The node number of the AC–DC hybrid sending system is set to m+n. After considering the dynamic process on the DC side of the battery storage, the state equation of the equivalent synchronous power source for the AC–DC hybrid system considering the excitation characteristics is δ ˙ i = ω i − 1 ω 0 , ω ˙ i = P m i − P e i − D i ω i − 1 / T J i , E ˙ q i ′ = E f i − E q i ′ − X d i − X d i ′ I d i / T d 0 i ′ , (1) where δ ˙ i , ω ˙ i , and E ˙ q i ′ are the power angle, rotor angular velocity, and transient potential, respectively, of the equivalent synchronous power source at node i of the renewable energy sending system during the transient process; ω 0 is the rated synchronous angular velocity of the renewable energy sending system; I d i is the direct-axis component of the output current of the equivalent synchronous power source at node i of the renewable energy sending system; P m i is the input energy of the equivalent synchronous power source at node i of the renewable energy sending system; E f i is the excitation voltage of the equivalent synchronous power source at node i of the renewable energy sending system; X d i and X d i ′ are the direct-axis equivalent reactance and direct-axis transient reactance of the equivalent synchronous power source at node i of the renewable energy sending system, respectively; T d 0 i ′ is the transient time constant of the equivalent synchronous power source at node i of the renewable energy sending system; T J i is the rotor inertia time constant of the equivalent synchronous power source at node i of the renewable energy sending system; and D i is the damping coefficient of the equivalent synchronous power source at node i of the renewable energy sending system.

When the battery storage is connected to the ith node of the renewable energy sending system, if the battery storage system is converting the DC power into AC power input to the grid, that is, when the battery energy storage system is in discharge state at this time, the active power P e i of the equivalent synchronous power source in Eq. 1 should be P e i = P a c i + P d c i + P L i , (2) where P a c i and P d c i are the AC input power and DC input power at node i of the renewable energy sending system, respectively; P L i is the total equivalent active load power at node i of the renewable energy sending system.

When the battery storage is connected to the ith node of the renewable energy sending system, if the battery storage system is converting the AC power into DC power input to the battery energy storage system, that is, when the battery energy storage system is in the charging state at this time, the active power P e i of the equivalent synchronous power source in Eq. 1 is P e i = P a c i − P d c i + P L i . (3)

In this study, a first-order inertial link model is used to describe the DC-side dynamic model of the battery energy storage system: P ˙ d c = 1 T d c − P d c + P d c r + u d c s , (4) where P d c r is the power value given at the DC side of the battery energy storage system when the renewable energy sending system is in a steady state; u d c s is the additional control amount exerted on the DC side of the battery energy storage system during the transient process of the renewable energy sending system; and T d c is the inertia time constant of the battery storage system DC-side power response during the transient process of the renewable energy sending system.

The total node number of the renewable energy sending system is set to m + n, where the node with an equivalent synchronous power source is m, and the node with the battery storage system and capable of DC damping control is n. It is assumed that all nodes relate to the load, that is, all nodes relate to equivalent load impedance, regardless of whether it has an equivalent synchronous power source or battery storage system.

After considering the input and output characteristics of the DC side of the battery storage system, the power balance equation of the renewable energy sending system can be expressed as S ˙ g S ˙ b S ˙ l S ˙ r = E ′ ˙ g q U ˙ B U ˙ L E ˙ r q ′ I ˙ g I ˙ b I ˙ l I ˙ r * = E ′ ˙ g q U ˙ B U ˙ L E ˙ r q ′ Y G G Y G H Y G L Y G R Y B G Y B H Y B L Y B R Y LG Y L H Y L L Y L R Y R G Y R H Y R L Y R R * E ′ ˙ g q U ˙ B U ˙ L E ˙ r q ′ * , (5) where S ˙ g , S ˙ b , S ˙ l , and S ˙ r are the complex power of the traditional equivalent synchronous power source, the complex power of the battery storage system, the complex power of the equivalent load, and the complex power of the renewable energy equivalent synchronous power source during the transient process of the renewable energy sending system, respectively; E ′ ˙ g q , U ˙ B , U ˙ L , and E ˙ r q ′ are the transient potential of the traditional equivalent synchronous power source, the voltage of the battery storage system, the voltage of the equivalent load, and the transient potential of the renewable energy equivalent synchronous power source during the transient process of the renewable energy sending system, respectively; I ˙ g , I ˙ b , I ˙ l , and I ˙ r are the current of the traditional equivalent synchronous power source, the current of the battery storage system, the current of the equivalent load, and the current of the renewable energy equivalent synchronous power source in the transient process of the renewable energy sending system, respectively; and Y is the network conductance matrix of the renewable energy sending system during the transient process.

