l t e r (s) {∆ v}_{d}

(7)

The formula for

{∆ i}_{q, r e f}

resembles that of

{∆ i}_{d, r e f}

. The reactive power, expressed as the negative value of the product of

|v|

and

i_{q}

, yields Equation (8).

i_{q, r e f} = - \frac{Q_{r e f}}{F i l t e r (s) v_{d}}

(8)

The small-signal component can be expressed as shown in Equation (9).

{∆ i}_{q, r e f} = \frac{Q_{r e f}}{v_{d}^{2}} F i l t e r (s) {∆ v}_{d}

(9)

By substituting Equations (7) and (9) into Equation (4), we can derive Equation (10), which represents a linear relationship between

∆ i

and

∆ v

[\begin{matrix} {∆ i}_{d} \\ {∆ i}_{q} \end{matrix}] = [\begin{matrix} y_{i} (s) - \frac{P_{r e f}}{v_{d}^{2}} F i l t e r (s) g_{c} (s) & 0 \\ \frac{Q_{r e f}}{v_{d}^{2}} F i l t e r (s) g_{c} (s) & y_{i} (s) \end{matrix}] [\begin{matrix} {∆ v}_{d} \\ {∆ v}_{q} \end{matrix}]

(10)

This equation illustrates how changes on one side lead to variations in the converter response expressed in the admittance form.

3.3. Constant DC Voltage Control

In contrast, in VSCs executing constant DC voltage control, obtaining the relationship between

{∆ i}_{r e f}

and

∆ v

is more sophisticated as

i_{d, r e f}

is not directly controlled by active power. In DC voltage control,

i_{d, r e f}

is calculated using Equation (11), where

k_{P, d c}

and

k_{I, d c}

represent the PI gains of the DC voltage control loop.

i_{d, r e f} = (k_{P, d c} + \frac{k_{I, d c}}{s}) (v_{d c, r e f} - v_{d c})

(11)

When computing the small-signal representation in this equation,

v_{d c, r e f}

remains at a fixed value, resulting in Equation (12).

{∆ i}_{d, r e f} = - (k_{P, d c} + \frac{k_{I, d c}}{s}) {∆ v}_{d c}

(12)

Then, the conversion of

{∆ v}_{d c}

∆ v

is necessary, accomplished through expressions involving the apparent power and DC voltage. First, the expression for the apparent power from the perspective of the AC grid is represented in Equation (13).

S = v i^{*} = v_{d} (i_{d} - j i_{q})

(13)

The small-signal component of active power can be derived as Equation (14).

\begin{matrix} S + ∆ S = (v_{d} + {∆ v}_{d}) (i_{d} + {∆ i}_{d}) - j (v_{d} + {∆ v}_{d}) (i_{q} + {∆ i}_{q}) \\ \Rightarrow ∆ S \approx (v_{d} {∆ i}_{d} + i_{d} {∆ v}_{d}) - j (v_{d} {∆ i}_{q} + i_{q} {∆ v}_{d}) \\ \Rightarrow ∆ P \approx v_{d} {∆ i}_{d} + i_{d} {∆ v}_{d} \end{matrix}

(14)

Active power can also be derived from the energy equation of the DC capacitor and its derivative from the perspective of a DC grid.

\begin{matrix} W_{d c} = \frac{1}{2} C_{d c} v_{d c}^{2} \\ \Rightarrow \frac{d W_{d c}}{d t} = P_{d c} = C_{d c} v_{d c} \frac{d v_{d c}}{d t} \end{matrix}

(15)

The small-signal component will then be calculated as Equation (16).

∆ P_{d c} = C_{d c} v_{d c} \frac{d {∆ v}_{d c}}{d t} = s C_{d c} v_{d c} {∆ v}_{d c}

(16)

The terms

∆ P

in Equation (14) and

∆ P_{d c}

in Equation (16) both signify the same value. Combining these equations allows us to derive a final expression for

{∆ v}_{d c}

in terms of

{∆ i}_{d}

and

{∆ v}_{d}

, suitable for substitution for Equation (12).

{∆ v}_{d c} = \frac{F i l t e r (s) v_{d} {∆ i}_{d} + i_{d} {∆ v}_{d}}{s C_{d c} v_{d c}}

(17)

We obtain the following expression after incorporating the calculated

{∆ i}_{d, r e f}

from Equation (12) in terms of

{∆ i}_{d}

and

{∆ v}_{d}

into the upper equation of Equation (4).

