Distributed Peer-to-Peer Energy Trading for Residential Fuel Cell Combined Heat and Power Systems

This research studies a SOFC system using gas (whether natural gas or liquefied petroleum gas) as the input fuel. The gas energy consumed by the FC at dwelling $i$ is computed by

$\displaystyle E_{i,fc}(t)=\dfrac{P_{i,fc}(t)\Delta t\xi_{e}}{\eta_{i,e}(t)\eta_{i,g2h}}$

(1)

Hence, the cost of gas usage at dwelling $i$ is

$\displaystyle C_{i,g}(t)=p_{g}(t)E_{i,fc}(t)=p_{g}(t)\dfrac{P_{i,fc}(t)\Delta t\xi_{e}}{\eta_{i,e}(t)\eta_{i,g2h}}$

(2)

SOFC-CHP system is the main source of power generation for dwellings, which continuously runs throughout the day [25]. The exhausted heat from the FC is employed to heat water from city water network, which is then charged to a hot water storage tank. Hot water from this storage tank is utilized for fulfilling the dwelling hot water demand including air conditioner, floor heating, kitchen, bath, washing machine, toilet, etc. Any lack of hot water is compensated by a backup boiler inside the FC-CHP system (see Fig. 1). The hot water amount charged to the storage tank is computed by

$\displaystyle HT_{i,in}(t)=\frac{E_{i,fc}(t)\eta_{i,hr}(t)}{q_{w}(T_{i,ht}-T_{i,cn})}=\zeta_{i}\frac{\eta_{i,hr}(t)}{\eta_{i,e}(t)}P_{i,fc}(t)$

(3)

where $\zeta_{i}\triangleq\frac{\Delta t\xi_{e}}{q_{w}(T_{i,ht}-T_{i,cn})\eta_{i,g2h}}$ is a positive constant. Because of the nonlinear dependence of the power and heat efficiencies $\eta_{i,e}(t)$ and $\eta_{i,hr}(t)$ to the FC output power $P_{i,fc}(t)$, $HT_{i,in}(t)$ is also a nonlinear function of $P_{i,fc}(t)$. Nevertheless, in the following we show that $HT_{i,in}(t)$ can be well approximated by a linear function of $P_{i,fc}(t)$.

For a variety of FCs, the power and heat efficiencies $\eta_{i,e}(t)$ and $\eta_{i,hr}(t)$ have the exponential forms [27, 32, 2] described as follows.

	$\displaystyle\eta_{i,e}(t)$	$\displaystyle=a_{i,e}-b_{i,e}e^{-k_{i,e}\frac{P_{i,fc}(t)}{P_{i,fc}^{\max}}},$		(4)
	$\displaystyle a_{i,e}$	$\displaystyle=b_{i,e}+\eta_{i,e}^{0},b_{i,e}=\frac{\eta_{i,e}^{\max}-\eta_{i,e}^{0}}{1-e^{-k_{i,e}}}$
	$\displaystyle\eta_{i,hr}(t)$	$\displaystyle=a_{i,hr}-b_{i,hr}e^{-k_{i,hr}\frac{P_{i,fc}(t)}{P_{i,fc}^{\max}}},$
	$\displaystyle a_{i,hr}$	$\displaystyle=b_{i,hr}+\eta_{i,hr}^{0},~{}b_{i,hr}=\frac{\eta_{i,hr}^{\max}-\eta_{i,hr}^{0}}{1-e^{-k_{i,hr}}}$

where $P_{i,fc}^{\max}$ is the maximum rated output power of the FC in dwelling $i$; and $k_{i,e},\eta_{i,e}^{0},\eta_{i,e}^{\max},k_{i,hr},\eta_{i,hr}^{0},\eta_{i,hr}^{\max}$ are constant parameters. To linearize $HT_{i,in}(t)$, we approximate

$$\frac{\eta_{i,hr}(t)}{\eta_{i,e}(t)}P_{i,fc}(t)\approx\alpha_{i,wt}P_{i,fc}(t)+\beta_{i,wt}$$

(5)

where $\alpha_{i,wt}$ and $\beta_{i,wt}$ are constants. As an example, using the data of a SOFC provided in [27], the linear approximation above, conducted in MATLAB, gives us $\alpha_{i,wt}=0.9439$, $\beta_{i,wt}=0.006502$, with $95\%$ confidence bounds, which is depicted in Figure 2.

