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IEEE Trans. on Smart Grid, 3(1), pp.162-173, 2012.

Optimal Power Allocation Under Communication Network Externalities --M.G. