Devices in the IES can be divided into two types: certain and uncertain devices. Devices that can be controlled are certain devices; for example, CHP and energy conversion devices are certain devices, whereas devices that cannot be controlled are uncertain devices. In the IES, the uncertain devices are renewable energy devices and terminal energy loads. It should be noted that uncertain devices do not have any operating cost functions because the operating cost for energy devices and terminal energy loads is considerably low. These devices have only an energy input or output value x i , j m . The limits for uncertain devices are x i , j m   ∈ 0 , x i , j m − u p . (1)

x i , j m − u p represents the operating limit. Because the performance of uncertain devices including but not limited to lines and transformers is not infinite, we need to establish an operating limit.

The operating cost function and limits of CHP are C i , j , t k = a i , j x i , j 1 2 + b i , j x i , j , t k 1 + α i , j x i , j , t k 2 2 + β i , j x i , j , t k 2 + χ i , j (2)

where a i , j , b i , j , α i , j , β i , j , and χ i , j represent constants and a i , j is positive. We regard the energy conversion efficiency of gas or coal to power and heat as 100%. The constant values in Equation 2 and other equations determine the character of devices. The constant values affect the relationship between the operating cost and the rate of energies. The main motivation of the APDA is the scalability because the APDA suits all kinds of convex functions. How to match cost functions is not the emphasis of the APDA.

The limits are − P i , j ramp ≤ x i , j , t k 1 − x i , j , t k − 1 1 ≤ P i , j ramp , (3) d i , j x i , j , t k 1 + e i , j x i , j , t k 2 + f i , j ≥ 0 , (4) d i , j x i , j , t k 1 + e i , j x i , j , t k 2 ≤ g i , j , (5) x i , j m   ∈ x i , j m − down   , x i , j m − u p   . (6)

x i , j m − down   and x i , j   m − u p   indicate the operating limits of CHP, and P i , j ramp denotes the ramp limit of power. Because the output of’ CHP cannot change too fast, we should add a ramp limit. However, we only consider the ramp limit of power because the time-scale of heat is exceedingly large. In addition, the CHP also has a maximum operating limit and a start–stop limit. The reason for the maximum operating limit is very easy to understand. However, what is the reason for the start–stop limit? If a CHP stops operating, it is too hard to restart it. So we need to establish a start–stop limit to keep the CHP in operation. Zhang et al. (2017) have already shown that the maximum operating limit and the start-–stop limit for CHP should be linear as in (4) and (5). d i , j , e i , j , f i , j , and g i , j are positive constants. In addition, because the power and heat output transmission of CHP is not infinite, we establish limit (6), and x i , j m − down and x i , j m − u p indicate the transmission limits of CHP.

Natural gas is a fossil fuel; hence, we cannot produce it. Gas price is decided by other departments, and not by the IES.

The model and limits of energy storage devices are O i , j , t k i , t m = a i , j x i , j , t k m − S x i , j , t k m − S − μ i , j m + b i , j C = O i , j , t k i , t m − O i , j , t k i , t − 1 m , (7)

It is worth noting that there are two energy values in the energy storage device. One is the energy that is stored in the energy storage device, for which we use x i , j , t k m − S to express, and the other is the energy output (input) to (from) the IES, for which we adopt x i , j , t k m to express. In addition, because if the energy in the storage device is too much or too little will reduce the service life of the device, each energy storage device has an optimization store value, and we adopt O i , j , t k i , t m to express the optimization of energy storage value. So, the cost of the energy storage device is the loss of store optimization. a i , j , μ i , j m , and b i , j are constants and a i , j is negative. Thus, we obtain x i , j , t k m = x i , j , t k − 1 m − S − x i , j , t k m − S . (8)

The limits are − x i , j m in − S D ≤ x i , j , t k m ≤ x i , j m out − S D | x i , j m − S − min ≤ x i , j , t k m − S ≤ x i , j m − S − max . (9)

Not only the energy exchanged from the IES but also the energy stored in the energy storage devices has maximum and minimum limits, so there are four limits in one energy storage device. x i , j m i n − S D and x i , j m o u t − S D are the energy input and output limits, respectively; and x i , j m − S − min and x i , j m − S − max indicate minimum and maximum energy store limits, respectively.

Furthermore, there is a coupling item of energy load because one load can choose more than one energy to realize its function. In this study, we considered that the power to gas devices and electric boilers are supplied by energy transformation devices that can change energy loads. It is worth mentioning that the energy transformation devices need to satisfy the energy input required to be a price-changeable energy, so gas-based CHPs are not classified under such devices. The energy load changing and its cost is given as ∆ x a = − η × ∆ x b C = w × ∆ x a . (10)

Essentially, we can adopt load of energy a to replace load of energy b. The energy conversion efficiency is η , and the operating cost price is w .

