A Multi-Timescale Characteristics Analysis of the Network Power Response Excited by Voltage with Time-Varying Amplitude and Frequency

Jiabing Hu; Weizhong Wen; Yingbiao Li; Xing Liu; Jianbo Guo

doi:10.1016/j.eng.2024.12.015

Engineering ›› 2025, Vol. 51 ›› Issue (8) :52 -65. DOI: 10.1016/j.eng.2024.12.015

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A Multi-Timescale Characteristics Analysis of the Network Power Response Excited by Voltage with Time-Varying Amplitude and Frequency

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Abstract

The dynamics of network power response play a crucial role in system stability. However, the integration of power electronic equipment leads to amplitude and angular frequency (abbreviated as “frequency”) time-varying characteristics of the node voltage during dynamic processes. As a result, traditional calculation methods for and characteristics of the power response of the network based on phasor and impedance lose their validity. Therefore, this paper undertakes mathematical calculations to reveal the power response of a network under excitation by voltage with time-varying amplitude and frequency (TVAF), relying on the original mathematical relationships and superimposed step response. Then, the multi-timescale characteristics of both the active and reactive power of the network are explored physically. Additionally, this paper reveals a new phenomenon of storing and releasing the active and reactive power of the network. To meet practical engineering requirements, a simplified power expression is presented. Finally, the theoretical analysis is validated through time-domain simulations.

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Power response of network / Voltage with time-varying amplitude and frequency / Multi-timescale characteristics / Power electronic equipment

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Jiabing Hu, Weizhong Wen, Yingbiao Li, Xing Liu, Jianbo Guo. A Multi-Timescale Characteristics Analysis of the Network Power Response Excited by Voltage with Time-Varying Amplitude and Frequency. Engineering, 2025, 51(8): 52-65 DOI:10.1016/j.eng.2024.12.015

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1. Introduction

Renewable energy power generation and its grid-connected electronic equipment have gained prominence in the power system. In some areas, the proportion of renewable energy generation has surpassed that of traditional synchronous generators (SGs) [1]. However, this transformation of physical objects in the power system has brought about stability issues with unclear mechanisms [2], [3]. Starting from the first principles of power system operation, the dynamic characteristics of the system are jointly determined by the internal voltage dynamics of the equipment and the power dynamics of the network under internal voltage excitation. However, existing research mostly focuses on equipment characteristics and neglects analysis of the power dynamics of the network response. Therefore, it is necessary to conduct a thorough analysis of the power response of the network under the excitation of power electronic equipment.

In a traditional power system, most of the pieces of power equipment are SGs. Due to their substantial rotor inertia, the rotor speed of SGs can be assumed to be constant during dynamic processes. Thus, the amplitude and angular frequency (abbreviated as “frequency”) of the internal voltage provided by SGs can also be considered constant in the system dynamic process. In traditional power system stability analyses, impedance and phasor-based network power calculation methods have been widely applied. However, because of the increasing proportion of grid-connected power electronic equipment, converter-based resources will determine the characteristics of the amplitude and frequency of the node voltage in a power–electronics-dominated power system. Significantly different from SGs, the low inertia characteristics of converter-based resources lead to time-varying amplitude and frequency (TVAF) characteristics of the internal voltage during dynamic processes [4], [5]. For example, in a 110-kV artificial asymmetric short-circuit test of a wind power base in Xinjiang Uygur Autonomous Region (China), the amplitude and frequency of the fault current presented significant time-varying characteristics, as shown in Fig. 1(a) [6]. Similarly, at a renewable energy base in Zhangbei County, Hebei Province (China), shown in Fig. 1(b), a 6-Hz disturbance induced noticeable dynamic phenomena in both the amplitude and frequency of the network voltage. Thus, the time-varying characteristics of amplitude and frequency are receiving increasing attention [7]. In other words, the TVAF features of the internal voltage are non-negligible during the system’s dynamic processes. Furthermore, the TVAF of the internal voltage induces dynamic changes in the active/reactive power within the network. These changes, in turn, provide feedback to the power electronic equipment, resulting in further intricate dynamic alterations to the internal voltage.

The power-response characteristics of the network have undergone significant changes in comparison with traditional power systems. However, traditional calculation methods for the network power response based on the phasor and impedance models have become ineffective, as the amplitude and frequency of the internal voltage are no longer constant. In conclusion, for a power–electronics-dominated power system, recalculating and analyzing the network power response under voltage excitation with TVAF holds great significance in system dynamic analysis.

From the analysis above, it is evident that the traditional quasi-steady-state assumption does not hold for a dynamic analysis of power–electronics-dominated power systems [8], [9]. Although the dynamic phasor method based on Fourier series can capture amplitude and frequency dynamics using frequency domain components, its applicability is limited to periodic signals, and the physical interpretation of the obtained current and power lacks clarity [10], [11]. Similarly, in Ref. [12], the internal voltage is decomposed into the superposition form of the frequency components by the Bessel function to obtain network power. The method is also limited to periodic signals and cannot directly reflect the influence of the dynamic amplitude/frequency of the internal voltage on the power response. In Ref. [13], phasor and vector models with TVAF are presented but are limited to the description of electrical quantities. On this basis, in Ref. [14], the integration-by-parts method is employed to obtain the network current and power response of the voltage with TVAF. However, due to the difficulty of calculating quadratic and high-order terms, the method in Ref. [14] is more suitable for dynamic analyses with an oscillation frequency of less than 10 Hz and lacks a comprehensive understanding of the influence of different nodes on the branch power response. It is evident that existing methods fail to meet the requirements of dynamic analysis in power–electronics-dominated power systems. In addition, the characteristics of the power response have not been focused on in overall research. Therefore, this paper focuses on the characteristics of the network power response excited by voltage with TVAF. The main contributions of this paper are as follows:

(1) The multi-timescale characteristics of the network power response excited by voltage with TVAF are revealed both mathematically and physically.

(2) A new phenomenon of the storage and loss of active and reactive power in inductance networks excited by voltage with TVAF is discovered.

The remainder of this paper is organized as follows. First, the response current of a one-node network is described using a superimposed step response method based on the original mathematical relationships, and its time-varying characteristics are analyzed. Then, the power transfer characteristics and multi-timescale characteristics are revealed both mathematically and physically. The influence of voltage amplitude and frequency on the multi-timescale network power is clarified. Moreover, a simplified expression of the current and active/reactive power is provided for ease of practical engineering application. Subsequently, the power response of the complex network is calculated, and the power components and characteristics are analyzed. Finally, case studies are provided to verify the mathematical derivation and theoretical analysis.

