Figure 1 shows a typical MCPS architecture, which consists of two parts: the physical layer and the cyber layer. The physical layer is composed of energy production units (photovoltaic, wind power, energy storage, etc.), energy conversion units, various types of loads and other physical equipment, and the power network connecting these physical devices. The cyber layer is composed of cyber equipment such as sensing equipment, distributed computing equipment, control decision equipment and a communication network connecting these cyber devices.

In MCPS, the cyber layer and the physical layer have the function of the energy management of the microgrid through the tight coupling of information flow and energy flow. Cyber-physical coupling is manifested in that the cyber layer perceives the operating status of the equipment in the physical layer through sensors, converts the energy flow into an information flow, transmits it to the cyber layer through the communication network and the information flow is processed by the cyber processing unit as the input of the decision-making control unit. The decision-making control unit converts the input information into control instructions and sends them to the physical equipment in the physical layer to control the operation of the physical equipment. It can be seen that the information input of the cyber layer to make decisions comes from the operating status of the equipment in the physical layer. The operating status of the physical layer equipment depends on the control commands issued by the cyber layer, indicating that the cyber layer and the physical layer in the MCPS are tightly coupled.

The hybrid system (HS) can be formally described by the following multi-element tuple (Petreczky and Schuppen, 2297). H = { M ,   Σ ,   P ,   δ ,   R ,   X } (1) Where: M denotes the set of discrete states of the system (also called the mode of operation of the system), and for any one discrete state, there are m = m ( t ) ∈ M ;

Σ = { e 1 ,   e 2 ,   ⋯ ,   e N } denotes the set of discrete events of the system, and for any one discrete event, it can be expressed as e = e ( t ) ∈ Σ ;

P denotes the parameter variable of the system;

δ denotes the system discrete state transition function;

R denotes the reset function, which determines the initial value of the continuous variable after the discrete state transition of the system occurs;

X = { A ( m ) } m ∈ M is the set of state spaces of the CVDS, Among them, A ( m ) = ( X ( m ) ,   G ( m ) ,   f ( m ) ,   λ ( m ) ) , the elements of A ( m ) is defined as follows:

X ( m ) denotes the continous state space of the system in mode m ∈ M , and x ( t ) ∈ X ( m ) denotes the state variables of the CVDS;

G ( m ) : X ( m ) × M × Σ → { t r u e ,   f a l s e } is the transition condition under the mode m ∈ M ;

f ( m ) denotes the evolution rule of the CVDS under mode m, characterizing the dynamic behavioral properties of the CVDS;

λ ( m ) indicates the output function of the system.

The dynamics of the HS can be described by the following equations. x ˙ ( m ) ( t ) = f ( m ) ( x ( m ) ( t ) ,   u ( t ) ,   p ) (2) y ( t ) = λ ( m ) ( x ( m ) ( t ) ,   u ( t ) ,   p ) (3) m + ( t ) = δ ( m − ( t ) ,   x − ( m − ) ( t ) ,   e ) (4) x + ( m + ) ( t ) = R ( m − ( t ) ,   m + ( t ) ,   x − ( m − ) ( t ) ,   e ) (5) Where the subscripts "+" denotes the state after the event e occurs and subscripts "−" denotes the state before the event e occurs, m − ( t ) and x − ( m − ) ( t ) denote the state mode and continuous state before the event e occurs, respectively, m + ( t ) and x + ( m + ) ( t ) denote the state mode and continuous state after the event e occurs, respectively.

Let t be the initial moment and the initial state of the system is ( m 0 ,   x 0 ( m 0 ) ) . In the discrete state m(t) = m ₀, the continuous dynamics are described by the differential Equation 2. At any time t ∈ T , the occurrence of an event may lead to a change in the operating mode of the system. The change of system operation mode is described by Eq. 3, which indicates that at time t, the event e changes the system from mode m − ( t ) to m + ( t ) . Also, as the system changes to operate in the successor mode, the state variables of the system may undergo a jump, i.e., change from system state ( m − ( t ) ,   x − ( m − ) ( t ) ) to system state ( m + ( t ) ,   x + ( m + ) ( t ) ) , and the change of state is described by Eq. 5. After the HS transitions to the successor mode, the system will continue to evolve along the trajectory described by Equations 2–5 if a new event occurs.

