## Abstract

We present a novel oblivious routing economic dispatch (ORED) algorithm for power systems. The method is inspired by the oblivious network design which works perfectly for networks in which different sources (generators) send power flow toward their destinations (load points) while they are unaware of the current network state and other flows. Basically, our focus is on the economic dispatch while managing congestion and mitigating power losses. Furthermore, we studied a loss-minimizing type of the economic dispatch which aims to minimize the emission by optimizing the total power generation rather than system cost. Comparing to state-of-the-art economic dispatch methods, our algorithm is independent of network topology and works for both radial and non-radial networks. Our algorithm is thus suited for large-scale economic dispatch problems that will emerge in the future smart distribution grids with host of small, decentralized, and flexibly controllable prosumers; i.e., entities able to consume and produce electricity. The effectiveness of the proposed ORED is evaluated via the IEEE 57-bus standard test system. The simulation results verify the superior performance of the proposed method over the current methods in the literature in terms of congestion management and power loss minimization.

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## Notes

Asymptotic analysis is a method of describing limiting behaviors of functions. Assuming that

*f*(*n*) and*g*(*n*) are two functions defined on some subset of the real numbers, \(f(n)=O(g(n))\) if there exists constants*c*and*N*such that \(f(n)\le cg(n)\) for all \(n>N\). In this definition,*g*(*n*) is called an asymptotic high-threshold (upper-bound) for*f*(*n*), or*f*(*n*) is called to be asymptotically bounded to*g*(*n*).As this paper focuses on the novel method for solving the economic dispatch problem, we assume the throughput of the generator to be deterministic. This way, the method can be illustrated more clearly.

Diameter of a graph is defined as the maximum distance between any pair of nodes in the graph.

Partition of set

*A*is a set of*A*subsets \(\{A_i|i\in [1,k]\}\) such that \(\bigcup \nolimits _{i=1}^k A_i = A\), \(\forall i\in [1,k]:A_i\ne \emptyset \), and \(\forall i\ne j: A_i\cap A_j = \emptyset \).The distance \(d_{\mathcal {G}}(u,v)\) between nodes

*u*,*v*is defined as the length of shortest path between*u*and*v*in \(\mathcal {G}\). The length of a path in a weighted graph is defined as the total weight of the edges participating in the path.Set \(B_{\mathcal {G}}(v,r)\in V\) is called a ball of center \(v\in V\) of radius

*r*if \(u\in B_{\mathcal {G}}(v,r)\) iff \(d_{\mathcal {G}}(u,v)\le r\).Permutation \(\pi \) on finite set

*A*of size*n*is a one-to-one and onto function mapping set \(\{1,2,\ldots ,n\}\) to set*A*.In this paper, path of a graph is considered as a simple path and represented by a subset of edge set

*E*such that there exist a permutation of edges in a path where the first edge is incident to the start node of the path, each two consecutive edges are incident to a common node, and the last edge is incident to the end node of the path.

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## Appendix

### Appendix

In contrast to the traditional MCFPs that are defined with a well-defined set of commodities (with specific size and source/destination nodes) and given flow cost function for every edge, oblivious network routing aims to solve an MCFP in which either the flow cost function is not specified (flow cost is oblivious), or the commodities size, source, and destination are not specified (commodities are oblivious).

Oblivious network routing approach solves oblivious MCFPs by employing a practical method not guaranteed to be perfect, but sufficient for the immediate goal which is obtaining an acceptable approximation of the optimal solution. In this regard, oblivious network routing is considered to be a heuristic method since it can be used to speed up the process of finding a satisfactory solution (while computing the optimal solution is impractical).

Oblivious network routing is different with Dynamic (adaptive) Routing (DR) since DR proposes a different solution in response to any change of the MCFP oblivious components; while, oblivious routing deploys a single routing scheme for an oblivious MCFP with the aim of approximating the optimal solution of the problem with oblivious parameters.