2. Current response of a one-node inductance network

To investigate the network power, it is necessary to first reveal the characteristics of the network current response. In an actual power system, the forms of networks are complex and varied. However, complex networks can be simplified into equivalent networks in the form of Fig. 2, utilizing the star-delta network transformation and lumped parameter model [15]. According to the superposition theorem, the current response of a complex network is simplified as the superposition of a single branch current excited by the voltage of the nodes at both sides. Thus, to analyze the network power response, it is imperative to calculate the branch current under the excitation of a single voltage.

Due to the amplitude and frequency dynamics of the voltage, the phasor model and the impedance model are no longer applicable. Under these conditions, solving the current response based on the mathematical relationship between the voltage and current in the network element becomes challenging. As shown in Eq. (1), the oscillatory integral in the current response expression cannot be mathematically solved using a finite and exact expression, which is a problem that has been known in applied mathematics for hundreds of years [14], [16]. Therefore, the superimposed step response method is improved to calculate the current response [17].

(1)

I_{L} (t) = \frac{1}{L} \int \{E (t) \cos [\int ω (t) d t + φ]\} d t

where E(t) and ω(t) are the amplitude and frequency, and t is time; φ is initial phase angle; I_L(t) is current response of inductance network and L is the inductance value of the network.

2.1. Mathematical expression of the current response

2.1.1. Vector model of the voltage

Vector modal is a fundamental mathematical model that enhances the efficiency of calculations. The vector model with TVAF is established in Ref. [13]. The vector model of the voltage, E(t), rotates in the abc coordinate plane, whose length and rotational velocity are time-varying. Its mathematic expression is obtained as follows:

(2)

E (t) = E (t) e^{j [\int ω (t) d t + φ]} = E (t) e^{j θ (t)}

where j is imaginary unit; θ(t) is time-varying phase angle of the voltage.

(3)

\{\begin{array}{l} E (t) = E_{0} + υ (t) \\ ω (t) = ω_{0} + μ (t) \end{array}

in which E₀ and ω₀ are the direct current (DC) component of the amplitude and frequency, respectively; and υ(t) and μ(t) are the time-varying component of the amplitude and frequency, respectively.

2.1.2. Mathematical calculations

For the alternative network, under the excitation of a step voltage E·ε(t − τ) at time τ, where ε(t − τ) is the Heaviside function, the response current i(t) is expressed as follows:

(4)

i (t) = E \cdot A (t - τ)

where A(t − τ) is the ratio of the voltage to the network current and can be termed the “indicial admittance” [17].

Fig. 3 shows an alternative voltage waveform, e(t). The initial voltage is e(0). After each time step, Δt, the step voltage with amplitude Δ_me is superimposed based on the original voltage to form a complete e(t) waveform until time t (where m is step number). Thus, the response current can be expressed as

(5)

i (t) = \lim_{Δ t \to 0} \sum_{m = 0}^{\infty} Δ_{m} e \cdot A (t - m Δ t) = \lim_{Δ t \to 0} \sum_{m = 0}^{\infty} e^{'} (m Δ t) \cdot A (t - m Δ t) \cdot Δ t = \int_{0}^{\infty} e^{'} (τ) A (t - τ) d τ = \int_{0}^{\infty} e (τ) A^{'} (t - τ) d τ

in which the theory of the integral is used to transform the infinite series to an integral expression, as follows:

(6)

e^{'} (m Δ t) \cdot A (t - m Δ t) = f (m Δ t)

(7)

\lim_{Δ t \to 0} \sum_{m = 0}^{\infty} f (m Δ t) Δ t = \int_{0}^{\infty} f (τ) d τ

where e′ is the differential of the voltage, A′ is the differential of the indicial admittance, and f(·) represents for a function.

Since it does not involve the specific waveform of the excitation voltage and the voltage/current relationship of specific elements, Eq. (5) is applicable for calculating the response current of a network under any form of excitation. Therefore, the network current response I(t) can be obtained by bringing Eq. (2) into Eq. (5), as follows:

(8)

I (t) = [\int_{0}^{\infty} H (t, τ) e^{- j ω_{0} τ} A^{'} (τ) d τ] \cdot E (t) e^{j [\int_{0}^{t} ω (τ_{1}) d τ_{1} + φ]}

in which

(9)

H (t, τ) = [E (t - τ) / E (t)] e^{- j \int_{0}^{τ} μ (t - τ_{1}) d τ_{1}}

where H(t,τ) is the Taylor series expansion and performed at τ = 0. Then, Eq. (8) can be transformed into

(10)

I (t) = E (t) [Y (j ω_{0}) + \frac{1}{E (t)} \sum_{n = 1}^{\infty} \frac{{(- j)}^{n}}{n!} G_{n} (t) {\frac{d^{n} Y (j ω)}{d ω^{n}}|}_{ω = ω_{0}}]

where Y(·) is steady-state admittance, n represents the order of the Taylor series expansion, n∈N⁺. G_n(t) can be expressed by the time-varying terms of amplitude and frequency of E(t) and its high-order differentials, as follows:

(11)

\{\begin{array}{l} G_{n} (t) = \sum_{k = 0}^{n} j^{k} C (n, k) \cdot M_{k} (t) \frac{d^{n - k} E (t)}{d t^{n - k}} \\ M_{k} (t) = \{\begin{matrix} 1 (k = 0) \\ μ (t) (k = 1) \\ [μ (t) - j \frac{d}{d t}] M_{k - 1} (t) (k = 2, 3, 4 \dots) \end{matrix} \end{array}

where C(n,k) is the binomial coefficient and k∈N; M_k(t) is an intermediate transition term containing the voltage’s time-varying frequency and its differentials.

In Eq. (10), Y(jω₀) can be obtained by bringing the sinusoidal steady-state voltage vector, Eq. (12), into Eq. (5). The current expression obtained is shown as Eq. (13).

(12)

E = E e^{j (ω_{0} t + φ)}

(13)

I = [\int_{0}^{\infty} e^{- j ω_{0} τ} A^{'} (τ) d τ] \cdot E e^{j (ω_{0} t + φ)}

Using Eq. (13), a new form of steady-state admittance can be obtained by the definition of impedance model shown in Eq. (14).