The MCPS presents a hybrid system characteristic during operation. For example, the energy flow in the physical system varies continuously with time according to a certain rule, which has the characteristics of a continuous variable dynamic system (CVDS). The cyber system controls the physical system through the information flow, including information collection, communication, processing, and control processes, all of which work digitally and consist of various types of events and their responses, which has the characteristics of a discrete event dynamic system (DEDS). Moreover, there is an interaction between the above two characteristics.

From the above analysis, it is clear that the physical system in the MCPS can be regarded as a CVDS, while the cyber system in the MCPS can be regarded as a DEDS. Therefore, the analysis of the interplay of information flow and energy flow in the MCPS can be converted into the analysis of the interplay of the CVDS and the DEDS in the MCPS.

To accurately characterize the interplay of the information flow and energy flow in the MCPS, a generic hierarchical modeling framework for cyber-physical integration modeling of MCPS is proposed, as shown in Figure 2.

In Figure 2, the modeling framework of MCPS includes two layers: the physical device layer and the controller layer.

1) Physical device layer: This layer mainly includes some PV unit, WT unit, BSS unit, etc., which can be considered as a unit-level CPS. The unit-level CPS can optimize the allocation of resources within the working capacity of the device through the physical device, its control system, and communication modules. At this layer, each device in the physical device layer transmits information related to its state and sensed environmental information to the controller layer and executes control commands issued by the controller layer.

2) Controller layer: As the upper layer of the microgrid CPS, this layer, on the one hand, mainly issues control commands to the physical device layer to drive the actions of the units in the physical device layer. On the other hand, it mainly uses data fusion, distributed computing, big data analysis, and other technologies to monitor, analyze and control the whole working status of the system in real-time, to achieve the optimal configuration, decision, and operation of the system.

In Figure 2, both the physical device layer and the controller layer consist of two parts: the continuous part and the discrete part. The continuous part is used to describe the characteristics of the continuous change of the device state over time. While the discrete part is used to describe the transfer process of the device between different operating states. The continuous part and the discrete part interact with each other through interface functions.

In this section, to clearly present the operation status, state transfer conditions, and state transfer process of each unit in the microgrid under the conditions of external control commands or various events, the cyber-physical integration modeling of MCPS is established using hybrid automaton (HA) (Lin et al., 2021). The HA is an extension of the original Finite State Machine (FSM) (Li et al., 2014), that is, continuous variables are added to the variables of the system and the influence of continuous variables is added to the state transfer process of the system.

The HA can describe the continuous dynamic behavior of the system as well as the discrete dynamic behavior of the system. An HA usually includes the following elements: 1) present state (the current state of the system), 2) conditions (also called events, when a condition is met, it will trigger the state transition of the system), 3) action (referring to the action executed after the system state transition condition is met) 4) second state (referring to the next state to be transitioned after the system state transition condition is met).

An HA is often represented by a directed graph and the state of the HA is represented by a circle. In each state, there is a function that describes how the continuous state changes. A directed curve is used to represent the state transition process of the system. The starting point of the directed curve represents the current state of the system and the end point represents the next state after the transition. At the same time, mark the transition conditions of the system state transition on the edge of the directed curve. Take an HA with two states as an example to illustrate the working principle of HA, as shown in Figure. 3.

The HA in Figure. 3 includes two states S1 and S2. The arrow pointing to S1 on the left indicates that the initial state of the system is S1. Above S1, draw a directed curve with the start and end points on S1. The condition a of state transition is marked on the curve, which means that when the system state is S1 and condition a is met, the state of HA remains unchanged; the directed curve marked with condition c from S1 to S2 shows that when the system state is S1 and condition c is satisfied, the system state is converted to S2. Similarly, according to the directed graph, when the system is in the S2 state, and the condition b is satisfied, the state of HA remains unchanged; when the condition b is satisfied, the system state changes to S1.

Taking the three states of a device in the MCPS as an example, the CPS model of the device is established using HA as shown in Figure. 4.