As seen in Figure 2, this linear approximation is very close to the original nonlinear curve. Therefore, the hot water amount charged to the storage tank is mostly linearly proportional to the FC output power. This obviously will reduce significantly the complexity of the FC-CHP optimal energy management problem when the hot water trading [3, 2, 27] is considered, because the optimization problem is convexified and no efficiency matching algorithm is needed, unlike that in [27]. This point is further elaborated in Section 3.1, where the nonlinear term $\frac{P_{i,fc}(t)}{\eta_{i,e}(t)}$ in the gas usage cost is also shown to be reasonably estimated as a linear function of the FC output power $P_{i,fc}(t)$.

Having the linear approximation of hot water storage – FC output power relation, we can estimate quite precisely the storage tank level at each time step $t$. Accordingly, a smart FC-CHP controller, which memorizes electricity and water consumptions in previous days, will notify dwellings’ owners once the hot water storage tanks are near their limits through in-dwelling monitors or applications in smart phones, tablets, etc., to release hot water in advance of users’ desire, e.g. for bath tub, washing machines, etc. Such actions can also be automatically executed by the controller through pre-programmed functions. Thus, in this study we do not consider hot water in the FC-CHP optimal energy management, instead we focus on gas-electricity management with electricity trading between FC-CHP equipped dwellings.

Considering the P2P energy trading between $n$ dwellings/peers during the time interval $[1,\mathbb{T}]$. Denote $P_{ij}(t)$ the energy to be traded at time step $t$ between the $i$-th and $j$-th peers, where $P_{ij}(t)>0$ means peer $i$ buys electricity from peer $j$, and vice versa, $P_{ij}(t)<0$ means peer $i$ sells electricity to peer $j$. To simplify the trading of dwellings/peers, it is assumed that at each time step one dwelling/peer only buys or sells energy, but not to do both. For each dwelling/peer $i$, denote $\mathcal{N}_{i}$ its neighboring set, i.e. the set of other peers it is communicated for energy trading. Denote $\mathcal{G}$ the inter-peer communication graph. Due to the bilateral trading between peers, $\mathcal{G}$ is undirected. Next, let $a_{ij}$ be elements of the adjacency matrix $\mathcal{A}$, i.e. $a_{ij}=1$ if peers $i$ and $j$ are connected, and $a_{ij}=0$ otherwise. The degree matrix $\mathcal{D}$ is defined by $\mathcal{D}=\mathrm{diag}\{d_{i}\}_{i=1,\ldots,n}$, where $d_{i}\triangleq\sum_{j\in\mathcal{N}_{i}}{a_{ij}}$. Then the Laplacian matrix $\mathcal{L}$ associated to $\mathcal{G}$ is defined by $\mathcal{L}=\mathcal{D}-\mathcal{A}$.

Let $n_{i}\triangleq|\mathcal{N}_{i}|$, $P_{i}\in\mathbb{R}^{n_{i}}$ be the vector of all $P_{ij}$ with $j\in\mathcal{N}_{i}$, $P_{i,tr}(t)$ be dwelling $i$’s total traded power. Then $P_{i,tr}(t)=\mathbf{1}_{n_{i}}^{T}P_{i}(t)$. Next, denote $C_{i}(P_{i,tr}(t))$ the total cost of dwelling/peer $i$ for trading in the P2P market, which composes of the following two components. The first component is the utility function,

$$C_{i,1}(P_{i}(t))=a_{i}(t)P_{i,tr}^{2}(t)+\tilde{b}_{i}(t)P_{i,tr}(t)+c_{i}(t)$$

(6)

The parameters $a_{i}(t),\tilde{b}_{i}(t),c_{i}(t)$ are only known for peer $i$, which are presented here as time-dependent parameters to reflect the time-varying and complex behaviors of dwellings/peers. The second element is the implementation cost for the traded powers to be physically executed through the power network,

$$C_{i,2}(P_{i}(t))=\gamma P_{i,tr}(t)$$

(7)

where $\gamma>0$ is a fixed rate. Thus, summing up (6) and (7) gives us the following total cost of each peer in the P2P market

$$C_{i}(P_{i}(t))=a_{i}(t)P_{i,tr}^{2}(t)+\hat{b}_{i}(t)P_{i,tr}(t)+c_{i}(t)$$

(8)

where $\hat{b}_{i}(t)\triangleq\tilde{b}_{i}(t)+\gamma$.