(14)

Y (j ω_{0}) = \int_{0}^{\infty} e^{- j ω_{0} τ} A^{'} (τ) d τ

In the inductance network, its form is obtained as follows:

(15)

Y (j ω_{0}) = \frac{1}{j ω_{0} L}

Based on the method proposed above, differential equations can be transformed into algebraic equations in an infinite series form for an approximate solution.

2.2. Physical characteristics of the current response

A single-node three-phase system with an equivalent inductance branch is shown in Fig. 4, which is the basic structure in Fig. 2.

The internal voltage E(t) excites the network to generate a response current I(t). By bringing Eq. (15) into Eq. (10), the current response in Fig. 4 can be obtained in the form of Eq. (16).

(16)

I (t) = \sum_{n = 0}^{\infty} I_{n} (t) = \sum_{n = 0}^{\infty} \{\frac{Im [j^{n} G_{n} (t)]}{ω_{0}^{n + 1} L} + \frac{Re [j^{n} G_{n} (t)]}{j ω_{0}^{n + 1} L}\} e^{j θ (t)}

where Im[·] represents taking the imaginary part, Re[·] represents taking the real part.

The vector diagram of I(t) in Eq. (16) is depicted in Fig. 5(a). The initial term of current is named I₀(t). With increasing order of Taylor series expansion, subsequent current terms are labeled from I₁(t) to I_n(t). Each current can be divided into two terms, which are respectively perpendicular and parallel to the voltage vector, as shown in Eq. (16). The concept of “perpendicular or orthogonal” mentioned in this paper refers to two vectors that maintain a phase angle difference of 90° at any given time. Similarly, the concept of “parallel” refers to two vectors that maintain a phase angle difference of 0/180° at any given time. This concept applies equally to both voltage/current with TVAF and sinusoidal quantities, independent of the time-varying characteristics of the amplitude and frequency of the vector.

The current vector diagram after classification is shown in Fig. 5(b). In relation to the spatial position of E(t), I(t) contains two parts of the current vector:

(17)

\begin{matrix} I (t) = I_{pa} (t) + I_{pe} (t) = A_{pa} (t) e^{j θ (t)} + A_{pe} (t) e^{j [θ (t) - π / 2]} = \sqrt{A_{pa}^{2} (t) + A_{pe}^{2} (t)} \cdot e^{j [θ (t) - \arctan \frac{A_{pe} (t)}{A_{pa} (t)}]} \end{matrix}

where I_pa(t) is the sum of the vectors parallel to E(t), and I_pe(t) is the sum of the vectors perpendicular to E(t). The time-varying frequencies of I_pa(t) and I_pe(t) remain consistent with the frequency of E(t). In addition, the time-varying amplitudes of I_pa(t) and I_pe(t) are respectively A_pa(t) and A_pe(t), which are expressed as follows:

(18)

\{\begin{matrix} A_{pa} (t) = \sum_{n = 0}^{\infty} \frac{Im [j^{n} G_{n} (t)]}{ω_{0}^{n + 1} L} \\ A_{pe} (t) = \sum_{n = 0}^{\infty} \frac{Re [j^{n} G_{n} (t)]}{ω_{0}^{n + 1} L} \end{matrix}

According to Eqs. (17), (18), the response current of an inductance network under the excitation of voltage with TVAF has the following characteristics:

Firstly, the amplitude of the response current exhibits time-varying characteristics, which are related to the time-varying terms of amplitude/frequency, as well as their higher-order differentials, of the voltage.

Secondly, the frequency of the response current also varies with time. The current can be decomposed into components that are perpendicular and parallel to the voltage, both of which share the same instantaneous frequency with the voltage. The overall frequency is affected by the time-varying terms of amplitude and frequency, along with their higher-order differentials, of the voltage.

Thirdly, the angle between the voltage and the response current varies over time due to the time-varying characteristics of the amplitude of the perpendicular and parallel components of the current. As a result, the vectors of the voltage and the response current are no longer orthogonal.

The amplitude/frequency time-varying characteristics of the excitation voltage challenge the traditional understanding of the relationship between the voltage and response current in an inductance network. The changes in the phase angle between the voltage and current inevitably alter the system characteristics concerning the active and reactive power.

3. Multi-timescale characteristics of the power response in a one-node inductance network

To delve deeper into an analysis of the power characteristics of the network, the oscillation forms of the amplitude and frequency are considered to be sinusoidal modulations over time [14]:

(19)

\{\begin{matrix} E (t) = E_{0} + υ (t) = E_{0} + A_{E} \sin (ω_{E} t + φ_{E}) \\ ω (t) = ω_{0} + μ (t) = ω_{0} + A_{ω} \sin (ω_{ω} t + φ_{ω}) \end{matrix}

where A_E, ω_E, and φ_E are the amplitude, frequency, and initial phase angle of the time-varying component of the voltage amplitude, respectively, and A_ω, ω_ω, and φ_ω are the amplitude, frequency, and initial phase angle of the time-varying component of the voltage frequency, respectively.

When Eq. (19) is brought into Eq. (18), according to the properties of trigonometric functions, the frequencies of A_pa(t) and A_pe(t) comprise three types of components: ω_E, kω_ω, and (ω_E ± kω_ω), in which k∈N⁺. A_pa(t) and A_pe(t) can be formulated in the following form:

(20)

\{\begin{matrix} A_{pa} (t) = a_{E} \cos (ω_{E} t + φ_{k 1}) + \sum_{k = 1}^{\infty} a_{k ω} \cos (k ω_{ω} t + φ_{k ω}) + \sum_{k = 1}^{\infty} a_{k \pm} \cos [(ω_{E} \pm k ω_{ω}) t + φ_{n \pm}] \\ A_{pe} (t) = \frac{E_{0}}{ω_{0} L} + b_{E} \sin (ω_{E} t + φ_{k 1}) + \sum_{n = 1}^{\infty} b_{k ω} \sin (k ω_{ω} t + φ_{k ω}) + \sum_{k = 1}^{\infty} b_{k \pm} \sin [(ω_{E} \pm k ω_{ω}) t + φ_{k \pm}] \end{matrix}

where a_E, a_kω, a_k±, b_E, b_kω, b_k±, φ_k₁, φ_kω, and φ_n± are time-invariant coefficients.