The common characteristics of the oblivious routing schemes includes being pretty flexible to the obliviousness of the environment in which the commodities are flowing and making the traffic flows distributed over the network and preventing the flow-congestion in some specific nodes or edges. Additionally, these types of routing schemes provide a low-cost flow routing in long term even if a wide range of unpredictable events occur in the network like bursty flow derived from a specific node or failure of some node in forwarding the flow through the network. In fact, the versatile routing schemes best fit to those networks that we have little/no knowledge regarding their current and future states.

### Preliminaries to oblivious network routing

At first, we provide some basic definitions and preliminary data structures needed to explain the \(\mathcal {ORED}\) algorithm. In the rest of the paper, \(\mathcal {G}=(V,E,w)\) is considered as a weighted connected graph of diameter^{Footnote 3}
\(2^h\) (for some integer *h*) where \(w_e>1\) for every \(e\in E\). The mentioned conditions on the graph diameter and its weight function don’t reduce its generality because any weighted connected graph can be converted to \(\mathcal {G}\) by linearly scaling its weight function. We prove this claim here (diameter of graph \(\mathcal {G}\) is represented by \(diam(\mathcal {G})\)):

### Lemma 1

For any connected graph \(\mathcal {G}=(V,E,w)\), there is a connected graph \(\mathcal {G}_c=(V,E,w_c)\) such that for every edge \(e\in E\), \(w_c(e)=c\cdot w(e)\) (\(c\in \mathbb {R}^+\)), \(diam(\mathcal {G}_c)=2^h\) (for some \(h\in \mathbb {Z}^+\)), and \(w_c(e)>1\) for every \(e\in E\) [21].

### Proof

Consider graph \(\mathcal {G}'=(V,E,w')\) where:

such that \(\varepsilon \in \mathbb {R}^+\) is an arbitrary positive number. It is clear that \(w'(e)>1\) for every \(e\in E\). We define graph \(\mathcal {G}''=(V,E,w'')\) such that:

Since \(w''(e)>1\) and diam\((\mathcal {G}'')=2^{\left\lceil \log _2(diam(\mathcal {G}'))\right\rceil }\), for the following value of *c*, graph \(\mathcal {G}_c=(V,E,w_c)\) satisfies the conditions mentioned in the Lemma 1.

\(\square \)

Assuming \(\Delta \) as a positive real number, \(\Delta \)-partition of \(\mathcal {G}\) is defined as a partition^{Footnote 4} of *V* into a number of subsets *S* such that there exist no pair of nodes in *S* with distance^{Footnote 5} of more than \(\Delta \) from each other.

Hierarchical decomposition sequence (HDS) \(\bar{H}\) of graph \(\mathcal {G}\) is the vector \(\bar{H}=(H_0\), \(H_1\), ..., \(H_h)\) where \(H_h = \{V\}\), \(H_i \text { is a }2^i\text {-partition}\), and if *S* belongs to \(H_i\), there exists some \( S'\in H_{i+1}\) such that \( S\subseteq S'\) for every \(i=0,1,\ldots ,h-1\). Partition \(H_i\) is called the *ith* level partition of \(\bar{H}\). Node *v* is called \(\alpha \)-padded in \(\bar{H}\) (\(0<\alpha <1\)) if for every *i*, the *ith*-level partition \(H_{i}\) contains a set \(S\subseteq V\) such that *S* completely covers ball^{Footnote 6}
\(B_{\mathcal {G}}(v,\alpha \cdot 2^i)\); in other words, for every level *i*, there exists some \(S\in H_i \text{ such } \text{ that } B_{\mathcal {G}}(v,\alpha \cdot 2^i)\subseteq S \) [21]. See Fig. 8 for illustration.