The cost function of the equivalent gas producer is C i , t DGP = a i , t DGP G i , t DGP 2 + b i , t DGP G i , t DGP + c i , t DGP . (11)

How to model the network limits? Because the information and the energy flow are disconnected in the APDA and the privacy protection in the APDA is entire, the energy flow in the APDA can choose every path, and each we-energy can link to every other we-energies. In other words, the adjacency matrix of the APDA is an all-one matrix. For this reason, the energy flow in the APDA is more free than that in existing distribution methods, so the possibility of energy violating the limit is much less in the APDA than in other distribution methods. The only two possibilities of that harm are as follows. on one hand, if the hole energy flow in the hole IES is very large, all networks cannot handle the energy, and the energy violating the limit may happen. On the other hand, if the energy input or output in one we-energy is large, the energy transmission will go wrong. The complex network can change the energy flow to another path in the network, but if too much energy will transmit from (to) one we-energy, the complex network will be meaningless. The two cases can be handled together. We can establish an energy output or input limit for each energy because the energy that the IES network can contain is much larger than the total absolute value of each we-energy limit. In addition, because the operation of each device is dependent on the energy price, we can change the energy limit to the price limit of each we-energy. Each we-energy will compute its two price limits by the island model before energy management. Then, if the dispatch result of the price is beyond the energy limit, the we-energy will stop dispatch and operate by its price limit. The IES will consider it as a device that cannot respond to dispatch.

First, the communication path of the APDA is not a traditional centralized communication path or traditional distributed communication path. It is an annular-directed communication path from Zhang et al. (2017) shown in Figure 1A–C, which illustrates the traditional centralized and distributed IES. There are several advantages of the annular-directed communication path, of which the biggest advantage is that it is compatible with the APDA. The difference between the three communication paths is listed in Table 4.

Remark 1: In traditional research energy ratios, energy input or output and energy prices had to be known among neighboring agents; hence, they could not completely protect privacy. However, in the annular communication in the APDA, all data are cryptographic, which ensures that all data are completely protected even if among neighbor agents, no information is known by each other. The reason behind this will be introduced in the proceeding sections.

Privacy protection in the IES means that designers should make information, including but not limited to energy input, energy output, energy store values, and parameters in one we-energy, unknown to other we-energies because they may be business competitors. The APDA only needs to transmit some parameters, so protecting parameters is the content of privacy protection in the APDA. Existing methods established some neighbor agents and then transmitted data between them and did not transmit data to non-neighbor agents to protect privacy. Although privacy protection in existing studies can protect users’ privacy to a certain degree, private data are still obtained by neighbor we-energies. However, in the annular-directed communication path of Figure 1C, privacy protection can protect all data, including data between neighbor agents. In the annular-directed communication path, all we-energies can only get the summation of corresponding parameters before them in the annular path so they do not know the parameter values in every previous we-energy. Furthermore, we will establish some fictitious parameter values to protect the privacy of the first agent. By this method, all privacy will be protected so there are not any neighbor agents. It is worth noting that the cyber information communicating path for the APDA is a simple annular path. However, the energy transmitting path is not the circle. The cyber information communicating path only transmits information but not energy because if energies transmit in a simple circle, the energy transmission cost will be very high. After the running of the APDA, the balance of all energies will be realized. Then, how to transmit corresponding energies to corresponding we-energies will be an easy elementary math problem that does not need any algorithms. So, this article does not discuss the energy-transmitting path. However, the equivalent gas producer does not have the gas limit because it does not belong to the IES, although we regard it as a we-energy. It can transmit gas to corresponding places directly.

For some IESs with a strong ability for computation and communication, adopting their model may be the best choice. However, for some IESs with a weak ability for computation and communication, the APDA needs to reduce the order of models to reduce communication and computation by using the method in Lai et al. (2017). We assume that the model order limit of the IES is 2.

If the function f x is a polynomial function but the order is too high, we can adopt Chebyshev approximation to reduce the order, as shown below.

Herein, we introduce how one order can be reduced. If several orders need to be reduced, the APDA will reduce the order repeatedly until the order meets the standard. We assume that there is only one independent variable in the function. If there is more than one independent variable, the APDA will handle them sequentially. T 0 x = 1 ; T 1 x = x ; T 2 x = 2 x 2 − 1 ; … … T n + 1 x = 2 x T n x − T n − 1 x (12) P x = f x − a 2 n − 1 T n x , where f x represents the primitive function; P x represents the new function; and a indicates the coefficient of the highest order in f x .

If the function f x is an exponential function or another convex function that is not primitive, we can fit it into a primitive function and handle it as a primitive function using the Lagrange interpolation polynomial, as shown below.