The first constraint is the power balance

$$P_{i,fc}(t)+P_{i,tr}(t)+P_{i,grid}(t)=P_{i,dem}(t)\;\forall\;t=1,\ldots,\mathbb{T}$$

(9)

The priority for compensating any lack of demand is from the P2P energy market. Grid power is bought only when powers from neighboring dwellings are not enough.

Next, the electricity generated by a FC is bounded by

$$P_{i,fc}^{\min}(t)\leq P_{i,fc}(t)\leq P_{i,fc}^{\max}\;\forall\;t=1,\ldots,\mathbb{T};\;i=1,\ldots,n$$

(10)

Since FC is the main power source for dwellings, it is expected that the FC output power is enough for a dwelling power demand, i.e.,

$$P_{i,fc}^{\min}(t)\geq P_{i,dem}(t)$$

(11)

However, this is not always true, because there is a possibility that a dwelling power demand could be greater than the FC maximum rated output power, i.e., $P_{i,dem}(t)\geq P_{i,fc}^{\max}$. In this case, we have

$$P_{i,fc}^{\min}(t)=P_{i,fc}^{\max}$$

(12)

The combination of (11) and (12) therefore gives us

$\displaystyle P_{i,fc}^{\min}(t)\triangleq\min\{P_{i,dem}(t),P_{i,fc}^{\max}\}$

(13)

Lastly, the constraints on the maximum and minimum P2P electricity trading are determined by

$$P_{i,tr}^{\min}(t)\leq P_{i,tr}(t)\leq P_{i,tr}^{\max}(t)\;\forall\;t=1,\ldots,\mathbb{T}$$

(14)

where

	$\displaystyle P_{i,tr}^{\min}(t)$	$\displaystyle\triangleq\min\{0,P_{i,dem}(t)-P_{i,fc}^{\max}\},$
	$\displaystyle P_{i,tr}^{\max}(t)$	$\displaystyle\triangleq\max\{0,P_{i,dem}(t)-P_{i,fc}^{\max}\}$		(15)

Dwellings adjust the functionality of their FCs to minimize their total energy costs during the considered time period $[1,\mathbb{T}]$, subject to the specified constraints, which results in the following optimization problem.

	$\displaystyle\min~{}$	$\displaystyle\sum_{t=1}^{\mathbb{T}}\sum_{i=1}^{n}C_{i,g}(t)+C_{i}(P_{i}(t))$		(16a)
	s.t.	$\displaystyle\eqref{e-fc},\eqref{e-2}$		(16b)

It is worth emphasizing again that electric demand in each dwelling is time-varying and inconsistent from one day to another. Moreover, electric demand prediction is often inexact. Therefore, solving (16) for the whole time period $[1,\mathbb{T}]$ at once is not a good choice in practice for intra-day energy management. Instead, it is better to solve(16) at each time step $t$ so that the FC-CHPs in dwellings can be scheduled properly to the variation of electric demands. As such, the cooperative energy management problem (16) should be solved one-time-step ahead. This means in the optimization problem to be solved for the next time step $t$, the values of all variables and costs up to the current time steps are known. Therefore, we only need to find the variables and costs at time step $t$, as shown in (17).

	$\displaystyle\min~{}$	$\displaystyle\sum_{i=1}^{n}C_{i,g}(t)+C_{i}(P_{i,tr}(t))$		(17a)
	s.t.	$\displaystyle P_{i,fc}^{\min}(t)\leq P_{i,fc}(t)\leq P_{i,fc}^{\max}$		(17b)
		$\displaystyle P_{i,tr}^{\min}\leq P_{i,tr}(t)\leq P_{i,tr}^{\max}$		(17c)
		$\displaystyle P_{ij}(t)+P_{ji}(t)=0~{}\forall\,j\in\mathcal{N}_{i},i,j=1,\ldots,n$		(17d)

where $P_{i,fc}^{\min}(t)$, $P_{i,tr}^{\min}$, and $P_{i,tr}^{\max}$ are specified in (13) and (2.3).

Note that (17) is a non-convex problem due to the nonlinearity of the FC power efficiency $\eta_{i,e}(t)$ which causes the non-convexity of the cost function $C_{i,g}(t)$ in (2). Therefore, in this research we propose an approximation of $C_{i,g}(t)$ as a linear function of $P_{i,fc}(t)$, hence (17) becomes convex.