From the aforementioned power balance equation, the total equivalent active power of the equivalent synchronous power source and battery storage system connected to node i in the renewable energy sending system can be obtained as P e i = E q i ′ ∑ j = 1 m E q j ′ G i j ′ ⁡ cos ⁡ δ i j + B i j ′ ⁡ sin ⁡ δ i j + E q i ′ ∑ j = m + 1 n I B j G i j ″ ⁡ cos ⁡ ψ i j + B i j ″ ⁡ sin ⁡ ψ i j , (6) where δ i j = δ i − δ j is the phase angle difference between the power angles δ i and δ j of the two equivalent synchronous power sources connected at nodes i and j in the renewable energy sending system; ψ i j = δ i − φ B j is the phase angle difference between the power angle δ i of the equivalent synchronous power source connected at node i and the AC side voltage phase angle φ B j of the battery energy storage system in the renewable energy sending system.

After considering the DC damping characteristics of the battery storage, the state equation of the renewable energy sending system under the discrete-time system can be expressed as the AC–DC hybrid state equation: x k + 1 = f x k , ξ k + G x k , ξ k u k , β x k , ξ k = 0 , (7)

where x k represents the state variables such as the power angle, frequency, transient potential, and DC power of the battery storage system of each power node in the renewable energy sending system; ξ k represents the active and reactive components of the output power of the equivalent synchronous generators of hydropower, thermal power, wind power, and photovoltaic power sources in the renewable energy sending system, that is, the straight-axis and cross-axis components of the output current of each equivalent synchronous power source; u k represents the equivalent synchronous generator excitation voltage control variables of hydropower, thermal power, wind power, and photovoltaic power sources, and DC control variables of the battery storage system in the renewable energy sending system; f is the relationship function of the state variables between the equivalent synchronous generator compound power, power angle, frequency, transient voltage, and DC power of the battery storage system at each power node in the renewable energy sending system; G is the relationship function between the controllable input variables in the system and the state variables of the sending system, where the state variables include equivalent synchronous complex power, power angle, frequency, transient voltage, and DC power of the battery energy storage system at each power node of the transmission system; and β is the equation of complex power balance of the sending system.

Assume that the equivalent synchronous power source in the renewable energy sending system is node 1,2 , … , m and the battery energy storage system is node m + 1 , m + 2 , … , m + n . The relationship function of the state variables between the equivalent synchronous generator compound power, power angle, frequency, transient voltage, and DC power of the battery storage system at each power node in the renewable energy sending system can be described as shown in Eq. 8, which is determined by the AC–DC hybrid state equation: f = f 1 x , ξ f 2 x , ξ ⋮ f m x , ξ f m + 1 x , ξ ⋮ f m + n x , ξ = ω 1 − 1 ω 0 P m 1 − P e 1 − D 1 ω 1 − 1 T J 1 − E q 1 ′ − X d 1 − X d 1 ′ I d 1 T d 01 ′ ω 2 − 1 ω 0 P m 2 − P e 2 − D 2 ω 2 − 1 T J 2 − E q 2 ′ − X d 2 − X d 2 ′ I d 2 T d 02 ′ ⋮ ⋮ ⋮ ω m − 1 ω 0 P m m − P e m − D m ω m − 1 T J m − E q m ′ − X d m − X d m ′ I d m T d 0 m ′ 0 0 − P d c m + 1 + P d c r m + 1 T d c m + 1 ⋮ ⋮ ⋮ 0 0 − P d c m + n + P d c r m + n T d c m + n . (8)

According to the AC–DC hybrid state equation, the relationship function between the controllable input variables in the system and the state variables of the sending system can be described as shown in Eq. 9: G = G 1 x , ξ ⋮ G m x , ξ G m + 1 x , ξ ⋮ G m + n x , ξ = 0 0 1 T d 01 ′ ⋱ 0 0 1 T d 0 m ′ 1 T d 0 m + 1 ′ ⋱ 1 T d 0 m + n ′ . (9)

The complex power balance equation of the renewable energy sending system can be described as β x , ξ = β 1 x , ξ ⋅ ⋅ ⋅ β i x , ξ ⋅ ⋅ ⋅ β m + k − p x , ξ T , (10) where β i x , ξ = I q i − ∑ j = 1 m E q j ′ G i j ′ ⁡ cos ⁡ δ i j + B i j ′ ⁡ sin ⁡ δ i j − ∑ j = 1 k I B j G i j ″ ⁡ cos ⁡ ψ i j + B i j ″ ⁡ sin ⁡ ψ i j I d i − ∑ j = 1 m E q j ′ G i j ′ ⁡ sin ⁡ δ i j − B i j ′ ⁡ cos ⁡ δ i j − ∑ j = 1 k I B j G i j ″ ⁡ sin ⁡ ψ i j − B i j ″ ⁡ cos ⁡ ψ i j . (11)

