This paper aims to minimize the operating cost of polymorphic low-carbon port microgrid. The objective function includes three parts: one is the operating cost of the power generation and energy storage device; one is the cost of trading with the main grid; the other is the carbon cost. The function is as follows: F = min { F 1 + F 2 + F 3 } (1) where, F is the total operating cost of the polymorphic low carbon port microgrid, F is the operating cost of the power generation and energy storage device, F 2 is the cost of trading with the main grid and F 3 is the carbon cost.

To ensure the reliable operation of the port microgrid, the following constraints should be followed.

1) Power balance constraint    ∑ i = 1 n P i + P M = ∑ i = 1 n P l o a d + P c c + P c s (9) where, P M is the main grid input and output power, and P l o a d is the load of the port microgrid.

2) Output power constraints for power generation devices P i min ≤ P i ≤ P i max (10) where, P i min and P i max are the minimum output power and maximum output power of the i th generation device, respectively.

3) Energy consumption constraints for carbon capture devices P c c min ≤ P c c ≤ P c c max (11) where, P c c min and P c c max are the minimum and maximum energy consumption of the carbon capture device, respectively.

4) Energy storage constraints S o C min ≤ | 1 − P s P e | ≤ S o C max (12) where, P e is the rated power of the energy storage device, S o C min and S o C max are the upper and lower limits of the capacity of the energy storage device, respectively.

A directed graph can be represented as G = { V , E } , N is the number of nodes. Where V = { v 1 , v 2 , v 3 , ...... v N } is the set of nodes and E = { e 1 , e 2 , e 3 , ...... e M } is the set of lines connecting two points, that is, the set of relations between two points. a i j denotes the edges ( v i , v j ) , when ( v i , v i ) ∈ E , a i j = 1 ; when ( v i , v i ) ∉ E , a i j = 0 . The adjacency matrix represents the relations between vertices, and A = ( a i j ) N × N , is an N-order square matrix (Li et al., 2020).

In the grid-connected mode, since the electricity price of the main grid is not affected by other factors, the main grid does not receive information, but only transmits information. However, in the energy flow, the main grid is able to exchange energy with the generation plant, carbon capture and storage plant. In the event that the port microgrid generates too much power, or does not generate enough power to meet the required load, it can sell power or buy power from the main grid.

Since port microgrids exhibit distributed characteristics, this section proposes a distributed energy management approach for port microgrids in grid-connected mode, where the model for the energy management problem can be written as the following equation. min     { ∑ i = 1 N F 1 + F 2 + F 3 } s .t .       ∑ i = 1 n P i + P M = ∑ i = 1 n P l o a d + P c c + P c s              P i min ≤ P i ≤ P i max (13)

Taking (1)–(8) into (13), the objective function is further expressed as: min    { ∑ i = 1 N ( ( P i − α i ) 2 2 β i + φ i ) + λ 0 P M + C t a x E e + σ E c c } (14)

Since in the objective function, E e and E e in the carbon cost function are both linearly related to P i , which can be combined with the power generation cost, the objective function (14) can be organized as the following formula. min { ∑ i = 1 N ( ( P i − α i ′ ) 2 2 β i ′ + φ i ′ ) + λ 0 P M } (15) where, α i ′ , β i ′ and φ i ′ are the cost coefficients of the collapsed function.

The Lagrangian function of the model for this problem takes the form of: L ( P 1 , P 2 ... , P N , P M , λ , ν ¯ , ν ¯ ) = ∑ i = 1 N ( ( P i − α i ′ ) 2 2 β i ′ + φ i ′ ) + λ 0 P M + λ ( ∑ i = 1 n P i + P M − ∑ i = 1 n P l o a d − P c c − P c s )    + ∑ i = 1 n ν i ¯ ( P i − P i max ) + ∑ i = 1 n ν i ¯ ( P i min − P i ) (16) where, λ , ν ¯ and ν ¯ are all Lagrangian multipliers, and the carbon capture and carbon storage loads can be expressed as P c c + P c s = D i P i , D i is a coefficient matrix.