As aforementioned, we aim at linearizing $C_{i,g}(t)$ to make (17) a convex problem. To do so, $\frac{P_{i,fc}(t)}{\eta_{i,e}(t)}$ is linearized as follows,

$\displaystyle\frac{P_{i,fc}(t)}{\eta_{i,e}(t)}\approx\alpha_{i,fc}P_{i,fc}(t)+\beta_{i,fc}$

(18)

where $\alpha_{i,fc}$ and $\beta_{i,fc}$ are constants. Therefore, the FC-CHP gas energy consumption is approximated by

$\displaystyle E_{i,fc}(t)\approx(\alpha_{i,fc}P_{i,fc}(t)+\beta_{i,fc})\dfrac{\Delta t\xi_{e}}{\eta_{i,g2h}}$

(19)

As an example, using the data of a SOFC provided in [27], we first obtain the nonlinear curve for the ratio $\frac{P_{i,fc}(t)}{\eta_{i,e}}$ as shown by the solid black line in Figure 3. Next, a linear fit conducted in MATLAB gives us the linear approximation depicted by the dash-dot blue line in Figure 3, where $\alpha_{i,fc}=2.042$ and $\beta_{i,fc}=0.06323$, with $95\%$ confidence bounds.

Note that the P2P trading between dwellings pushes their FCs to produce higher output powers so that they have powers to trade with the others, hence increases the FC power and heat efficiencies. Also, our proposed linear approximation is more accurate when the FC output power is higher (see Figures 2–3), therefore it is suitable and reasonable for the system we are considering. Further, this approximation allows us to obtain a convexified problem without any efficiency matching algorithm (e.g. that in [27]), hence the complexity as well as the running time are much lower.

The convexified optimal energy management problem is described below, where the constant terms are removed from the objective function.

	$\displaystyle\min~{}$	$\displaystyle\sum_{i=1}^{n}\frac{\alpha_{i,fc}\Delta t\xi_{e}}{\eta_{i,g2h}}P_{i,fc}+a_{i}P_{i,tr}^{2}+\hat{b}_{i}P_{i,tr}$		(20a)
	s.t.	$\displaystyle P_{i,fc}^{\min}\leq P_{i,fc}\leq P_{i,fc}^{\max}$		(20b)
		$\displaystyle P_{i,tr}^{\min}\leq P_{i,tr}\leq P_{i,tr}^{\max}$		(20c)
		$\displaystyle P_{ij}+P_{ji}=0~{}\forall\,j\in\mathcal{N}_{i},i,j=1,\ldots,n$		(20d)

We proceed by further elaborating on the optimization problem (20) with two scenarios of dwelling demand. First, if $P_{i,dem}\geq P_{i,fc}^{\max}$, then the constraint (20b) implies that $P_{i,fc}=P_{i,fc}^{\max}$, due to (2.3), hence the first term on gas consumption cost in (20a) is constant. As a result, the local optimization problem at dwelling $i$ becomes a problem for only variables $P_{ij}$, i.e., for only P2P energy trading, as follows, where (2.3) is explicitly employed.

	$\displaystyle\min~{}$	$\displaystyle a_{i}P_{i,tr}^{2}+\hat{b}_{i}P_{i,tr}$		(21a)
	s.t.	$\displaystyle 0\leq P_{i,tr}\leq P_{i,dem}-P_{i,fc}^{\max}$		(21b)
		$\displaystyle P_{ij}+P_{ji}=0~{}\forall\,j\in\mathcal{N}_{i},i,j=1,\ldots,n$		(21c)

Second, if $P_{i,dem}<P_{i,fc}^{\max}$, then dwelling $i$ becomes a potential seller in the P2P energy market with $P_{i,tr}=P_{i,dem}-P_{i,fc}$. Then the local optimization problem at dwelling $i$ again is an optimization problem of only variables $P_{ij}$ as in the following, where $P_{i,fc}$ in (20) is substituted by $P_{i,dem}-P_{i,tr}$, and (2.3) is explicitly utilized.

	$\displaystyle\min~{}$	$\displaystyle a_{i}P_{i,tr}^{2}+\left(\hat{b}_{i}-\frac{\alpha_{i,fc}\Delta t\xi_{e}}{\eta_{i,g2h}}\right)P_{i,tr}$		(22a)
	s.t.	$\displaystyle P_{i,dem}(t)-P_{i,fc}^{\max}\leq P_{i,tr}\leq 0$		(22b)
		$\displaystyle P_{ij}+P_{ji}=0~{}\forall\,j\in\mathcal{N}_{i},i,j=1,\ldots,n$		(22c)

Thus, our considering optimal energy management for a group of dwellings equipped with FC-CHPs is in fact reduced to an optimal P2P energy trading which is a quadratic convex problem having the following form.