First, we select three points on f x , and we use independent variables to solve function variables. Subsequently, we solve f x as w k x = ∏ m = 0 k − 1 x − x m , (13) L k x = ∑ p = 0 k y p w k + 1 x x − x p w k + 1 ′ x k , (14) where x m , y m represents a point on f x ; L k x represents the new polynomial function.

All theories about model adjusting can be found in Ma et al. (2022). Additionally, model adjusting is rarely observed. In most cases, models can satisfy the standard of the IES because the APDA can substantially reduce the communication and computational pressure.

Second, we solve the first-order partial differentials of the cost functions of each type of energy (considering other energy types as constants). Accordingly, we solve their inverse functions. These functions are named as f w x , w ⊂ 1,2,3 , … . The independent variables are the first-order partial differentials and the dependent variables are energy values. There may be more than one f w x in one device. The number of f w x indicates the energy types in the devices. Moreover, some devices do not have cost functions. Their operating conditions are unchangeable. Therefore, they cannot solve first-order partial differentials. The f w x values of such devices are constant functions; the constant values represent their operating conditions. Additionally, we do not consider the energy transmission devices in this step. It is worth mentioning that the APDA must first reduce orders before changing function shapes. This is because if function shapes are changed before reducing orders, the errors will be much larger. Furthermore, this study shows only the circumstance that the order limit is 2. However, in practice, the order limit can be any value owing to the different performances of energy routers.

It is worth mentioning that if the order limit of C x is 2, then the high-order differential of f w x will be 0. However, if the order limit of C x is higher than 2, then the high-order differential of f w x will not be 0. However, the independent variable order of f w x will decrease with the increase of differential order of f w x . If the independent variable order is less than –5, we can regard the high-order differential of f w x as 0.

The operations of the APDA are listed in Table 5; some complex operations are in the remarks.

Remark 1: Here, jj denotes the price-changeable energy type. In this article, jj is 2 because the power and heat price can be changed; however, the gas price cannot be changed. We should add some initialization values into Q and store Q into the control center. The reason behind that will be introduced later. Moreover, kk indicates the order limit. In this study, kk is 2. The order limit analysis will be introduced in the proceeding sections.

Remark 2: The rows of Q represent devices with each single price-changeable energy. The model in this article comprises two rows, which correspond to power and heat. The first column of Q contains the function values of f w x corresponding to energy. The second column of Q contains the first-order partial differential of the corresponding independent variable in the first column. The APDA will add corresponding energy and corresponding partial into the corresponding place. It is worth mentioning that two or more different values can be added into one place. We can use the model in this study as an example. Q + = A A 1 B B 1 , (15) A = ∑ j ⊂ 1,2,8,11,4,7 x i , j 1 , (16) A 1 = ∑ j ⊂ 1,2,8,11,4,7 ∂ C ∂ x i , j 1 , (17) B = ∑ j ⊂ 3,9,12,4,7 x i , j 2 , (18) B 2 = ∑ j ⊂ 3.9,12,4,7 ∂ C ∂ x i , j 2 , (19) where Q − is the Q before step 2.

Remark 3: First, the Q in step 4 should be settled. There are two Q s in the control center: first, we store an initialization Q in the control center in step 1, which we named Q 1 ; second, we use the algorithm to compute a Q and transmit it to the control center, which we named Q 2 . Q = Q 1 − Q 2 . (20)

Subsequently, we establish j j functions. These functions are f w w x , w w ⊂ 1,2,3 , … , j j . We establish a system of partial differential equations as ∂ k k f w w ∂ x k k = Q w w , k k + 1 k k ≥ 0 . (21)

Accordingly, all f w w can be solved.

Subsequently, we should use these f w w to ensure energy balance. Because all function values of each f w w denote the total energy in the IES, if all f w w values are 0, the energy balance is reached. However, that is not the optimization condition because the energy conversion is not considered. There are three circumstances that could be considered in this article

Circumstance 1: One price-changeable energy a changes to another price-changeable energy b.

Circumstance 2: One price-changeable energy a changes to one price-unchangeable energy b.

Circumstance 3: One price-unchangeable energy b changes to one price-changeable energy a.

In circumstance 1, we should solve these functions. x a = x b , (22) f w w a = − f w w b ≥ 0 , (23) where x indicates the independent variable of f w w ; f w w a and f w w b are the f w w of energy a and b, respectively.

In circumstance 2, we should solve the following functions. x a = p r b , (24) f w w a ≥ 0 , (25) where prb indicates the energy price of b.

In circumstance 3, we are required to solve the following functions. x a = p r b , (26) f w w a ≤ 0 . (27)

Other f w w without energy transformation are all 0.