As seen from Figure 4, the transfer process between different states of the device can be described as follows: the trajectory of the device starts from the initial state, and the transfer process includes continuous state evolution and discrete state transfer. The initial discrete state of the device is q1, then the continuous variable x evolves according to the following differential equation. d x d t = f ( q 1 , x ) ,   x ( 0 ) = x 0 (16)

As long as the state transfer condition function g () is not satisfied, the device keeps evolving in the current operating state, i.e., the discrete state q remains unchanged with q(t) = q ₁. And when the state transfer condition function G (q ₁, q ₂) is satisfied for some reason, the discrete state transfer of the system is triggered and the system changes from the discrete state q ₁ to the discrete state q ₂ according to the state transfer function δ . The x jumps from the current value of the continuous variable to the new value according to the reset function and evolves according to the corresponding differential equation in the new discrete state q ₂. The switching between other modes is similar to the above process, and by repeating this process continuously, the device runs continuously.

The evolution of the above device can be described by the evolutionary trajectory shown in Figure 5.

The transfer conditions between the different modes can be divided into two categories: internal state events due to the evolution of internal state variables across a specific threshold or state plane, and external state events due to the influence of external inputs (e.g., command issuance) on the system. These two types of events can act on the MCPS either individually or simultaneously.

By connecting CPS models of the multiple devices established using HA in the MCPS through the CPS data bus, the CPS model of the entire microgrid system (i.e., the system-level CPS model) can be obtained, as shown in Figure 6.

As can be seen from Figure 6 that the operating process of the microgrid system can be characterized and analyzed by multiple state transfer models, and the transfer between different states is achieved using HA. In each operating state, the device’s continuous variation characteristics are described using differential or algebraic equations.

In summary, the microgrid model established in this paper is completely different from the conventional microgrid state space model in terms of the modeling approach and thinking and has more advantages in terms of observability. On the one hand, no matter what operating state the microgrid is in, the operating state of each unit in the microgrid can be known. On the other hand, during the transfer of the microgrid operating state, the operation state transfer process of each unit in the microgrid and the conditions of transfer can be clearly presented and observed, which is not available in the conventional state space model.

This paper takes the operating mode of Interlinking Converter (ILC) in hybrid AC/DC microgrids (HMs) as an example to verify the modeling method proposed in this paper. Figure 7 corresponds to the topological structure diagram of the HMs in this example, which consists of an AC subgrid, a DC subgrid, and ILC connecting the AC subgrid and the DC subgrid.

Among them, an unbalanced load is connected to the AC bus. The unbalanced load will make the AC bus voltage u _ac and the ILC current i _ILC unbalanced. The DC bus voltage u _dc will produce ripple through the coupling effect of ILC. Since ILC does not run at rated power most of the time, we can make full use of the remaining available capacity of ILC to realize HMs power quality control, including balancing u _ac, i _ILC and suppressing ripple in u _dc. The specific principles are beyond the scope of this paper and are referred to the references (Nejabatkhah et al., 2018; Wang et al., 2018). Among them, ILC can be divided into different operating modes according to different control objectives.

The different operating modes of ILC are realized by adjusting the equivalent impedance zILC of ILC. Different zILC corresponds to different operating modes. The switching between the different operating modes of ILC is determined by the imbalance degree of u _ac, i _ILC and the ripple ratio of u _dc. B _iILC represents the unbalance degree of i _ILC. B _uac represents the unbalance degree of u _ac. R _udc represents the ripple ratio of u _dc. B _iILC-up represents the upper limit of B _iILC. B _uac-up represents the upper limit of B _uac. R _udc-up represents the upper limit of R _udc. When B _iILC is greater than B _iILC-up, which means that the current imbalance of ILC exceeds the upper limit, ILC needs to switch to current imbalance compensation mode. Other situations are similar. The situation that two or three of i _ILC, u _ac, and u _dc exceed the upper limit is defined as an ILC fault state, and it is necessary to switch to the ILC fault state.

B _iILC-stan represents the standard value of B _iILC . B _uac-stan represents the standard value of B _uac. R _udc-stan represents the standard value of R _udc When ILC works in the current unbalance compensation mode, the ILC current balances. The condition that B _iILC equals to B _iILC-stan is satisfied. Other situations are similar and not mentioned here.

According to the modeling method of part of the MCPS model in Section 4.2, the modeling process of ILC can be obtained as follows.