As the low carbon port microgrid distributed energy management model contains inequality constraints, in order to find its optimal solution, its KKT (Karush Kuhn Tucker) condition is analyzed. { ∂ L ( P 1 ∗ , P 2 ∗ ... , P N ∗ , P M ∗ , λ ∗ , ν ∗ ¯ , ν ∗ ¯ ) ∂ { P 1 , P 2 ... , P N , P M } = 0 ∑ i = 1 n P i + P M − ∑ i = 1 n P l o a d − P c c − P c s = 0 ν i ∗ ¯ ( P i − P i max ) = 0 ν i ∗ ¯ ( P i min − P i ) = 0 (17)

According to (17), we get: { λ ∗ = P i − α i ′ β i ′ ( 1 − D i ) λ ∗ = λ 0 (18)

Bringing (18) into (17) gives the global optimum result as: P i ∗ = { λ ∗ β i ′ ( 1 − D i ) + α i ′      P i ¯ ≤ λ ∗ β i ′ ( 1 − D i ) + α i ′ ≤ P i ¯    P i ¯                               λ ∗ β i ′ ( 1 − D i ) + α i ′ > P i ¯    P i ¯                               λ ∗ β i ′ ( 1 − D i ) + α i ′ < P i ¯ (19) P M ∗ = ∑ i = 1 n P l o a d + ∑ i = n N D i P i − ∑ i = 1 n P i (20)

When in grid-connected mode, the iterative approach of electricity prices follows the leader-following consistency (Chen and Li, 2021), with the following iterative process. According to (21), the amount of power generated by each generating unit can be obtained as:   λ i ( k + 1 ) =   λ i ( k ) + μ i [ ∑ j ∈ N i a i j ( λ j ( k ) − λ i ( k ) ) + a i 0 ( λ 0 − λ i ( k ) ) ] (21)

The above equation represents that the λ of each node follows the leader-following consensus to the main grid price λ 0 and the node a i 0 = 1 , which is capable of receiving direct main grid communication information from the main grid.

According to (21), the power generation capacity of each generation unit can be obtained as follows.

It can be obtained that the power generation capacity of each generation unit is as follows. P i ( k ) = { λ ( k ) β i ′ ( 1 − D i ) + α i ′      P i ¯ ≤ λ ( k ) β i ′ ( 1 − D i ) + α i ′ ≤ P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ > P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ < P i ¯ (22) where, P i ( k ) is the amount of electricity generated by the i th node at the k th iteration.

Then, based on the known estimated local mismatch values P i ( k ) , the formula is as follows. { ρ i ( k + 1 ) = Δ P i ^ ( k ) + ε [ ∑ j ∈ N i a i j ( Δ P j ^ ( k ) − Δ P i ^ ( k ) ) ] + Δ P i ( k + 1 ) − Δ P i ( k ) Δ P i ^ ( k + 1 ) = ( 1 − a 0 i ) ρ i ( k + 1 ) (23) where, Δ P i ( k ) is the actual local mismatch at the k th iteration and Δ P i ^ ( k ) is the estimated local mismatch at the k th iteration. Δ P i ^ ( k + 1 ) = ( 1 − a 0 i ) ρ i ( k + 1 ) means that for the neighbor unit of the main grid, the power complement of the main grid is obtained directly in each iteration. At the k + 1 th iteration, the estimation of the local power mismatch becomes zero. P M ( k + 1 ) = P M ( k ) + ∑ i = n N a 0 i ρ i ( k + 1 ) (24) where, P M ( k + 1 ) is the supplementary power value of the main grid to neighbor nodes at the k + 1 th iteration, which equals the sum of the supplementary power value of the k th iteration and the mismatch value of all neighbor nodes.

For the above algorithm, for a connected undirected graph G, when the step size of the algorithm satisfies μ i ∈ ( 0 , 1 ∑ j = 0 N a i j ) and ε ∈ ( 0 , 1 max i = 1,2,3... N ∑ j = 0 N a i j ) , it can be obtained that: lim k → ∞ λ i ( k ) = λ ( 0 ) ,    i = 1,2,3... N lim k → ∞ P i ( k ) = P i ∗ ,    i = 1,2,3... N lim k → ∞ Δ P i ^ ( k ) = 0 ,    i = 1,2,3... N lim k → ∞ P M ( k ) = ∑ i = 1 n P l o a d + ∑ i = n N D i P i ∗ − ∑ i = 1 n P i ∗ = P M ∗ (25) where, the value of P M ( k ) can be positive or negative.