This paper studies and establishes the power and excitation characteristics of the equivalent synchronous power source during the transient process of the renewable energy sending system, as well as the transient frequency–voltage equation of state for the renewable energy sending system considering the DC side characteristics of the battery energy storage system. Based on this finding, when the renewable energy sending system is impacted by transient port energy, based on the propagation characteristics of transient port energy in the system, further judgment is made on whether the frequency and voltage changes in the equivalent power supply at each node of the delivery network will deviate too much from the expected trajectory. Based on the frequency and voltage trajectory evolution characteristics of each equivalent power source node, the voltage and frequency of the equivalent power source are controlled at each node of the renewable energy sending system.

The voltage frequency state evolution trajectory of the transient stability equilibrium points x e , y e of the renewable energy sending system is defined as x ˙ e = f x e , y e + G x e , y e u r , 0 = ρ x e , y e . (12)

When the renewable energy sending system is in a steady state ( x ˙ e = 0 ), the corresponding steady-state input control quantity is u r . When the renewable energy sending system is in a transient process, the deviation equation of state Eq. 7 under the state evolution trajectory is Δ x ˙ = Δ f x , y + y x , y Δ u , 0 = Δ ρ x , y , (13) where Δ x ˙ = x ˙ − x ˙ e , Δ u = u − u r , Δ f = f x , y − f x e , y e , and Δ ρ = ρ x , y − ρ x e , y e .

The output function of voltage and frequency of the renewable energy sending system corresponding to the dynamic system in Eq. 7 is g = h x , y . (14)

The frequency and voltage control target deviation ξ in the transient process that needs to be controlled and constrained under the transient stability condition of the renewable energy sending system is ξ = Δ g = g − g r , (15) where g r represents the evolution trajectory of frequency and voltage output functions for each equivalent power source node when the renewable energy sending system is in a transient stable equilibrium state ( x ˙ e = 0 , u r ).

According to the aforementioned analysis, the frequency and voltage control target deviation ξ in the transient process of the renewable energy sending system is independent of the stable equilibrium state of the system and its corresponding state variables. On this basis, it can be concluded that ξ ˙ = Δ g ˙ = g . ˙ (16)

According to the aforementioned two equations, this paper constructs a multi-objective dynamic control system for the frequency and voltage of the equivalent synchronous power source of the sending network node under the transient stability of the renewable energy sending system. ξ ˙ = A ξ + B u , (17) where u represents the control input of active power and excitation voltage of each node equivalent synchronous power source, and the DC-side power control input of the battery energy storage system. The coefficient matrices A and B are as follows: A = A 1 ⋱ A i ⋱ A m + n , B = B 1 ⋱ B i ⋱ B m + n . (18)

The diagonal matrices A i and B i are of the Brunovsky standard form: A i = 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 0 0 0 ⋯ 0 μ i × μ i , B i = 0 0 ⋮ 0 1 μ i × 1 , (19) where μ i is the subsystem dimension of the ith input of the multi-objective dynamic control system.

According to Lyapunov theory, the feedback control law of a linear system is designed. u = − K ξ , (20) where ξ = A − B K ξ , A − B K , and R e λ i A − B K < 0 . K = R − 1 B T P is the control matrix. R is the positive definite weight matrix, P is the solution of the Riccati equation P A + A T P − P B R − 1 B T P = − Q , and Q is the positive definite matrix.

Therefore, the linear control law of the ith subsystem designed to make the system stable can be expressed as u i = ξ i μ i = − k i 1 ξ i 1 − k i 2 ξ i 2 − ⋯ − k i μ i ξ i μ i = − k i 1 Δ g i 1 − k i 2 Δ g i 2 − ⋯ − k i μ i Δ g i μ i . (21)

Then, the nonlinear equivalent input μ i is solved. If the algebraic variable in the algebraic equation is an explicit function, the algebraic variable is derived. If the algebraic variables in the algebraic equation are implicit functions, the calculation is as follows: Γ f ξ i μ i = Γ f Δ h i μ i x , y = ∂ Δ h i μ i x , y ∂ x ∂ Δ h i μ i x , y ∂ y Λ x , y Δ f x , y , Γ G Γ f 0 h i μ i x , y = ∂ Γ f 0 Δ h i μ i x , y ∂ x ∂ Γ f 0 Δ h i μ i x , y ∂ y Λ x , y G x , y , (22) Λ x , y = I n − ∂ Δ ρ ∂ y − 1 ∂ Δ ρ ∂ x , (23) where Λ x , y is the relation transformation matrix and I n is an n × n identity matrix. The equivalent nonlinear control input corresponding to the linear input u i is obtained as u i = Γ G Γ f 0 Δ h u i x , y − 1 v i − L f Δ h i u i x , y = Γ G Γ f 0 Δ h u i x , y − 1 − k i 1 ξ i 1 − ⋯ − k i u i ξ i u i − Γ f Δ h i u i x , y , (24) where Δ h u i = Δ h 1 u 1 ⋯ Δ h m + k u m + k T .

