# Designing a Marketplace for the Trading and Distribution of Energy in the Smart Grid

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5 polynomial complexity. Due to lack of space, we formulate the lemmas and theorem and leave the proofs in an anonymous technical report [1]. Finally, section 5.3 introduces RadPro and compares the communication and computational complexity with that of Acyclic-Solving. 5.1 Direct message passing solution Note that the value of an allocation, as defined in equation 5, is a sum of utility functions, one per prosumer. Thus, we can directly map the EAP into a COP. When the EAP is acyclic, so is the constraint network and we can directly apply Acyclic-Solving as it is described in section 2. Note that the separator s j between each node j and its parent p j corresponds either with the single variable y ip j if p j is an outneighbor of j or with y p j j if p j is an in-neighbor of j. In any case the size of the message is the number of possible states of the link between j and p j, and the message can be understood as communicating the utility of each of these states to the network composed by the prosumers in the subtree rooted at j. Thus, the amount of communication needed to run Acyclic-Solving is very small. The expression of the message from j to its parent can be obtained by particularizing the general Acyclic-Solving equation 1. When the parent node p j is an out-neighbor, j sends the following message: (y jp j ) = = max v j(y jp j, Y j p j ) + Y j p j k out( j)\{p j } µ j k (y jk ) + µ i j (y i j ), i in( j) and when the parent node p j is an in-neighbor, j sends the following message: (y p j j) = = max v j(y p j j, Y j p j ) + µ j k (y jk ) + Y j p j k out( j) i in( j)\{p j } µ i j (y i j ). where Y j p j stands for an assignment of values to each variable in Y j except y jp j (or y p j j). Note that in agreement with section 2, the assessment of messages takes time exponential in the number of variables in the scope of the constraint, which in this case corresponds with the number of neighbors of the prosumer. The computational cost for a prosumer j of assessing the message to its parent is bounded O((2C j + 1) N j), where C j is the capacity of the most powerful link between i and a neighboring agent and N j is the number of neighbors of j. Again, as explained in section 2, we can use bookkeeping during the assessment of the messages to reduce the complexity of the assessment of the best assignment according to equation 2 to constant time. Thus Acyclic-Solving will perform efficiently as long as the degree of the nodes in the energy network is small. This is likely to be so in rural scenarios [17], however in urban areas [25] the degree of nodes has been seen to follow a geometric distribution with some nodes degree reported to be over 35. This hampers the direct application of Acyclic-Solving in such scenarios. Similar dynamic programming approaches for the problem of CO 2 reduction in energy networks have been reported functional for nodes with a branching factor up to four [14]. In the following section we develop an algebra of valuations that allows for a much more efficient computation of messages. (6) (7) 5.2 Efficient message computation The assessment of messages using equations 6 and 7 takes time exponential in the number of neighbors that the prosumer is connected to. Thus, for densely connected energy networks, it can severely hinder the applicability of the algorithm. To overcome this problem, in this section we introduce an algebra of valuations that the prosumers can take advantage of when assessing their messages. As a result, the message assessment complexity is severely reduced. Objects and Operations of the algebra of valuations. Our objective is to build an algebra that enables us to perform the assessment of messages faster. The main mathematical object of the algebra, namely the valuation has already been defined in definition 1. Both the offers o j of each prosumer and the messages can be directly mapped into valuations. Valuations can be efficiently represented computationally by means of hash tables. In the following we will introduce the operations on valuations required to assess the messages. First consider the scenario of a prosumer j with a single inneighbor p j. The message to send according to equation 7 will basically copy the offer o j. However, since the possible states of y p j j are limited by the capacity of the link, we will need to have an operation in the algebra that removes from valuation o j, those values which are out of the range. This operation is called restriction. Definition 2. Given a valuation α, and a subset D Z we define α[d], the restriction of α to D as α(k) k D α[d](k) = otherwise. Thus, the message of a prosumer j to its single in-neighbor p j can be written as = o j [D p j j]. Now consider the scenario of a prosumer j with a single outneighbor p j. In this case, the equation of the net energy flow for j shows that net(y jp j ) = y jp j. Thus, the message to send according to equation 6 is a symmetric version of o j, restricted to the capacity of the link. We refer to the operation that assesses the symmetric version of a valuation as the complement. Definition 3. Given a valuation α, we define α, the complement of α as α(k) = α( k) Thus, the message of a prosumer j to its single out-neighbor p j can be written as = o j [D p j j]. Finally take the scenario of a prosumer j such that its parent p j is an in-neighbor and that it has an additional out-neighbor k. In that case equation 7 is reduced to (y p j j) = max y jk ( v j (y p j j, y jk ) + µ j k (y jk ) ) = max y jk ( o j (y p j j y jk ) + µ j k (y jk ) ) (8) Note that for some of the assignments y p j j, y jk, it will happen that o j is undefined for the net energy balance y p j j y jk. We can avoid considering these assignments by introducing a new operation for valuations, the aggregation. Definition 4. Given two valuations α, β, we define α β, the aggregation of α and β as (α β)(k) = max α(i) + β( j) (9) i, j k=i+ j 1289