3.1. A mathematical expression of the network power

In this work, we conduct research on the power characteristics of three-phase networks. The active power P(t) represents the portion of energy transmitted from the electrical equipment to the network, which is subsequently converted by the load into other forms of energy. The reactive power Q(t) represents the energy used in establishing a magnetic field and maintaining a three-phase dynamic balance, which is the energy transmitted within the network, as illustrated in Fig. 6 [18]. Thus, the current flowing in the network consists of two components: The first is the active current I_P(t), which transfers energy between the electrical equipment and the network. The corresponding power of this part is the same as the instantaneous power of the three-phase system. The second is the reactive current I_Q(t), which exchanges energy within the network. The sum of the three-phase power corresponding to this part should be zero. According to the basic physical concept of power, the active and reactive currents have the following characteristics:

(21)

\{\begin{matrix} E (t) \cdot I_{P} (t) = E (t) \cdot I (t) \\ E (t) \cdot I_{Q} (t) = 0 \\ I_{P} (t) + I_{Q} (t) = I (t) \end{matrix}

Therefore, with the spatial position of the voltage vector as a reference, the component of the current vector parallel to it represents the active current component. In contrast, the component perpendicular to the voltage vector represents the reactive current component. Consequently, the two current components in Eq. (17) correspond to the active current and reactive current of the network: I_pa(t) = I_P(t) and I_pe(t) = I_Q(t). The active and reactive power of the network under voltage excitation with TVAF can be expressed as Eqs. (22), (23):

(22)

P (t) = \frac{3}{2} |E (t)| |I_{pa} (t)| = \frac{3}{2} E (t) A_{pa} (t)

(23)

Q (t) = \frac{3}{2} |E (t)| |I_{pe} (t)| = \frac{3}{2} E (t) A_{pe} (t)

3.2. The multi-timescale characteristics of network power

According to Eqs. (22), (23), both the active and reactive power are time-varying. The characteristics of the active/reactive power are jointly determined by the time-varying characteristics of E(t), A_pa(t), and A_pe(t). A_pa(t) and A_pe(t) are composed of three types of frequency components, as shown in Eq. (20), while E(t) contains both a DC component and a time-varying component with the frequency of ω_E. Hence, the active and reactive powers include five categories of frequency components: ω_E, 2ω_E, kω_ω, (ω_E ± kω_ω), and (2ω_E ± kω_ω). Fig. 7 retraces the formation process of the power components at various frequencies through the calculation process, where lines of the same sequence number represent the formation path of one frequency component.

The transmission characteristics of the active and reactive power are fundamentally determined by the TVAF characteristics of the excitation voltage.

3.2.1. Characteristics of active power

The active power comprises five types of frequency components, as depicted in Fig. 7. When k increases, the frequency of the component increases. When the amplitude and frequency of the voltage are time-invariant, all frequency components are 0, which aligns with the conclusion of a traditional power system analysis. However, under TVAF of the voltage, active power storage and release occur in the inductor, which is time-varying.

3.2.2. Characteristics of reactive power

The reactive power response contains the time-invariant component Q₀ and the time-varying component Q_t. Q₀ can be expressed as follows:

(24)

Q_{0} = \frac{3}{2} [\frac{E_{0}^{2}}{ω_{0} L} + \frac{A_{E}^{2}}{2 ω_{E} L} \sum_{n = 0}^{\infty} {(\frac{ω_{E}}{ω_{0}})}^{2 n + 1}]

Mathematically, Q₀ arises from the presence of trigonometric components in E(t) and A_pe(t) that are in phase and have the same frequency, while the in-phase components in A_pa(t) are phase-shifted by π/2.

The frequency components of Q_t are the same as those in the active power. The reactive power also contains a variety of fast scale components based on k. When the amplitude and frequency of the voltage are time-invariant, Q_t becomes 0, and the value of Q₀ aligns with the conclusion of a traditional circuit analysis. However, when A_E and A_ω are not 0, under the excitation of voltage with TVAF, Q₀ increases and is only related to the time-varying characteristics of the amplitude—that is, the second term of Eq. (24)—while Q_t becomes time-varying and is related to the time-varying characteristics of both the amplitude and frequency.

3.2.3. Analysis of energy consumption in inductance networks

To summarize, under the excitation of voltage with TVAF, the energy transmission within an inductance network exhibits the following characteristics.

Firstly, an active power storage and release process occurs on the inductive network. Time-varying active power exists in the inductance network excited by voltage with TVAF, challenging our understanding of traditional power system analyses. This indicates an energy exchange between the electrical equipment and the inductance network, influenced by the time-varying parameters of voltage amplitude and frequency.

Secondly, the energy exchanged between the three phases increases. Time-varying reactive power exists under the excitation of voltage with TVAF, signifying an ongoing energy interaction within the three-phase network. The amplitude of the interaction energy comprises both time-invariant and time-varying components, with both parts being influenced by the TVAF characteristics of the voltage.

Thirdly, under the influence of the internal voltage with TVAF at a certain timescale, ω_E, and ω_ω, both the active and reactive power exhibit broadband characteristics: ω_E, 2ω_E, kω_ω, (ω_E ± kω_ω), and (2ω_E ± kω_ω). Therefore, under the excitation of voltage with TVAF, the network generates power responses at multiple timescales.

In conclusion, the TVAF dynamics of the voltage after a disturbance modulate the network to generate multi-timescale dynamic active and reactive power. The intensified energy exchange between the electrical equipment and the network, as well as that within the networks, poses significant challenges to fault protection and system stability.

4. Amplitude characteristics of the power response and a simplified calculation method

The calculated power is represented in an infinite-series superposition form, which is challenging to apply in practical engineering applications. Therefore, the calculation is simplified within the range where the current series converges. The simplified current expression is described using a general formula, and the characteristics of the simplified power are analyzed.

4.1. The simplified current expression

From Eqs. (16), (18), the convergence of the overall current amplitude series depends on the convergence of A_pa(t) and A_pe(t). According to the properties of infinite series, the convergence interval of A_pa(t) and A_pe(t) can be determined by

(25)

\sum A_{n} (t) = \sum_{n = 0}^{\infty} \frac{G_{n} (t)}{ω_{0}^{n + 1} L} = \sum_{n = 0}^{\infty} \frac{\sum_{k = 0}^{n} C (n, k) M_{k} (t) E {(t)}^{(n - k)}}{ω_{0}^{n + 1} L}

in which j of the current series is ignored, as it does not affect the convergence.