	$\displaystyle\min~{}$	$\displaystyle\sum_{i=1}^{n}a_{i}P_{i,tr}^{2}+b_{i}P_{i,tr}$		(23a)
	s.t.	$\displaystyle P_{i,tr}^{\min}\leq P_{i,tr}\leq P_{i,tr}^{\max}$		(23b)
		$\displaystyle P_{ij}+P_{ji}=0~{}\forall\,j\in\mathcal{N}_{i},i,j=1,\ldots,n$		(23c)

where $b_{i}$, $P_{i,tr}^{\min}$, and $P_{i,tr}^{\max}$ are determined in (21) or (22). A distributed P2P energy trading mechanism will be proposed in the next section to solve (23).

In the following, a distributed and parallel ADMM approach is proposed to solve the mathematical programming (23). The advantage of this approach is that it allows each peer/agent to solve its own local optimization problem while negotiating with other peers/agents to eventually reach the solution of the global optimization problem (23). Thus, the communication and computation burden at a centralized entity is avoided, and the privacy of each peer/agent can be guaranteed.

Denote $m\triangleq\sum_{i=1}^{n}n_{i}$ and $P\in\mathbb{R}^{m}$ the vector of all $P_{i},i=1,\ldots,n$. Then let us define the sets of coupling and local constraints as in (24) and (25), respectively.

$\displaystyle\Omega_{g}\triangleq$

$\displaystyle\left\{P\in\mathbb{R}^{m}:P_{ij}(t)+P_{ji}(t)=0~{}\forall\,j\in\mathcal{N}_{i}\right\}$

(24)

$\displaystyle\Omega_{\ell}\triangleq$

$\displaystyle\left\{P\in\mathbb{R}^{m}:P_{i,tr}^{\min}\leq\mathbf{1}_{n_{i}}^{T}P_{i}\leq P_{i,tr}^{\max}\right\}$

(25)

For those sets, the following indicator functions are defined.

$\displaystyle I_{g}(P)\triangleq\left\{\begin{array}[]{rl}0:&P\in\Omega_{g}\\ +\infty:&P\notin\Omega_{g}\end{array}\right.,~{}I_{\ell}(P)\triangleq\left\{\begin{array}[]{rl}0:&P\in\Omega_{\ell}\\ +\infty:&P\notin\Omega_{\ell}\end{array}\right.$

(30)

Now, by utilizing a new variable $X\in\mathbb{R}^{m}$, the optimization problem (17) is rewritten such that equality and inequality constraints are separated into different sets corresponding to different variables $P$ and $X$, as follows.

	$\displaystyle\min\,$	$\displaystyle\sum_{i=1}^{n}C_{i}(P_{i})+I_{g}(P)+I_{\ell}(X)$		(31a)
	s.t.	$\displaystyle P-X=0$		(31b)
		$\displaystyle P\in\Omega_{g},~{}X\in\Omega_{\ell}$		(31c)

Obviously, (31) is in the standard form of the ADMM method [7], which solves (31) iteratively. Nevertheless, the classical ADMM method in [7] is centralized and the updates of variables are in order. Therefore, in this research, we propose a novel ADMM approach that solves (31) in a fully distributed manner, and variables are updated in parallel. Define the following augmented Lagrangian,

$\displaystyle L_{\rho}(P,X,u)=$

$\displaystyle\sum_{i=1}^{n}C_{i}(P_{i})+I_{g}(P)+I_{\ell}(X)+\frac{\rho}{2}\|P-X+u\|_{2}^{2},$

where $\rho>0$ is a scalar penalty parameter and $u\in\mathbb{R}^{m}$ is called the scaled Lagrange (dual) multiplier [7]. Next, the variables $P,X,u$ are computed at each algorithm iteration $k+1$ by solving the following sub-problems,