In 2003, Fakcharoenphol et al. [50] proposed a randomized algorithm to generate a random HDS in which a minimum quotient of nodes are asymptotically guaranteed to be \(\alpha \)-padded with high probability (\(\alpha \le 1/8\)). The main idea of this algorithm is to construct the *ith*-level partitions by dividing each \((i+1)^{th}\)-level set \(S\in H_{i+1}\) into smaller subsets \(S_{1}, S_{2},\ldots \) such that there exists no node \(r\in S_{j}\) such that *r* is further than \(\delta _i\) from an *S*’s special member called its *representative*. Value of \(\delta _i\) is chosen randomly in the interval \([2^{i-2},2^{i-1})\). For more details, see Algorithm 1 [21].^{Footnote 7}

Finally, we define the Hierarchical Decomposition Tree (HDT) which is the main data structure utilized for the novel \(\mathcal {ORED}\) algorithm and is constructed based on the HDS \(\bar{H}\). HDT is an *h*-level tree where its *ith* level nodes are the members of \(H_i\) (for every \(i=0,1,\ldots ,h\)) and any ith level tree node \(S\in H_i\) is connected to its parent \(S'\in H_{i+1}\) in upper-level (\(i+1\)) if \(S\subseteq S'\).

### Construcing the oblivious routing scheme

First, we show how Algorithm 2 constructs the oblivious routing scheme (mentioned in [21]) which plays the main role in simplification of the couple optimization problems mentioned in Section II by placing more restrictions on the problems, shrinking their feasible areas, and subsequently reducing the computational complexity of them so that their solutions become computationally feasible.

Algorithm 2 gets the weighted graph \(\mathcal {G}\) as its input and returns function \( \mathbb {S}:\{G_i\}_1^n\times \{F_i\}_1^q\mapsto \mathcal {P}(E) \) which represents the oblivious routing scheme and specifies a path^{Footnote 8} in graph \(\mathcal {G}\) for every pair of source-sink nodes. The algorithm starts with generating \(27\log |V|\) random HDS’s utilizing randomized Algorithm 1. The reason for generating multiple random HDS’s is to assure the existence of at-least one HDS \(\bar{H}\) for every pair of nodes \(source\in G\) and \(sink\in F\). In fact, Iyengar et al. in [21] proved that the probability of existing at-least one \(\alpha \)-padding HDS among the \(c\log |V|\) HDS’s constructed by Algorithm 1 is more than \(1-e^{-\frac{(c-2)^2}{2c}}\) (for every \(\alpha \le 1/8\)). By considering \(c\ge 27\), the mentioned probability would be greater than \(1-10^{-5}\) which provides a reasonable theoretical guarantee that Algorithm 2 will find at-least one HDS for every source-sink pair.

After creating \(27\log |V|\) random HDS’s and their corresponding HDTs, Algorithm 2 runs its main loop in lines 5–11 for every pair of source-sink nodes. Assume *T* as the HDT corresponding to the \(\alpha \)-padding HDS of an arbitrary pair of source-sink nodes. Tree *T* would have a pair of leaves (level-zero nodes) corresponding to the source and sink (as \(\mathcal {G}\) is supposed to be a weighted graph of the weight function greater than one for every edge). Let *p* denote the only tree path connecting the mentioned leaves together. Finally, the resulted routing scheme is computed in line 11 where the path between source and sink nodes \(\mathbb {S}(source,sink)\) is obtained by projecting *p* on graph \(\mathcal {G}\). The projection of path *p* of HDT *T* on graph \(\mathcal {G}\) is defined in the following way:

*Consider*
\(\gamma _e\)
*as the shortest path between the incident nodes of arbitrary edge*
\(e\in p\)
*in graph*
\(\mathcal {G}\). *The projection of tree path*
*p*
*on the graph is obtained by concatenating all of the shortest paths*
\(\gamma _e\)s *back to back. In the case that the concatenation result is not a simple path and has crossed some nodes more than once, the projected path will be the shortest simple path corresponding to the concatenation result.*

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Boroojeni, K.G., Amini, M.H., Iyengar, S.S. *et al.* An economic dispatch algorithm for congestion management of smart power networks.
*Energy Syst* **8**, 643–667 (2017). https://doi.org/10.1007/s12667-016-0224-6

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DOI: https://doi.org/10.1007/s12667-016-0224-6

### Keywords

- Congestion management
- Economic dispatch
- Loss minimization
- Oblivious network design
- Smart power network