A_n(t) is taken as an absolute value and scaled as follows:

(26)

\sum |A_{n} (t)| < \sum_{n = 0}^{\infty} 2^{n} (n + 1) \cdot \frac{n!}{{(\frac{n}{2}!)}^{2}} \frac{(E_{0} + A_{E}) ω_{E}^{n}}{ω_{0}^{n + 1} L}

where n is considered to be even, and ω_E is considered to be larger than ω_ω and A_ω. Cases in which n is considered to be odd and ω_ω or A_E is considered to be large exhibit similar characteristics. The scaling process in detail is provided in Eq. (S1) in Appendix A.

As in Eq. (27), the current series is convergent according to the D’Alembert test rule of infinite series when the condition in Eq. (28) is met. Hence, the fundamental frequency ω₀ has a decisive influence on the convergence rate of the series.

(27)

\lim_{n \to \infty} \frac{|A_{n + 1} (t)|}{|A_{n} (t)|} = \frac{4 ω_{E}}{ω_{0}} < 1

(28)

\max [A_{ω}, ω_{ω}, ω_{E}] < ω_{0} / 4

In a case where the convergence conditions are met, the current in Eq. (16) can be approximately described by the determinant term in the series. Firstly, as n increases, the corresponding amplitude of the current term decreases, occupying a smaller proportion of the overall current. Secondly, in practical engineering scenarios, because E₀ is relatively large compared with other time varying parameters, when n is fixed, the current term containing E(t) dominates. Therefore, by retaining the dominant current components, a simplified expression I_s(t) can be obtained in the form of Eq. (29). The relation between the terms of Eq. (29) and Eq. (16) can be known by the values of n and k: from the kth term of the terms obtained by the expansion of the nth Taylor series in Eq. (16). The superscript “(n)” represents the nth differential, and the time-varying forms of E(t) and μ(t) are shown in Eq. (19).

(29)

I (t) ≐ I_{s} (t) = \frac{E (t)}{ω_{0} L} e^{j [θ (t) - \frac{π}{2}]} + E (t) \sum_{n = 0}^{\infty} \{\frac{1}{ω_{0}^{n + 2} L} \frac{E {(t)}^{(n + 1)}}{E (t)} e^{j [θ (t) + \frac{(n - 1) π}{2}]} + \sum_{k = 1}^{\infty} {(- 1)}^{k} \frac{{[μ^{k} (t)]}^{(n)}}{j ω_{0}^{k - 1}} e^{j [θ (t) + \frac{(n - 1) π}{2}]}\}

The simplified current expression has the following characteristics and advantages. Firstly, the dominant term for the current is retained, with a more concise form, and the truncation order can be selected to meet the requirements of engineering calculations. Eq. (29) completely contains the term that accounts for the significant proportion of the overall current. Consequently, within the application range defined by Eq. (28), the simplified expression provides high accuracy. In practical engineering applications, when A_ω, ω_E, and ω_ω are small, a small value of k is enough to meet the accuracy requirements. As A_ω, ω_E, and ω_ω increase, the proportion of higher-order current terms in the overall current becomes more significant, necessitating a larger value of n to describe the current accurately. Secondly, all frequency components in the current are retained to meet the requirements of the stability mechanism analysis. The mathematical characteristics of Eq. (29) accurately reflect the influence of the time-varying characteristics of the voltage TVAF on the current. Thirdly, the effects of the TVAF—as well as their higher-order differentials—on the current can be reflected. The first term in Eq. (29) characterizes the influence of the traditional definition of impedance on the current. The subsequent terms—that is, the second term in Eq. (29)—represent the influence of the high-order differentials of the TVAF on the current, which cannot be described using traditional impedance definitions.

By bringing Eq. (19) into Eq. (29), the amplitude of I_s(t) sorted by frequency can be obtained, as shown in Eq. (S2) in Appendix A. The high-order frequency components of the original current are ignored, as only k = 1 is considered. Therefore, the dominant frequencies of A_pa(t) and A_pe(t) comprise DC components and three types of frequency components: ω_E, ω_ω, and ω_E ± ω_ω.

For a current where n = 2 and k = 1, the dominant component of I_s(t) can be expressed as follows:

(30)

I_{s} (t) = A_{pe} (t) e^{j [θ (t) - π / 2]} + A_{pa} (t) e^{j θ (t)} ≐ [\frac{E (t)}{ω_{0} L} - \frac{E (t) μ (t)}{ω_{0}^{2} L}] e^{j [θ (t) - \frac{π}{2}]} + [\frac{E^{'} (t)}{ω_{0}^{2} L} - \frac{E (t) μ^{'} (t)}{ω_{0}^{3} L}] e^{j θ (t)}

As in Eq. (30), A_pe(t) is mainly affected by the time-varying component of the TVAF of E(t). A_pa(t) is mainly generated by the differentials of the TVAF of E(t).

4.2. The simplified power expression

According to Eqs. (22), (23), and (29), a simplified expression of the active power P_s(t) and reactive power Q_s(t) is provided in Eq. (31) that reflects the overall active and reactive power characteristics listed in the previous section well and has relatively high precision.

(31)

\{\begin{matrix} P_{s} (t) = \frac{3}{2} E (t) \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} \{\frac{E {(t)}^{(2 n - 1)}}{ω_{0}^{2 n} L} + \sum_{k = 1}^{\infty} {(- 1)}^{k} \frac{E (t) {[μ^{k} (t)]}^{(2 n - 1)}}{ω_{0}^{k + 2 n} L}\} \\ Q_{s} (t) = \frac{3}{2} E (t) \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} \{\frac{E {(t)}^{(2 n - 2)}}{ω_{0}^{2 n - 1} L} + \sum_{k = 1}^{\infty} {(- 1)}^{k} \frac{E (t) {[μ^{k} (t)]}^{(2 n - 2)}}{ω_{0}^{k + 2 n - 1} L}\} \end{matrix}

By bringing Eq. (19) into Eq. (31) and considering k = 1, the first four terms of the detailed expression, sorted by frequency, are shown in Eqs. (S3) and (S4) in Appendix A. We then proceed to analyze the five frequency components that constitute a relatively large proportion, as illustrated in Table 1. Here, the physical and mathematical meanings of variable n are consistent with that in Eq. (29). The active power and reactive power contain the same frequency components, and the corresponding amplitudes of each frequency component have similar characteristics, as follows.

Firstly, the amplitude of the frequency component related to ω_E is only related to the time-varying parameters of the amplitude of the voltage.

Secondly, the amplitude of the frequency component related to ω_ω is only related to each time-varying parameter of the voltage frequency.