When the system is subjected to large disturbances, the excitation control can maintain the stability of the system voltage and frequency. The power balance and rotor power angle of the system are the keys to determine the frequency and voltage stability. Therefore, the angular frequency, active power, and terminal voltage of the generator are selected as the output functions, and the target deviation can be expressed as ξ i = Δ ω i Δ P t i Δ U g i . (25)

The multi-objective equation is constructed based on the target deviation, and the multi-objective sub-equation of generator excitation control corresponding to the ith subsystem is Δ ω ˙ Δ P ˙ t i Δ U ˙ g i = 0 1 0 0 0 1 0 0 0 Δ ω i Δ P t i Δ U ˙ g i + 0 0 1 v i . (26)

The linear feedback control law is designed to stabilize the system: u = Δ U ˙ g i = − K ξ = − k i 1 Δ ω i − k i 2 Δ P t i − k i 3 Δ U ˙ g i . (27)

Then, the Γ derivative of the target deviation Δ U g i is obtained as follows: Γ f = Δ U g i = ∂ Δ U g i ∂ x   ∂ Δ U g i ∂ y Λ x , y Δ f i x , y = − Δ U g i 2 − X q i Δ I q i 2 X d i Δ I d i + T d 0 ′ X d i ′ Δ I ˙ d i / T d 0 ′ U g i + X q i 2 Δ I q i I ˙ q i U g i / − Δ U g i 2 − X q i Δ I q i 2 / T d 0 ′ U g i , (28) Γ G Γ f 0 Δ U g i = ∂ Γ f 0 Δ U g i ∂ x   ∂ Γ f 0 Δ U g i ∂ y Λ x , y G i x , y = Δ U g i 2 − X q i Δ I q i 2 U g i T d 0 i ′ (29)

The equivalent additional excitation control input is E f i = Γ G Γ f 0 Δ U g i − 1 u i − Γ f Δ U g i = Γ G Γ f 0 Δ U g i − 1 − k i 1 ξ i 1 − k i 2 ξ i 2 − k i 3 ξ i 3 − Γ f Δ U g i Γ G Γ f 0 Δ U g i − 1 − k i 1 Δ ω i − k i 2 Δ P t i − k i 3 Δ U ˙ g i − Γ f Δ U g i . (30)

For the system without a wide-area signal acquisition, the control signal can be controlled by using the local conventional acquisition signal as follows: I d i = Q e i + X q i I g i 2 / U g i + Q e i X q i / U g i 2 + P t i X q i / U g i 2 I q i = I g i 2 − I d i 2 . (31)

The control law designed in this paper can implement control behavior in advance based on the trend of target quantity changes, which makes the control law have advanced predictability.

The nonlinear additional control of the battery energy storage can suppress the power fluctuation of the power grid, increase the system damping, and improve the transient stability of the system through the fast power control of the DC system. Therefore, the control target deviation is determined as the deviation of system frequency Δ ω between the actual operation and stable operation, the power deviation of AC tie lines Δ P a c , and the DC transmission power deviation Δ P d c . The deviation power relationship obtained by replacing the constant load with impedance equivalent is Δ P c n i = Δ P t i − Δ P d n i . (32)

The target equation based on the target deviation is as follows: Δ ω ˙ i j Δ P ˙ c n i Δ U ˙ d n i = 0 1 0 0 0 1 0 0 0 Δ ω i j Δ P c n i Δ U ˙ d n i + 0 0 1 v i . (33)

According to the control algorithm proposed in this paper, the DC linear control law is u i = Δ P d n i = − k 1 Δ ω i j − k d n i 2 Δ P c n i − k d n i 3 Δ P d n i , (34) Δ P d n i = − 1 T d d Δ P d n i + 1 T d d Δ u d c c , (35) where K d n i = k d n i 1   k d n i 2   k d n i 3 T is designed according to the Lyapunov method.