① Determine the input and output variables of ILC:

Input variables include external input control command e _in and continuous input variable u _in. Among them, e i n = { o f f ,   o n } represents the start and stop running commands of ILC, and.

u i n = { B i I L C − u p ,   B u a c − u p ,   R u d c − u p ,   B i I L C − s ⁡ tan ,   B u a c − s ⁡ tan ,   R u d c − s ⁡ tan } .

The output variables include y _out and the output event signal e _out which indicates ILC failure, among them y o u t = { u a c ,   u d c ,   i I L C } .

② Determine the state variables of the ILC:

Discrete variable q = { q p o w e r ,   q s t a t e ,   q mod } includes:

q p o w e r = { o f f ,   o n } represents the running and stopping status of ILC;

q s t a t e = { f a u l t ,   n o r m a l } represents the fault and normal running status of the ILC;

q mod = { q mod 1 ,   q mod 2 ,   q mod 3 ,   q mod 4 } represents that ILC respectively works in uncontrolled mode, AC bus voltage negative sequence compensation mode, current imbalance compensation mode and DC bus voltage ripple suppression mode.

Continuous variables x = { u a c ,   u d c ,   i I L C ,   z I L C } .

③ Determine the transition function of continuous system and discrete system and the function of interaction between the two: define the value of parameter m as 5, and continuous system has different update functions under different values. The realization of the multi-mode operation of ILC is essentially achieved by adjusting the equivalent impedance z _ILC of ILC, so the update function of the continuous system is the function of adjusting the ILC equivalent impedance z _ILC, that is f _m (z _ILC, i _ILC, u _ac). The discrete system acts on the continuous part by the value of m.

Define five hypersurfaces h _i, which are used to generate five continuous events e _c, and influence the discrete system through e _c. Among them, e c = { e c 1 , e c 2 , e c 3 , e c 4 , e c 5 } includes: e _c1 represents the fault of ILC. e _c2 represents uncontrolled event. e _c3 represents ILC current balance. e _c4 represents voltage balance in AC bus. e _c5 represents no ripple in DC bus voltage.

The model of ILC can be obtained as shown in Figure 8 according to the definition of the above elements. The above model can describe the effect of external input on ILC, and the relationship between internal state and input-output. The transition process and conditions of ILC between different states can be described by the model in Figure 9.

The conversion conditions between the different modes in Figure 9 are shown in Table 1

Take the transition from state 3 to state 4 as an example to illustrate the transition process of ILC between different modes. When ILC is in state 3, ILC works in the current imbalance compensation mode. The output current of ILC is basically balanced, as shown in Figure 8C, but the AC bus voltage u _ac is unbalanced, as shown in Figure 8B. The DC bus voltage u _dc produces ripple, as shown in Figure 8D. When B _uac is greater than B _uac-up for some reason, adjust the impedance z _ILC of the ILC to make ILC run from state three to state four. At this time, ILC works in AC bus voltage negative sequence compensation mode, AC bus voltage u _ac is basically balanced, but ILC output current i _ILC becomes unbalanced, and DC bus voltage u _dc produces ripple. The transition process between the other modes is similar. The waveform of the conversion process of ILC in different working states in Figure 7 above is obtained as shown in Figure 10, based on the simulation model shown in Figure 9 built by the PSCAD simulation software. Among them, state 1, state 2, state 3 and State 4 respectively correspond to the uncontrolled mode, ILC output current imbalance compensation mode, AC bus voltage imbalance compensation mode and DC bus voltage ripple suppression mode.

Take the transition from state 3 to state 4 as an example to illustrate the transition process of ILC between different modes. When ILC is in state 3, ILC works in the current imbalance compensation mode. The output current of ILC is basically balanced, as shown in Figure 10C, but the AC bus voltage u _ac is unbalanced, as shown in Figure 10B. The DC bus voltage u _dc produces ripple, as shown in Figure 10D. When B _uac is greater than B _uac-up for some reason, adjust the impedance z _ILC of the ILC to make ILC run from state three to state four. At this time, ILC works in AC bus voltage negative sequence compensation mode, AC bus voltage u _ac is basically balanced, but ILC output current i _ILC becomes unbalanced, and DC bus voltage u _dc produces ripple. The transition process between the other modes is similar.