Moreover, if n = 2, the dominant power component of Eq. (31) can be expressed as follows:

(32)

\{\begin{matrix} P_{s} (t) = \frac{3}{2} [\frac{E (t) E^{'} (t)}{ω_{0}^{2} L} - \frac{E^{2} (t) μ^{'} (t)}{ω_{0}^{3} L}] \\ Q_{s} (t) = \frac{3}{2} [\frac{E^{2} (t)}{ω_{0} L} - \frac{E^{2} (t) μ (t)}{ω_{0}^{2} L}] \end{matrix}

where Q_s(t) is mainly affected by the time-varying component of the TVAF of E(t), while P_s(t) is mainly affected by the differential of the TVAF of E(t).

5. Power characteristics in a two-node inductance network

According to Fig. 2, the power transmission of each branch is affected by the internal voltage of the nodes at both ends. An inductance branch in Fig. 2, as shown in Fig. 8, is studied, which is excited by the E_x(t) and E_y(t) of node x and y, and

(33)

\{\begin{matrix} E_{x} (t) = E_{x} (t) e^{j (\int ω_{x} (t) d t + φ_{x})} = E_{x} (t) e^{j θ_{x} (t)} \\ E_{y} (t) = E_{y} (t) e^{j (\int ω_{y} (t) d t + φ_{y})} = E_{y} (t) e^{j θ_{y} (t)} \end{matrix}

in which

(34)

\{\begin{matrix} E_{x} (t) = E_{0_x} + ε_{x} (t) = E_{0_x} + A_{E_x} \sin (ω_{E_x} t + φ_{E_x}) \\ E_{y} (t) = E_{0_y} + ε_{y} (t) = E_{0_y} + A_{E_y} \sin (ω_{E_y} t + φ_{E_y}) \\ ω_{x} (t) = ω_{0} + μ_{x} (t) = ω_{0} + A_{ω_x} \sin (ω_{ω_x} t + φ_{ω_x}) \\ ω_{y} (t) = ω_{0} + μ_{y} (t) = ω_{0} + A_{ω_y} \sin (ω_{ω_y} t + φ_{ω_y}) \end{matrix}

and the subscripts x and y represent the amplitude/frequency parameters of the voltage in nodes x and y.

Based on Section 2, the current response of the branch can be obtained as follows:

(35)

I (t) {= I}_{x} (t) - I_{y} (t) = [I_{P_x} (t) + I_{Q_x} (t)] - [I_{P_y} (t) + I_{Q_y} (t)] = [A_{pa_x} (t) - j A_{pe_x} (t)] e^{j θ_{x} (t)} + [A_{pa_y} (t) - j A_{pe_y} (t)] e^{j θ_{y} (t)}

in which I_x(t) and I_y(t) are the response current of the network excited by e_x(t) and e_y(t), respectively; I_P_{_x}(t) and I_Q_{_x}(t) are the active and reactive currents of the network excited by e_x(t), respectively; I_P_{_y}(t) and I_Q_{_y}(t) are the active and reactive currents of the network excited by e_y(t), respectively; A_pa_{_x}(t), A_pe_{_x}(t), A_pa_{_y}(t), and A_pe_{_y}(t) are the amplitudes of I_P_{_x}(t), I_Q_{_x}(t), I_P_{_y}(t), and I_Q_{_y}(t), respectively.

The time-varying feature of I(t) is determined by the amplitude/frequency characteristics of the voltages of both nodes. I(t) contains components of the current vectors that are perpendicular and parallel to the voltage vectors at both nodes. As a result, the power generated by node x can be calculated using Eqs. (36), (37).

(36)

P_{x} (t) = \frac{3}{2} |E_{x} (t)| \cdot [|I_{P_x} (t)| - |I_{pa P_y} (t)| + |I_{pa Q_y} (t)|] = \frac{3}{2} E_{i} (t) \cdot [A_{pa_x} (t) - A_{pa_y} (t) \cos δ + A_{pe_y} (t) \sin δ]

(37)

Q_{x} (t) = \frac{3}{2} |E_{x} (t)| \cdot [|I_{Q_x} (t)| - |I_{pe P_y} (t)| - |I_{pe Q_y} (t)|] = \frac{3}{2} E_{x} (t) \cdot [A_{pe_x} (t) - A_{pa_y} (t) \sin δ - A_{pe_y} (t) \cos δ]

where I_paP_{_y}(t) and I_peP_{_y}(t) are the projections of I_P_{_y}(t) onto the parallel and perpendicular directions of E_x(t), respectively; I_paQ_{_y}(t) and I_peQ_{_y}(t) are the projections of I_Q_{_y}(t) onto the parallel and perpendicular directions of E_x(t), respectively; δ is the time-varying phase difference of the voltage in the two nodes and

(38)

δ = θ_{x} (t) - θ_{y} (t)

From Eqs. (36), (37), the active power and reactive power generated by node x are determined by the projection of I(t) on the parallel and perpendicular lines of E_x(t), respectively, as shown in Fig. 9.

Suppose that n = 2 and k_x = k_y = 1; then, the dominate component of P_x(t) and Q_x(t) can be expressed as Eqs. (39), (40). The power generated by node y processes takes on a similar form as Eqs. (41), (42).

(39)

P_{x} (t) = \frac{3}{2} \{\frac{E_{x} (t) E_{y} (t)}{ω_{0} L} \sin δ + [\frac{E_{x} (t) E_{x}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{x}^{2} (t) μ_{x}^{'} (t)}{ω_{0}^{3} L}] - [\frac{E_{x} (t) E_{y}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{x} (t) E_{y} (t) μ_{y}^{'} (t)}{ω_{0}^{3} L}] \cos δ - \frac{E_{x} (t) E_{y} (t) μ_{y} (t)}{ω_{0}^{2} L} \sin δ\}

(40)

Q_{x} (t) = \frac{3}{2} \{\frac{E_{x}^{2} (t) - E_{x} (t) E_{y} (t) \cos δ}{ω_{0} L} - \frac{E_{x}^{2} (t) μ_{x} (t)}{ω_{0}^{2} L} - [\frac{E_{x} (t) E_{y}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{x} (t) E_{y} (t) μ_{y}^{'} (t)}{ω_{0}^{3} L}] \sin δ + \frac{E_{x} (t) E_{y} (t) μ_{y} (t)}{ω_{0}^{2} L} \cos δ\}

(41)