According to aforementioned equations, the DC nonlinear additional control of battery energy storage can be obtained as Δ u d c c = Δ P d n i − T d d − k d n i 1 Δ ω i j − k d n i 2 Δ P c n i − k d n i 3 Δ P d n i . (36)

The topological structure of the equivalent network model of the renewable energy sending system is shown in Supplementary Figure S1. The typical source-load curve of the sending system is shown in Supplementary Figure S2. The three-phase short-circuit fault at node 2 shown in Supplementary Figure S1 is taken as an example to verify the effectiveness of the transient energy balance robust control model of the sending-end system established in this paper.

The total load of the renewable energy sending system of node 2 is 12,000 MW, the AC transmission power is 2,300 MW, and the DC transmission power is 2,800 MW. The total load of the receiving system of node 1 is 15,200 MW, and the DC power accessed from the sending system is 3,000 MW. The total load of the receiving system at node 3 is 50,000 MW, and the AC power accessed from the sending system is 2,000 MW. In the renewable energy sending system shown in Supplementary Figure S1, PV1 and PV2 are photovoltaic power nodes, WP1 and WP2 are the wind power nodes, HG1 is the node of hydroelectric units, G1 and G2 are the nodes of thermal power units, and ES1, ES2, ES3, and ES4 are the nodes of energy storage systems.

When a three-phase short circuit occurs at node 2 of the renewable energy sending system and the fault is not cut within the limit time, the equivalent power angle instability curve and its corresponding transient port energy tracking situation at node 2 are shown in Supplementary Figure S3.

As shown in Supplementary Figure S3, when a three-phase short circuit occurs at node 2 of the sending system and the fault is not cut within the limit time, the system obtains a large amount of transient energy. Comparing the power angle variation characteristics of the system with and without the proposed transient energy balance robust control model, the following results can be obtained: although the system is equipped with battery energy storage, if the effective energy tracking control algorithm is not adopted, the fast energy balance characteristics of the battery energy storage in the sending system cannot be fully utilized, and the system will still be unstable. After adopting the energy balance control, the system has good stability performance and can quickly pull the power angle back to synchronization in the second pendulum.

At the same time, as shown in Supplementary Figure S3, the energy tracking control algorithm proposed in this paper can quickly judge the port energy obtained by the sending system when the transient fault starts. The port energy response is achieved through effective coordination between the energy storage and other power sources in the renewable energy sending system.

When a three-phase short circuit occurs at node 2 of the sending system and the fault is not cut within the limit time, the frequency and voltage change curves of the equivalent synchronous power source at nodes 1 and 2 of the sending system are shown in Supplementary Figure S4.

As shown in Supplementary Figure S4, when a three-phase short circuit occurs at the exit of the channel of the renewable energy sending system and the fault is not cut within the limit time, the frequency and voltage of the equivalent synchronous power source at nodes 1 and 2 of the grid are significantly affected due to the large unbalance of the system transient energy and its rapid propagation in the system. However, under the action of the voltage frequency control algorithm proposed in this paper, the exceedance time of the voltage and frequency is short, which has little impact on the system. On the contrary, if the frequency and voltage multi-objective nonlinear control model proposed in this paper is not adopted, the frequency and voltage of the grid-connected power supply at nodes 1 and 2 will have a large deviation. The transient energy obtained by the system is large.

When a three-phase short circuit occurs at node 2 of the sending system and the fault is cut within the limit cutting time, the equivalent power angle instability curve at node 2 of the sending system node and its corresponding transient port energy tracking are shown in Supplementary Figure S5.

As shown in Supplementary Figure S5, when a three-phase short circuit occurs at node 2 of the sending system and the fault is cut within the limit time, the transient energy obtained by the system is small. By comparing the power angle variation characteristics of the system with and without the proposed transient energy balance robust control model, the following results can be obtained: adopting effective energy tracking control algorithms can better utilize the fast energy regulation characteristics of battery energy storage. The fluctuation amplitude of the sending system can be effectively reduced, and synchronization can be restored in a shorter time.

When a three-phase short circuit occurs at node 2 of the sending system and the fault is cut within the limit time, the frequency and voltage change curves of the equivalent synchronous power source at nodes 1 and node 2 of the sending system are shown in Supplementary Figure S6.

As shown in Supplementary Figure S6, if a three-phase short circuit occurs at node 2 of the renewable energy sending system and the fault is cut within the limit time, the propagation of transient energy in the system has little influence on the frequency and voltage of the equivalent synchronous power source at nodes 1 and 2, due to the transient energy imbalance of the system being small. At the same time, the voltage frequency control algorithm can significantly reduce the amplitude of voltage and frequency fluctuations, and can realize the rapid suppression of voltage and frequency fluctuations.