6 Algorithm 2 Aggregation of two valuations 1: γ 2: for i FVD(α) do 3: for j FVD(β) do 4: γ(i + j) max (γ(i + j), α(i) + β( j)) 5: end for 6: end for Now the message of a prosumer j such that its parent p j is an inneighbor and that it has an additional out-neighbor k, can be written as = (o j µ k j )[D p j j]. Structure of the algebra of valuations. We have defined the mathematical objects (the valuations) and the operations (restriction, complement and aggregation) of our algebra. The following two results describe the algebraic structure we have created. First, we concentrate on the aggregation operation. Lemma 1. The set of valuations with the aggregation operation forms a commutative monoid with as the identity element. That is, for any valuations α, β, and γ we have that 1. α β = β α 2. (α β) γ = α (β γ) 3. α = α From a mathematical standpoint Lemma 1 enables us to write expressions such as α β γ that do not establish a specific order in which the aggregations should be performed. This allow us to extend the definition of aggregation from pairs of valuations to sets of valuations. We define the n-way aggregation of the valuations in A = {α 1,..., α n } as ( n i=1 α i )(k) = max n ni=1 j 1,..., j n i=1 α i ( j i ). Note j i =k that, as expected by the notation, we have that n i=1 α i = α 1... α n. Thus, the n-way aggregation of a finite set of valuations can be assessed using only the (2-way) aggregations from definition 4. From a computational perspective, Lemma 1 allow us to group the aggregation operations in the way we prefer. Thus, we can follow a sequential order, or assess them in a tree in case that is more efficient. Now we are interested in characterizing the relations between the different operations. Lemma 2. The complement defines a self-inverse automorphism of the commutative monoid of valuations. That is, for any two valuations α and β we have that α β = α β α = α = : Φ Φ is a bijective mapping. Furthermore, α[d] = (α)[ D] where D = { x x D}. Some of these properties will come in very handy when working with expressions that involve aggregations, complements and restrictions. The efficiency of aggregation. Next, we are interested in determining the computational cost of aggregating a set of valuations. Algorithm 2 can be used to implement aggregation, with the following result: Lemma 3. The aggregation of two valuations α and β can be done in time O(n α n β ). Furthermore n α β n α n β. From here we can show that assessing the aggregation of M valuations of size N can be done in time O(N M ). Our objective was to speed-up the computation by aggregating the messages. However we see that in the more general case, the aggregation of M messages takes exponential time. Next we take advantage of a particularity of the messages we need to aggregate that allow us to compute the aggregation in polynomial time. Notice that the FVD of a message µ i j always lies in a small range D i j = [ c i j..c i j ] around 0. We say that a valuation α is of restricted capacity C if FVD(α) [ C..C]. For restricted capacity valuations we get the following result. Lemma 4. The aggregation of two valuations α and β of restricted capacity C, can be done in time O(C 2 ). Furthermore α β is a valuation of restricted capacity 2C. Note that whilst for general valuations, the size of the aggregated valuation grows quadratically, for restricted capacity valuations, it only grows linearly. This turns out to be fundamental to prove the following result, which tell us that we can aggregate valuations in polynomial time as long as they are of restricted capacity. Lemma 5. Given a set of M valuations A of restricted capacity C we can assess its aggregation α A α in time O(M 2 C 2 ). The proof uses the associative property to divide the aggregation into two smaller aggregations each with one half of the valuations, to assess those smaller aggregations and to finally aggregate the results. Using the algebra of valuations to assess messages. The following theorem shows that we can provide an expression for Acyclic- Solving messages in equations 6 and 7 in terms of valuation algebra operations and that this can done in polynomial time. Theorem 1. The message from prosumer j to its in-neighbor parent p j defined in equation 7 can be assessed as = o j µ j k µ i j [D p j j] (10) k out( j) i in( j)\{p j } and the message from prosumer j to its out-neighbor parent p j defined in equation 6 can be assessed as = o j µ j k µ i j [ D jp j ]. (11) k out( j)\{p j } i in( j) Furthermore, the computational complexity of assessing each message is O(N j C j n o j + N 2 j C2 j ) where N j is the number of neighbors of j and C j is the largest capacity of any of the links connected to j. The proof relies on transforming the expressions in equations 6 and 7 into a k-way aggregation which is then mapped to 2-way aggregations and manipulated with the help of Lemma 2. The complexity result relies basically on Lemma RadPro message passing solution Theorem 1 shows that Acyclic-Solving messages can be assessed in polynomial time for the EAP. We name RadPro, the new algorithm resulting from running Acyclic-Solving as described in Algorithm 1 but assessing messages using equations 10 and 11, instead of the usual Acyclic-Solving expression in equation 1. Note that we are only modifying the algorithm that we use to assess the messages. Since the messages exchanged are exactly the same for both algorithms, the number and the size of the messages exchanged is exactly the same. Both Acyclic-Solving and RadPro, send 2n messages, being n the number of prosumers. Furthermore, the size of each message is bounded by 2C max + 1, where C max is the largest capacity of any link in the network. Thus the number of values exchanged by both algorithms is O(nC max ). Regarding computational complexity, as reported in section 5.1 in Acyclic-Solving the complexity for a prosumer j of assessing 1290

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