P_{y} (t) = \frac{3}{2} \{- \frac{E_{y} (t) E_{x} (t)}{ω_{0} L} \sin δ + [\frac{E_{y} (t) E_{y}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{y}^{2} (t) μ_{y}^{'} (t)}{ω_{0}^{3} L}] - [\frac{E_{y} (t) E_{x}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{y} (t) E_{x} (t) μ_{x}^{'} (t)}{ω_{0}^{3} L}] \cos δ + \frac{E_{y} (t) E_{x} (t) μ_{x} (t)}{ω_{0}^{2} L} \sin δ\}

(42)

Q_{y} (t) = \frac{3}{2} \{\frac{E_{y}^{2} (t) - E_{y} (t) E_{x} (t) \cos δ}{ω_{0} L} - \frac{E_{y}^{2} (t) μ_{y} (t)}{ω_{0}^{2} L} + [\frac{E_{y} (t) E_{x}^{'} (t)}{ω_{0}^{2} L} - \frac{E_{y} (t) E_{x} (t) μ_{x}^{'} (t)}{ω_{0}^{3} L}] \sin δ + \frac{E_{y} (t) E_{x} (t) μ_{x} (t)}{ω_{0}^{2} L} \cos δ\}

Beyond the understanding of traditional active and reactive power (i.e., the first term of Eqs. (39), (40)), when subjected to a voltage with TVAF in two nodes, the output power of one node exhibits the following characteristics:

Firstly, the transmitted active power on the network generated by one node is not only related to the amplitude and phase of the voltages at both ends, as in the first term of Eq. (39), but also related to the differentials of the time-varying components of the amplitude and frequency of the self-node voltage, as in the second set of terms of Eq. (39). Besides, it is also related to the time-varying terms and differentials of the amplitude and frequency of the voltage in the opposite node, as in the other terms of Eq. (39). The terms other than the first term result in unbalanced active power at both nodes, leading to active power storage in the network.

Secondly, the reactive power is not only related to the amplitude and phase of the voltages at both ends, as in the first term of Eq. (40), but also to the time-varying components of the frequency of the self-node voltage, as in the second set of terms of Eq. (40). Besides, it is also related to the time-varying terms and differentials of the amplitude and frequency of the voltage in the opposite node, as in the other terms of Eq. (40).

Thirdly, both the active and reactive power of the network are related to the sine and cosine values of the phase angle difference between the voltages of two nodes.

Moreover, because of the time-varying characteristics of the amplitude/frequency of the node voltage, Eqs. (39), (41) are not equal; in other words, there is time-varying power storage and release in the network.

6. Case study

The software PSCAD was used to build a two-voltage source converter (VSC) infinite bus system, as shown in Fig. 10, for simulation tests in order to validate the applicability of the proposed current and power equations. The VSCs are equivalent to Thévenin’s equivalent model [19]. A virtual synchronization generator (VSG) control method was applied to the VSCs [20]; control diagrams of VSC1 and VSC2 are provided in Fig. 10, and the parameters of the system are listed in Table 2. The internal voltages E₁(t) and E₂(t) of nodes 1 and 2 are provided by VSC1 and VSC2, respectively. The internal voltage E₃(t) of node 3 is provided by an infinite voltage source (V3). The currents i₁(t), i₂(t), and i₃(t) flow from the power sources to the nodes.

Case 1: A power disturbance occurs at 2 s, resulting in an unstable operation problem of the system. Fig. 11 illustrates the amplitude and frequency of the internal voltages of VSC1 and VSC2. Both the amplitude and frequency exhibit time-varying characteristics after the disturbance occurs. After 14 s, the system transitions into a period of constant amplitude oscillation (ω = ω_E = ω_ω =(10.67 × 2π) rad·s⁻¹).

Fig. 12 shows the waveform obtained by the entire expression in Eq. (16) when n = 4 (red dashed line), compared with the static relationship (green line), also known as the impedance expression, and the simulation result (blue line). The phase-a current of i₁(t) is illustrated in Fig. 12(a). The oscillations in the voltage amplitude and the frequency become more drastic over time, and the traditional impedance model can no longer fulfill the requirements for the current calculation. In contrast, the method proposed in this paper accurately describes the current throughout the process.

The active/reactive power expressions generated by VSC1 at node 1, P₁(t) and Q₁(t), and by VSC2 at node 2, P₂(t) and Q₂(t), are obtained by Eqs. (36), (37), as presented in Figs. 12(b)–(e). The power calculation method proposed in this paper accurately describes the active and reactive power within the nodes in comparison with the actual simulation results. Meanwhile, similarly to the current, as the oscillation amplitude of the voltage amplitude and frequency increase, there is a considerable discrepancy between the static expressions and actual simulation result. The proposed method is significantly more accurate than static methods for determining the active power, highlighting the limitations of impedance-based power calculation methods in characterizing network dynamics under drastic TVAF of the voltage.

Affected by the time-varying characteristics of the amplitude and frequency, the DC component Q₀ in the reactive power shows a significant change, which cannot be obtained using impedance-based calculation methods, as shown in Figs. 12(c) and (e). Figs. 12(f) and (g) show the frequency components of the active and reactive power in node 1 during the period of constant amplitude oscillation. According to the theoretical analysis, as n = 4, the power contains the components of DC, ω ((10.67 × 2π) rad·s⁻¹), 2ω ((21.34 × 2π) rad·s⁻¹), 3ω ((32.01 × 2π) rad·s⁻¹), and 4ω ((42.68 × 2π) rad·s⁻¹). The power simulation analysis is consistent with the theoretical analysis results, as shown in Figs. 12(f) and (g). The multi-timescale characteristics of the network power under the excitation of the voltage with TVAF are validated. Moreover, for the amplitude of each frequency component, the entire power expression is more accurate than the static expression.

Fig. 13(a) shows the active power injected into the network by nodes 1 (blue line), 2 (red line), and 3 (purple line), as well as their power differences (black line). During the steady state before 2 s, the active power injected by the three nodes balances out, resulting in no power being stored in the network. However, after the disturbance, due to the oscillation of the amplitude and frequency of the internal voltage at each node, the active power injected by the three nodes becomes unbalanced, generating time-varying power storage and release in the network. These conclusions cannot be captured by static methods. As shown in Fig. 13(b), the three-node power difference obtained by the static method remains at 0 after the disturbance.