	$\displaystyle X^{k+1}$	$\displaystyle\triangleq\operatorname*{argmin}_{X\in\Omega_{\ell}}\left[L_{\rho}(P^{k},X,u^{k})\right.$
		$\displaystyle\quad\left.+\frac{1}{2}(X-X^{k})^{T}\Psi(X-X^{k})\right]$
	$\displaystyle P^{k+1}$	$\displaystyle\triangleq\operatorname*{argmin}_{P\in\Omega_{g}}\left[L_{\rho}(P,X^{k},u^{k})\right.$
		$\displaystyle\quad\left.+\frac{1}{2}(P-P^{k})^{T}\Phi(P-P^{k})\right]$
	$\displaystyle u^{k+1}$	$\displaystyle\triangleq u^{k}-\kappa\rho(P^{k}-X^{k})$		(32)

in which $\Phi,\Psi,\kappa>0$ satisfy

$$\Phi\succ\rho(\frac{1}{\mu_{1}}-1)I,~{}\Psi\succ\rho(\frac{1}{\mu_{2}}-1)I,~{}\mu_{1}+\mu_{2}<2-\kappa$$

(33)

for some $\mu_{1}>0,\mu_{2}>0$. Condition (33) was proved to be sufficient for the convergence of the above variables update [11]. Note that the selection of $\Phi,\Psi,\kappa$ to fulfill (33) is not unique. One simple way is to let $\Phi=\phi I$, $\Psi=\psi I$ such that

$$\phi>\rho(\frac{1}{\mu_{1}}-1),~{}\psi>\rho(\frac{1}{\mu_{2}}-1),~{}\mu_{1}+\mu_{2}<2-\kappa$$

(34)

Unlike the ordered updates in the classical centralized ADMM [7] and the existing distributed ADMM methods (e.g. [24]), each variable in (3.2) at each iteration is completely independent of the other two variables at the same iteration. Therefore, problem variables are computed in parallel at every agent. Moreover, our proposed ADMM approach advances that in [11], which also updates variables in parallel, by allowing sparse connection between peers/agents, while that in [11] requires all-to-all inter-agent connection.

The updates of variables $P$ and $X$ in (3.2) are explicitly presented in the next sections.

The update for $X^{k+1}$ in (3.2) is derived by solving the following optimization problem.

	$\displaystyle\min\,$	$\displaystyle\frac{\rho}{2}\|P^{k}-X+u^{k}\|_{2}^{2}+\frac{\psi}{2}\|X-X^{k}\|_{2}^{2}$		(35a)
	s.t.	$\displaystyle P_{i,tr}^{\min}\leq\mathbf{1}_{n_{i}}^{T}X_{i}\leq P_{i,tr}^{\max}~{}\forall\,i=1,\ldots,n$		(35b)

Note that there is one more constraint on the positiveness or negativeness of each $X_{i},i=1,\ldots,n$, depending on whether the $i$-th peer/agent at the next time slot will perform as a buyer or seller. In any case, (35) is a quadratic convex problem and is decompsable to each peer/agent, i.e. it is fully decentralized, hence it can be easily solved by any off-the-self software embedded in each peer/agent, e.g., CVX [1].

To obtain the update for $P^{k+1}$ in (3.2), we need to solve the following mathematical programming.

	$\displaystyle\min\,$	$\displaystyle\sum_{i=1}^{n}C_{i}(P_{i})+\frac{\rho}{2}\|P-X^{k}+u^{k}\|_{2}^{2}+\frac{\phi}{2}\|P-P^{k}\|_{2}^{2}$		(36a)
	s.t.	$\displaystyle P_{ij}+P_{ji}=0~{}\forall\,i=1,\ldots,n;\ j\in\mathcal{N}_{i}$		(36b)

Denote $\lambda_{ij}>0$ the Lagrange multiplier associated with the constraint (36b), and $\lambda_{i}\in\mathbb{R}^{n_{i}}$ the vector of all $\lambda_{ij}$ with $j\in\mathcal{N}_{i}$. Since (36) is a convex optimization problem with quadratic cost function and linear equality, the strong duality holds and KKT conditions apply. Therefore, we obtain from (36) that

	$\displaystyle\lambda_{ij}^{k+1}=$	$\displaystyle\;\frac{\partial}{\partial P_{ij}}\left(\sum_{i=1}^{n}C_{i}(P_{i})+\frac{\rho}{2}\|P-X^{k}+u^{k}\|_{2}^{2}\right.$
		$\displaystyle\qquad\left.\left.+\frac{\phi}{2}\|P-P^{k}\|_{2}^{2}\right)\right|_{P_{ij}=P_{ij}^{k+1}}$
	$\displaystyle=$	$\displaystyle\;2a_{i}P_{i,tr}^{k+1}+(\rho+\phi)P_{ij}^{k+1}+v_{ij}^{k}$		(37)

where $v_{ij}^{k}\triangleq b_{i}+d_{ij}+\rho(-X_{ij}^{k}+u_{ij}^{k})-\phi P_{ij}^{k}$. Next, due to the bilateral trading constraint (36b), the Lagrange multipliers and the traded powers must satisfy the following constraints.