{∆ i}_{d} = - g_{c} (s) (k_{P, d c} + \frac{k_{I, d c}}{s}) \frac{v_{d} {∆ i}_{d} + i_{d} {∆ v}_{d}}{s C_{d c} v_{d c}} + y_{i} (s) {∆ v}_{d}

(18)

Equation (18) requires separating two terms, each associated with

{∆ i}_{d}

and

{∆ v}_{d}

, to compute the gradients of

{∆ i}_{d}

and

{∆ v}_{d}

{∆ i}_{d} = \frac{- g_{c} (s) (k_{P, d c} + \frac{k_{I, d c}}{s}) i_{d} + s C_{d c} v_{d c} y_{i} (s)}{s C_{d c} v_{d c} + g_{c} (s) (k_{P, d c} + \frac{k_{I, d c}}{s}) v_{d}} {∆ v}_{d}

(19)

No notable alterations were observed in the q-axis control or constant reactive power control, permitting their continued use in the same manner as before. Thus, Equation (20) provides the final relationship between

∆ i

and

∆ v

[\begin{matrix} {∆ i}_{d} \\ {∆ i}_{q} \end{matrix}] = [\begin{matrix} \frac{- g_{c} (s) (k_{P, d c} + \frac{k_{I, d c}}{s}) i_{d} + s C_{d c} v_{d c} y_{i} (s)}{s C_{d c} v_{d c} + g_{c} (s) (k_{P, d c} + \frac{k_{I, d c}}{s}) v_{d}} & 0 \\ \frac{Q_{r e f}}{v_{d}^{2}} F i l t e r (s) g_{c} (s) & y_{i} (s) \end{matrix}] [\begin{matrix} {∆ v}_{d} \\ {∆ v}_{q} \end{matrix}]

(20)

4. Perturb-and-Observe-Based Impedance Identification

Figure 3 illustrates the measurement of the VSC impedance using the perturb-and-observe method. This method initially injects a perturbing current source into a connection bus under steady-state conditions. Here, each VSC operates independently, so measurements are taken while disregarding the operation of the other. By modulating the frequency of this current source

f_{p}

, the frequency-dependent total admittance of the VSC and the connection system combination can be obtained by measuring the voltage fluctuations [20].

Considering the VSC is controlled using dq-axes, the injected current should be sine waves in dq-axes. Changes in the currents and voltages compared with those in the steady-state situation allow for the calculation of the total admittance at that VSC operating point, as expressed in Equation (21).

[\begin{matrix} {∆ i}_{d} \\ {∆ i}_{q} \end{matrix}] = Y_{d q, t o t a l}^{m} [\begin{matrix} {∆ v}_{d} \\ {∆ v}_{q} \end{matrix}]

(21)

As the total admittance has four variables and only two equations exist in Equation (21), two independent perturb-and-observe cases are required to calculate one admittance matrix. Two instances can use one sine wave in the d-axis and zero in the q-axis case and the other sine wave in the q-axis and zero in the d-axis case. In Equation (22), the variables in the first and second cases are denoted by subscripts 1 and 2, respectively.

Y_{d q, t o t a l}^{m} = [\begin{matrix} Y_{d d, t o t a l} & Y_{d q, t o t a l} \\ Y_{q d, t o t a l} & Y_{q q, t o t a l} \end{matrix}] = {[\begin{matrix} {∆ v}_{d 1} & ∆ v_{d 2} \\ {∆ v}_{q 1} & {∆ v}_{q 2} \end{matrix}]}^{- 1} [\begin{matrix} {∆ i}_{d 1} & {∆ i}_{d 2} \\ {∆ i}_{q 1} & ∆ i_{q 2} \end{matrix}]

(22)

Since

Y_{d q, t o t a l}^{m}

represents the combined admittance of the VSC and the connection system, it is necessary to separate each admittance to obtain the pure VSC admittance. This separation can be achieved using Equation (23), where

| |

denotes a harmonic mean.

\begin{matrix} Y_{d q, t o t a l}^{m} = Y_{d q, s y s t e m}^{m} | | Y_{d q, V S C}^{m} \\ \Rightarrow Y_{d q, t o t a l}^{m} = [\begin{matrix} Y_{d d, s y s t e m} | | Y_{d d, V S C} & Y_{d q, s y s t e m} | | Y_{d q, V S C} \\ Y_{q d, s y s t e m} | | Y_{q d, V S C} & Y_{q q, s y s t e m} | | Y_{q q, V S C} \end{matrix}] \end{matrix}

(23)

The magnitude of the perturbing current source is crucial in the perturb-and-observe method. If the perturbation current is too large, it may influence the operating point of the VSC, leading to the calculation of the admittance at different operating points. By contrast, a too-small perturbing current source makes it challenging to differentiate between the impact and noise, such as harmonic distortion in the VSC. Therefore, this study used a perturbing current equivalent to 5% of the steady-state current, as suggested in [6], to mitigate these concerns.