The reactive power difference between nodes 1 and 2, ΔQ₁₂(t), is obtained by the entire (red line) and the static methods (green line) expression in Fig. 14. Since the voltage of node 2 is always higher than that of node 1, as shown in Fig. 11(a), reactive power is always transmitted from node 2 to node 1 based on static methods, and the consumption of reactive power is always positive, as shown in Fig. 14. However, as described in Eqs. (40), (42), under the excitation of voltage with TVAF, the reactive power is not only related to the voltage amplitude but also related to the change rate of the amplitude and frequency. As shown in Fig. 14, this results in the transmission of reactive power from the lower voltage node to the higher voltage node at certain times, which cannot be captured by static methods. To summarize the contents of Fig. 12, Fig. 13, Fig. 14, the consumption of the active power and reactive power in the network under the excitation of a voltage with TVAF is validated.

As shown in Fig. 11, during the constant amplitude oscillation process after 14 s, the amplitude and frequency of the internal voltages at nodes 1 and 2 display sinusoidal time-varying characteristics with a frequency of (10.67 × 2π) rad·s⁻¹. Therefore, the simplified current and power expressions in Eqs. (29), (39), and (40) can be utilized to analyze the current and power. Fig. 15 shows the waveform obtained by the simplified expression. Figs. 15(a)–(c) show the waveform diagram of current and active/ reactive power obtained by the simplified current and power expression, where n = 4 and k = 1 (red dashed line), in comparison with the static relationship expression (green line) and simulation result (blue line). Figs. 15(a) and (d) show the phase-a current of i₁(t), as well as the frequency components of the current amplitude. For ease of observation, only the waveform within one oscillation period of the current is provided in Fig. 15(a). Since the oscillation frequency of the amplitude and frequency of the internal voltages in nodes 1 and 2 are the same, the frequency components in Fig. 15(d) contain the components DC, ω ((10.67 × 2π) rad·s⁻¹), and 2ω ((21.34 × 2π) rad·s⁻¹). The proposed simplified current expression maintains high accuracy in comparison with the static expression. It also includes all the frequency components of the actual simulation current.

Figs. 15(b) and (c) show the active and reactive power generated by VSC1 as obtained by the simplified expression during several oscillation periods. Similarly, the simplified power expression maintains high accuracy in the dynamic process and includes all the dominant frequency components. The frequency components are illustrated in Figs. 15(e) and (f). The frequency components of the active and reactive power are composed of DC, ω ((10.67 × 2π) rad·s⁻¹), 2ω ((21.34 × 2π) rad·s⁻¹), and 3ω ((32.01 × 2π) rad·s⁻¹). In this case, DC components are included in both the active and reactive power.

Case 2: The case above demonstrates the applicability of the proposed method and theory under extreme conditions. In order to have the simulation model conform to an actual working condition, a current limiting link of R_v = 0.4 Ω is added to the control of the VSCs, the form of which is shown in Ref. [21]. The inertia coefficients of VSC1/VSC2 are adjusted: T_j₁ = T_j₂ = 1. A power disturbance occurs at 2 s, resulting in an unstable operation problem of the system. The TVAF of the internal voltages of VSC1 and VSC2 are shown in Fig. 16. Unlike in Case 1, both the amplitude and frequency of the internal voltage of the VSCs are time-varying in a non-sinusoidal form in Case 2. Moreover, in comparison with those in Case 1, the oscillation amplitude is smaller and the oscillation frequency is lower in Case 2.

Fig. 17 shows the waveform obtained by the entire expression in Eq. (16) when n = 4 (red dashed line), comparing with the static relationship (green line) and simulation result (blue line). The phase-a current of i₁(t) is illustrated in Fig. 17(a). The active/reactive power expressions generated by VSC1 at node 1, P₁(t) and Q₁(t), and VSC2 at node 2, P₂(t) and Q₂(t), are determined using Eqs. (36), (37), as presented in Figs. 17(b) and (c). As shown in Fig. 17, the current and power calculation method proposed in this paper accurately describes the active and reactive power within the nodes, in compared with the actual simulation results. However, the static expressions fail to accurately describe the current and power of the network in this case. The simulation results in Case 2 also demonstrate that the proposed method breaks through the limitations of impedance-based methods in characterizing network dynamics under non-sinusoidal TVAF of the voltage.

Fig. 18 shows the active power of the network injected by three nodes. During the steady state before 2 s, no power storage occurs in the network. After the disturbance, due to the oscillation of the amplitude and frequency of the internal voltage at each node, the time-varying active power reveals the phenomenon of power storage and release in the network. The phenomenon can be captured by the entire expression (red dashed line), as well as the simulation result (blue line). However, it cannot be obtained by the static method (green line), as the three-node power difference obtained by the static method remains at 0 after the disturbance.

7. Conclusions

This paper discussed the power response characteristics of a network under excitation by voltage with TVAF. The amplitude and frequency dynamics of the network active/reactive power response were analyzed, and the multi-timescale characteristics of the network power response were revealed. Furthermore, the storage and transmission characteristics of the network power during dynamic processes were analyzed. On this basis, a network power response calculation formula that meets the requirements for engineering applications was provided.

(1) Excited by voltage with TVAF, the power response of the network exhibits multi-timescale characteristics. The dynamic frequency of the network power is related to the dynamic frequency of the internal voltage amplitude and frequency as ω_E, 2ω_E, kω_ω, (ω_E ± kω_ω), and (2ω_E ± kω_ω).

(2) The amplitude of the dynamic power response at different frequencies is related to the time-varying terms of the voltage amplitude and frequency, as well as their higher-order differentials.

(3) Unlike the traditional understanding in power systems that inductance networks serve only as a carrier for the transmission of active power, because of the TVAF characteristic, an inductance network experiences the phenomenon of active power storage and release.

(4) Unlike the traditional understanding that the reactive power in power systems is always transmitted from high-voltage nodes to low-voltage nodes, because of the TVAF characteristic, the reactive power transmission is not only related to the voltage amplitude but also related to the change rate of the amplitude and frequency. In certain situations, reactive power transmission from low-voltage nodes to high-voltage nodes can even occur.

CRediT authorship contribution statement

Jiabing Hu: Supervision, Methodology, Investigation, Funding acquisition. Weizhong Wen: Writing – original draft, Visualization, Methodology, Investigation, Formal analysis. Yingbiao Li: Writing – review & editing, Supervision, Methodology. Xing Liu: Writing – review & editing. Jianbo Guo: Writing – review & editing, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported in part by the National Natural Science Fundation of China (52225704 and 52107096).

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the author(s) used ChatGPT in order to improve readability and polish the language. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.12.015.

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