The low carbon port microgrid under the grid-connected mode considered in this paper can both buy electricity from and sell electricity to the port’s main grid, ensuring a balance between energy supply and demand and enabling the economic operation of the port microgrid.

In island mode, the port microgrid cannot buy power from the main grid and has to be powered by a generation device to supply the load. In this case, not only the economy of the port must be considered, but also its security. The economic optimization of the port microgrid is achieved on the basis that the port load demand can be met. In this section, the incremental cost of the islanding model is corrected by adding a penalty factor to meet the supply-demand balance of the port microgrid. The iterative formula used is as follows.   λ i ( k + 1 ) =   λ i ( k ) + μ i ′ [ ∑ j = 1 N a i j ( λ j ( k ) − λ i ( k ) ) ] + ζ ( k ) Δ P i ^ ( k ) (26) where, μ i ′ is the step size of the algorithm and ζ ( k ) is the feedback gain.

According to (26), the power generation capacity of each generation unit can be obtained by P i ( k ) = { λ ( k ) β i ′ ( 1 − D i ) + α i ′      P i ¯ ≤ λ ( k ) β i ′ ( 1 − D i ) + α i ′ ≤ P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ > P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ < P i ¯ (27) where, P i ( k ) is the amount of electricity generated by the i th node at the k th iteration.

Then, according to the known P i ( k ) , the local mismatch value is estimated, and the formula is as follows. { ρ i ( k + 1 ) = Δ P i ^ ( k ) + ε ′ [ ∑ j ∈ N i a i j ( Δ P j ^ ( k ) − Δ P i ^ ( k ) ) ] + Δ P i ( k + 1 ) − Δ P i ( k ) Δ P i ^ ( k + 1 ) = ρ i ( k + 1 ) (28) where, Δ P i ( k ) is the actual local mismatch value at the k th iteration and Δ P ^ i ( k ) is the estimated (local) mismatch value at the k th iteration.

For a connected undirected graph G, when the step size of the above algorithm satisfies μ i ′ ∈ ( 0 , 1 ∑ j = 0 N a i j ) and ε ′ ∈ ( 0 , 1 max i = 1,2,3... N ∑ j = 0 N a i j ) , for feedback gains ζ ( k ) , it satisfies lim k → ∞ ζ ( k ) = 0 and ∑ k = 0 ∞ ζ ( k ) = ∞ . λ i ( 0 ) is any given initial value, then we can obtain that, in the island mode, when the local estimate of the initial value of the mismatch value satisfies Δ P ^ i ( 0 ) = ∑ i = 1 n P l o a d + ∑ i = n N D i P i ( 0 ) − ∑ i = 1 n P i ( 0 ) , it can be obtained that: lim k → ∞ λ i ( k ) = λ i ∗ ,    i = 1,2,3... N lim k → ∞ P i ( k ) = P i ∗ ,    i = 1,2,3... N lim k → ∞ Δ P i ^ ( k ) = 0 ,    i = 1,2,3... N (29)

In the island model considered in this paper, security and the guarantee of supply and demand balance are more important as the port microgrid is not connected to the main grid. The essence of the algorithm is to follow the average consensus and add a penalty factor for feedback gain to correct the incremental cost in order to achieve safety and economy of port operation.

Low carbon port microgrid switching mode operation refers to the process of switching from grid-connected mode to island mode or from island mode to grid-connected mode. In the switching process, it is necessary to realize the economical operation of the port microgrid as much as possible on the basis of ensuring the supply and demand balance. At this time, the iterative process of the electricity price   λ is as follows. λ i ( k + 1 ) =   λ i ( k ) + μ i ′ [ ∑ j ∈ N i a i j ( λ j ( k ) − λ i ( k ) ) + m a i 0 ( λ 0 − λ i ( k ) ) ] + ζ ′ ( k ) Δ P i ^ ( k ) (30) where, m denotes the mode of operation of the port microgrid, when in grid-connected mode, m = 1 , otherwise, m = 0 ; when m = 1 , λ of each node follows the leader-following consensus to the main grid electricity price λ 0 , where a i 0 = 1 represents the node can directly receive the communication information from the main grid; when m = 0 , λ of each node follows the average consensus, and adopts the penalty factor correction method to tend to λ ∗ .