These two perturbations serve only one admittance matrix. Ensuring a trustworthy impedance model requires a range of admittance matrices that consider the perturbation frequencies and various operating points corresponding to the

P_{r e f}

. Consequently, numerous admittance matrices are required, highlighting the need for methods to craft an adequate impedance model with a minimal number of matrices.

5. Proposed Impedance and Stability Estimation Model

5.1. Physics-Informed Neural Network-Based Impedance Model of VSC

The structural overview of the proposed PINN model used to estimate the VSC impedance is shown in Figure 4. The model takes the perturbation frequency and representative values of

|v_{s}|

, the power factor, and

P_{r e f}

characterizing the grid state and VSC operating point as inputs. The inputs were used to calculate

|v_{s}|

i_{d}

, and

i_{q}

. Inputs are divided into two categories: inputs specific to each converter and inputs shared between both converters. Subsequently, an ANN was designed to generate the four elements of the VSC admittance in both magnitude and phase angle.

In a Physics-Informed Loss Function-based PINN, the topology of the neural network model is exactly the same as that of the conventional ANN method. The distinctiveness of the PINN emerges during training, enabling the creation of more realistic models with fewer data points. Conventional ANN training typically involves adjusting the hyperparameters to minimize errors between the calculated admittance data,

y_{t a r g e t}

, and the output of the ANN,

y_{N N}

, at fixed inputs

|v_{s}|

i_{d}

i_{q}

, and perturbation frequency; this function is known as Data Loss. Mean squared error, commonly represented by Equation (24), is often used as an error metric in the Data Loss function.

\min_{hyperparameter} \sum_{o u t p u t} {(y_{t a r g e t} - y_{N N})}^{2}

(24)

In a PINN model, physical equations play a crucial role in the training process. The errors from these equations are incorporated into the learning process using the expected admittance-like equations previously calculated in Equations (10) and (20). This integration of physical equations allows for training on the accuracy of model predictions and the accuracy of incremental changes in model predictions, thus facilitating the creation of a more rapid and accurate admittance model. The error stemming from using physical equations is referred to as Physical Loss. The derivative equations calculated from the control diagram, denoted by

\frac{d y}{d l}

, corresponding to the adopted control strategies of each VSC, can be employed in this approach in the form of Equation (25).

\underset{h y p e r p a r a m e t e r}{m i n} \sum_{o u t p u t} ({(y_{t a r g e t} - y_{N N})}^{2} + \sum_{l = i n p u t} {({\frac{d y}{d l}|}_{c o n t r o l} - \frac{{d y}_{N N}}{d l})}^{2})

(25)

Moreover, this approach of PINN simply substitutes values into pre-differentiated equations during the learning process, requiring only preparatory setup without a significant increase in training time due to added loss reflection.

5.2. Neural Network-Based Stability Estimation Model of the Back-to-Back HVDC

Once the impedance model tailored to the individual VSC’s operating points and frequencies is trained, it can be leveraged to develop a final stability estimation model using neural networks (Figure 5). The final model used PINN architectures tailored to each VSC’s corresponding controls to train the grid stability estimation model. Combined with the calculated equipment-specific impedances and present grid conditions, this approach enables the design of a neural network-based stability estimation. The stability estimation incorporated grid conditions, which included the SCR values of the individual AC systems and the electrical distances,

d_{e l e c}

, at the connection points of the VSC back-to-back HVDC, totaling three inputs. SCR values indicate each AC system’s strength, and

d_{e l e c}

implies possible interactions and their impact on the grid.

6. Case Study and Validation

In this section, we validate both the proposed PINN-based impedance model of the VSC and the ANN-based stability estimation model of the VSC back-to-back HVDC system through simulations using PSCAD/EMTDC. The similarity of the PINN structure to the actual impedance is compared with that of the ANN model. Additionally, the stability estimation model is tested under various grid situations. Its performance is evaluated using a confusion matrix.