$$\lambda_{ij}^{k+1}=\lambda_{ji}^{k+1},P_{ij}^{k+1}=-P_{ji}^{k+1}~{}\forall\;j\in\mathcal{N}_{i}$$

(38)

Here, $\lambda_{ij}^{k+1}$ is considered to be the trading price between the $i$-th and $j$-th peers. Interestingly, equation (3.4) reveals the relation between individually traded price between a pair of peers/agents with its associated traded power and the total available powers of those peers/agents.

Let us denote

	$\displaystyle\Gamma$	$\displaystyle\triangleq(\rho+\phi)\mathrm{diag}\{\frac{1}{a_{i}}\}_{i=1,\ldots,n},~{}\tilde{P}_{i,tr}^{k+1}\triangleq 2a_{i}P_{i,tr}^{k+1},$
	$\displaystyle\hat{v}_{i}^{k+1}$	$\displaystyle\triangleq\sum_{j\in\mathcal{N}_{i}}v_{ji}^{k+1},~{}\hat{v}^{k+1}\triangleq\begin{bmatrix}\hat{v}_{1}^{k+1},\cdots,\hat{v}_{n}^{k+1}\end{bmatrix}^{T},$
	$\displaystyle\tilde{v}_{i}^{k+1}$	$\displaystyle\triangleq\sum_{j\in\mathcal{N}_{i}}v_{ij}^{k+1},~{}\tilde{v}^{k+1}\triangleq\begin{bmatrix}\tilde{v}_{1}^{k+1},\cdots,\tilde{v}_{n}^{k+1}\end{bmatrix}^{T}$

Utilizing (38) and (3.4), we can easily obtain

	$\displaystyle P_{ij}^{k+1}$	$\displaystyle=\frac{v_{ji}^{k+1}+\tilde{P}_{j,tr}^{k+1}-v_{ij}^{k+1}-\tilde{P}_{i,tr}^{k+1}}{2(\rho+\phi)}$		(39a)
	$\displaystyle\lambda_{ij}^{k+1}$	$\displaystyle=\frac{v_{ji}^{k+1}+\tilde{P}_{j,tr}^{k+1}+v_{ij}^{k+1}+\tilde{P}_{i,tr}^{k+1}}{2}$		(39b)

The convergence of the proposed ADMM algorithm follows that provided in [11], hence we omit the proof here for brevity. Next, substituting the optimal solutions to (3.4) leads to $\lambda_{ij}^{\ast}=2a_{i}P_{i,tr}^{\ast}+b_{i}+d_{ij}+\rho u_{ij}^{\ast}$. Then summing up (39a) for all $j\in\mathcal{N}_{i}$ leads to

$\displaystyle P_{i,tr}^{k+1}=\frac{\displaystyle\sum_{j\in\mathcal{N}_{i}}v_{ji}^{k+1}+\sum_{j\in\mathcal{N}_{i}}\tilde{P}_{j,tr}^{k+1}-\sum_{j\in\mathcal{N}_{i}}v_{ij}^{k+1}-n_{i}\tilde{P}_{i,tr}^{k+1}}{2(\rho+\phi)}$

which is equivalent to

$\displaystyle 2(\rho+\phi+n_{i}a_{i})P_{i,tr}^{k+1}-\sum_{j\in\mathcal{N}_{i}}2a_{j}P_{j,tr}^{k+1}=\sum_{j\in\mathcal{N}_{i}}v_{ji}^{k+1}-\sum_{j\in\mathcal{N}_{i}}v_{ij}^{k+1}$

(40)

Stacking (40) with $i=1,\ldots,n$ results in

$$(\mathcal{L}+\Gamma)\tilde{P}_{tr}^{k+1}=\hat{v}^{k+1}-\tilde{v}^{k+1}$$

(41)

Equations (41), (39a), and (39b) give us the updates for variables $P_{ij}^{k+1}$ and the P2P energy prices $\lambda_{ij}^{k+1}$.