According to the above equation, the power generation capacity of each generation unit can be obtained by: P i ( k ) = { λ ( k ) β i ′ ( 1 − D i ) + α i ′      P i ¯ ≤ λ ( k ) β i ′ ( 1 − D i ) + α i ′ ≤ P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ > P i ¯    P i ¯                               λ ( k ) β i ′ ( 1 − D i ) + α i ′ < P i ¯ (31) where, P i ( k ) is the amount of power generated by the generation unit at the i th node at the k th iteration.

Then, based on the known estimated local mismatch values P i ( k ) , the formula is as follows. { ρ i ( k + 1 ) = Δ P i ^ ( k ) + ε ′ [ ∑ j ∈ N i a i j ( Δ P j ^ ( k ) − Δ P i ^ ( k ) ) ] + Δ P i ( k + 1 ) − Δ P i ( k ) Δ P M i ( k + 1 ) = m a 0 i ρ i ( k + 1 ) P M i ( k + 1 ) = m [ P M i ( k ) + a 0 i Δ P M i ( k + 1 ) ] Δ P i ^ ( k + 1 ) = ρ i ( k + 1 ) + a 0 i [ P M i ( k ) − P M i ( k + 1 ) ] (32)

Among them, ∑ i = 1 N Δ P ^ i ( 0 ) = ∑ i = 1 n P l o a d + ∑ i = n N D i P i ( 0 ) − ∑ i = 1 n P i ( 0 ) and P M i ( 0 ) = 0 .

Then, the power exchanged with the main grid can be expressed as: P M ( k ) = ∑ i = 1 N P M i ( k ) (33)

The switching mode considered in this paper is the switching of the port microgrid between grid-connected mode and island mode. With the algorithm used in this paper, the safe and economic operation of the port microgrid can be achieved on the basis of ensuring a balance between supply and demand in the port microgrid.

For this case, the operation of a polymorphic port microgrid in grid-connected mode is considered and divided into two scenarios: buying electricity from and selling electricity to the main grid.

For this case, the operation of a polymorphic port microgrid in island mode is considered. Unlike the grid-connected mode, in the island mode, it is not possible to purchase power from the main grid, so it is essential to ensure the economics and the security of the polymorphic port microgrid. In this section, the centralized algorithms and the distributed algorithms are used to solve the proposed model, and the obtained simulation results are shown in Figures 7A–D.

The results obtained using the centralized algorithm are shown in Figure 7A, which shows the power supply of each power supply device, and the lowest cost result is 4,035.78 ¥. By using distributed algorithms, the low-carbon port microgrid costs 4,116.80 ¥, and emits 4.61 t of C O 2 into the air, with 10.77 t of C O 2 treated by the carbon capture and storage device. The difference between the results obtained by the centralized algorithm and the distributed algorithm proposed in this paper is small, which can prove the accuracy of the proposed algorithm. In this mode, there is no leader and the values for each unit follow an average consensus under the penalty factor correction method. It can be seen in the graph that the values converge to consistency at k = 25 , with a faster convergence rate, λ = 0.78 at this time. The power supply for each device is [18.76, 17.78, 21.68, 10.53, 22.66, 26.01], and the total power used by the port microgrid is 117.42 MW, which is able to meet the required load of the port.

The cost is reduced compared to the grid-connected mode because the port’s power generation device powers all loads. At the same time, as the island mode cannot purchase power from the main grid, to ensure the safety of the port during island mode operation and to meet the balance of supply and demand of the port microgrid, the conventional power plant generates more power and emits more carbon dioxide into the air, illustrating that the grid-connected mode can effectively reduce carbon emissions from the port and is more friendly to the environment. In terms of convergence speed, the island model uses average consensus, which is significantly faster than the grid-connected model. The simulation results for both the grid-connected and island modes, which eventually reach convergence, prove the effectiveness of the proposed algorithm.

During the operation of the port microgrid, the possible emergencies will lead to its failure to connect with the main grid, changing it from grid-connected mode to island mode, which will make the supply-demand balance and security unable to be guaranteed. While the port, as an essential transport hub node, is obliged to operate continuously and reliably. Therefore, it is necessary to ensure the safety and reliability of the port microgrid during operation. In this section, a distributed energy management algorithm is used to study the switching mode of the port microgrid, which is divided into two switching modes: “grid-connected–island–grid-connected” and “island–grid-connected–island” for energy management. The simulation results are shown in Figures 8A–D and Figures 9A–D.