6.1. PINN-Based Impedance Model Validation

The impedance model was validated using two three-phase VSCs in a single back-to-back HVDC system. These VSCs operate at 230 kV AC with a capacity of 1200 MVA. In the PSCAD/EMTDC simulation environment, the active power output of the VSCs was varied from 200 MW to 1000 MW in increments of 20 MW. A perturbation dq-axes current ranging from 20 Hz to 100 Hz in 2 Hz increments was applied at each power level. The perturb-and-observe method was then used to measure the admittance. The grid configuration for these simulations utilized VSCs performing control, as depicted in Figure 2, and the perturb-and-observe method shown in Figure 3 was used for the measurements. The parameters for each VSC are detailed in Table 1.

The measured admittance matrices, totaling 861 (21 active power levels and 41 frequencies per VSC), were randomly divided for training the ANN and PINN models. Approximately 70% of the matrices (603) were used for the training set, 15% (129) for the validation set, and the remaining 15% (129) for the test set. These sets were used to train ANN and PINN models of the same size to develop the MOP admittance model, as illustrated in Figure 6.

Figure 6 presents four MOP admittance models in each subplot: the plane transitioning from yellow to green represents the admittance calculated from the admittance-like equation, the blue scatter points represent the admittance measured through PSCAD/EMTDC simulations, the red plane represents the MOP admittance model learned through ANN, and lastly the green plane represent those learned through PINN. In subplots (b) and (f), the plane for the admittance from the admittance-like equation is absent because the admittance from the equation is zero, resulting in a negative infinite value on the log scale.

Both ANN and PINN models predict admittance models closely match the measured admittance points, which serve as the training reference. This indicates that the models developed using these methods are highly accurate. The close resemblance of the admittance-like equation represented by the transitioning plane to the blue scatter points highlights its effectiveness in aiding the training of the PINN model. This consistency demonstrates that the admittance-like equation significantly enhances the learning process.

However, as seen in Figure 7, although the plane out of the PINN and the ANN appear very similar visually, the average data loss, which is the mean squared error with respect to the measured admittance data, reveals a significant difference. The PINN starts with less than half the error of the ANN from the first epoch and achieves a final error of

0.086 \times 10^{- 45}

, compared to

0.226 \times 10^{- 45}

for the ANN. This indicates that the accuracy of the PINN is much higher. Given that these results were obtained using the same data and neural network size, it suggests that using PINN can create an MOP admittance model with similar performance to that of ANN but with less data.

6.2. Neural Network-Based Stability Estimation Model Validation

To validate the neural network-based stability estimation model, the VSC back-to-back HVDC system, with the same specifications as the equipment used in Section 6.1, is configured, as illustrated in Figure 8. The SCRs of each AC system and the impedance linking both ends of the VSC are randomly varied, resulting in a total of 1000 stability simulation tests through PSCAD/EMTDC. This approach ensures the collection of comprehensive data under various grid conditions. Specifically, the SCRs are randomly adjusted between 1 and 30 to simulate both stable and unstable grid scenarios. Frequency sweep current perturbations are then applied to check whether the grid is stable or not. These data are utilized to train the neural network-based stability estimation model.

Subsequently, the performance of the model is validated using the IEEE-39 bus system (Figure 9). The line connecting buses 15 and 16 is replaced with a VSC back-to-back HVDC system to assess grid stability. Various grid conditions are simulated by altering load and generation levels, VSC operating points, and the line parameters of certain transmission lines. To identify issues across different frequency ranges, current perturbation sweeps are injected near the HVDC system to check grid stability. The results are then compared with those of the neural network-based stability estimation model, as shown in Table 2.

The stability estimation model achieved an accuracy of 93%, demonstrating a high level of precision. This indicates that the stability estimation model is both valid and reliable. The high accuracy underscores the effectiveness of the neural network-based approach in predicting grid stability under varying grid conditions.

7. Conclusions

This study addresses the challenges of validating impedance models and conducting stability analysis for VSC back-to-back HVDC systems, which are essential for maintaining grid stability amid the increasing integration of renewable energy sources. Using two three-phase VSCs, we validated an MOP impedance model under varying active power outputs and perturbation current frequency, obtaining detailed admittance data. These data are then used to train both ANN and PINN models, demonstrating that PINN models achieve high accuracy. The stability estimation model involves extensive testing with random variations in SCRs and connection impedance, yielding a stability model with high accuracy, thereby confirming its reliability and effectiveness. The high precision and robustness of the neural network-based models underscore their potential in complex power system stability assessments.