Accordingly, we can solve these functions to work out all x of each f w w and all conversion energies. We adopt each x of the corresponding energy as the energy prices and divide the energy conversion value to all energy transformation devices. We add all energy prices and divided the conversion energy value into I .

Remark 4: The optimization operating condition of each energy devices can be solved as max ⁡ W = ∑ w w = 1 m x i , j , t k w w p r w w − C , (28a) where C represents the cost function and prww indicates the energy price of the w w -th energy. Furthermore, all conditions of conversion devices are in I .

Remarkably, the reason for placing a random initialized value into Q in the first step is to protect privacy. In the APDA, each device can only know the sum of the data; hence, they cannot know the data of each device. However, the second device can know the data of the first device because the total data are the data of the first device. Accordingly, we place an initialized value into Q in the first step to protect the data of the first device.

First, we should prove some propositions.

Proposition 1: The cost price of energy is the differential of its cost function.

First, because all cost functions are convex, the energy cost price is dynamic. The average energy cost price between x and x 0 can be given as pr = C x − C x 0 x − x 0 (28b)

If for all positive values ε , x 0 − x < ε , we can draw a conclusion that pr = C x − C x 0 x − x 0 = C ′ x (29)

Accordingly, the cost price on x is the differential of C x .

Proposition 2: If there is only one type of energy in the IES, the best operating condition of each device is that all first-order differentials are the same.

We assume another operating condition in the IES. Some devices named condition 2 with less cost based on the condition that all first-order differentials are the same, which is named condition 1. The energy balances are satisfied in conditions 1 and 2.

There are some devices in condition 2 with more energy than that in condition 1. To ensure energy balance, there are also some devices in condition 3 with less energy than that in condition 1. Accordingly, the devices with more energy consume more cost, and vice-versa. Conditions 2 and 3 collectively constitute another operating condition of the IES.

For conditions 1 and 2, we assume x is energy in condition 1, x n is energy in condition 2. n ⊆ 1,2 , … , m , and m indicates all devices with more energy in condition 2. From the Lagrange mean value theorem, we can draw a conclusion that C ′ x ≤ C x n − C x x n − x = C ′ δ ⊂ x , x n δ ≤ C ′ x n . (30)

The cost between condition 1 and condition 2 is C more = ∑ n ⊆ 1,2 , … , m C x n − C x = ∑ n ⊆ 1,2 , … , m C δ ⊂ x , x n ′ δ x n − x . (31)

We name all the energy as x x and the smallest C δ ⊂ x , x n ′ δ in n ⊆ 1,2 , … , m is C ∇ . Thus, we obtain C more = ∑ n ⊆ 1,2 , … , m C δ ⊂ x , x n ′ δ x n − x ≥ C ∇ ∑ n ⊆ 1,2 , … , m x n − x = C ∇ x x > C ′ x x x . (32)

For conditions 1 and 3, we assume x is energy in condition 1, x k is energy in condition 3. k ⊆ 1,2 , … , mm , and mm indicates all devices with more energy in condition 3. From the Lagrange mean value theorem, we can draw a conclusion that C ′ x k ≤ C x − C x k x − x k = C δ ⊂ x , x n ′ ϕ ≤ C ′ x . (33)

The lesser cost between condition 1 and condition 2 is given as Cless = ∑ k ⊆ 1,2 , … , m m C x − C x k = ∑ k ⊆ 1,2 , … , m m C δ ⊂ x k , x ′ ϕ x − x k (34)

We name all less energy as x x x , and the largest C δ ⊂ x k , x ′ ϕ in k ⊆ 1,2 , … , m m is C ∇ ∇ . Thus, we obtain Cless = ∑ k ⊆ 1,2 , … , m m C ϕ ⊂ x k , x ′ ϕ x − x k ≤ C ∇ ∇ ∑ k ⊆ 1,2 , … , m m x − x k = C ∇ ∇ x x x < C ′ x x x x . (35)

To ensure energy balance, x x x = x x , which yields Cless < C ′ x x x x = C ′ x x x < C more . (36)

Thus, the best operating condition for all devices require that all first-order differentials be the same. Proposition 2 stands correct.

Similarly, if the price of energy a is the same as that of energy b, the energy transformation is the optimal condition. Accordingly, the optimization is proved.

Proposition 3: f w w is a sum of f w . f w w = Q w w , 1 = f w 1 + f w 2 + ⋯ + f w n f w w ′ = Q w w , 2 = f w 1 ′ + f w 2 ′ + ⋯ + f w n ′ f w w n n = Q w w , n n + 1 = f w 1 n n + f w 2 n n + ⋯ + f w n n n

Accordingly, f w w is a sum of f w .

Therefore, the value of f w w is the energy mismatch. Accordingly, the convergence is proved.