However, this study has several limitations. The simulations are conducted only in computer simulations, which may not fully capture real-world complexities. The models are tested on specific configurations, potentially limiting their generalizability. Future work should focus on validating these models under more realistic grid conditions using hardware-in-the-loop tests and extending the validation to other HVDC system configurations.

Author Contributions

Conceptualization, M.C. and Y.J.; methodology, M.C.; software, M.C. and S.K.; validation, Y.J. and G.J.; data curation, M.C.; writing, M.C.; visualization, M.C.; supervision, G.J.; project administration, G.J.; funding acquisition, G.J. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2023-00218377). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government (MOTIE) (No. 20210501010010).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. VSC back-to-back HVDC schematic diagram.

Figure 2. VSC back-to-back HVDC control diagram.

Figure 3. Perturb-and-observe-based impedance identification.

Figure 4. Proposed physics-informed neural network-based impedance model.

Figure 5. Proposed neural network-based stability estimation model of VSC back-to-back HVDC.

Figure 6. Magnitude of

Y_{d d}

Y_{d q}

Y_{q d}

, and

Y_{q q}

for VSC-1 (a–d) and VSC-2 (e–h).

Figure 6. Magnitude of

Y_{d d}

Y_{d q}

Y_{q d}

, and

Y_{q q}

for VSC-1 (a–d) and VSC-2 (e–h).

Figure 7. Average neural network data loss.

Figure 8. Neural network-based stability estimation model training setup.

Figure 9. Modified IEEE-39 bus with VSC back-to-back HVDC.

Table 1. VSC back-to-back HVDC parameters.

Symbol	Description	VSC-1 (Const. $v_{d c}$ )	VSC-2 (Const. $P_{r e f}$ )
$Q_{r e f}$	Reference reactive power	−100 MVar	−100 MVar
$k_{P, i}$	Proportional gain of the current controller	8	-
$k_{I, i}$	Integral gain of the current controller	10	-
$k_{P, d c}$	$Proportional gain of the v_{d c}$ controller	0.65	0.65
$k_{I, d c}$	$Integral gain of the v_{d c}$ controller	40	40
$v_{d c}$	Reference DC voltage	640 kV	-
$R$	System resistance	1 Ω	1 Ω
$L$	System inductance	0.05 H	0.05 H
$ζ$	Filter damping coefficient	0.7071	0.7071
$ω_{0}$	Filter cutoff frequency	1000 Hz	1000 Hz

Table 2. Confusion matrix of the stability model.

Actual	Stable	Unstable
Prediction	Stable	Unstable
Stable	802	41
Unstable	29	128

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Chang, M.; Jung, Y.; Kang, S.; Jang, G. Physics-Informed Neural Network-Based VSC Back-to-Back HVDC Impedance Model and Grid Stability Estimation. Electronics 2024, 13, 2590. https://doi.org/10.3390/electronics13132590

AMA Style

Chang M, Jung Y, Kang S, Jang G. Physics-Informed Neural Network-Based VSC Back-to-Back HVDC Impedance Model and Grid Stability Estimation. Electronics. 2024; 13(13):2590. https://doi.org/10.3390/electronics13132590

Chicago/Turabian Style

Chang, Minhyeok, Yoongun Jung, Seokjun Kang, and Gilsoo Jang. 2024. "Physics-Informed Neural Network-Based VSC Back-to-Back HVDC Impedance Model and Grid Stability Estimation" Electronics 13, no. 13: 2590. https://doi.org/10.3390/electronics13132590

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Physics-Informed Neural Network-Based VSC Back-to-Back HVDC Impedance Model and Grid Stability Estimation

Abstract

1. Introduction

2. VSC Back-to-Back HVDC Model Configuration

3. VSC Back-to-Back HVDC Small-Signal Impedance-like Equation

3.1. Control-Design-Based Small-Signal Impedance-like Equation

3.2. Constant Active Power Control

3.3. Constant DC Voltage Control

4. Perturb-and-Observe-Based Impedance Identification

5. Proposed Impedance and Stability Estimation Model

5.1. Physics-Informed Neural Network-Based Impedance Model of VSC

5.2. Neural Network-Based Stability Estimation Model of the Back-to-Back HVDC

6. Case Study and Validation

6.1. PINN-Based Impedance Model Validation

6.2. Neural Network-Based Stability Estimation Model Validation

7. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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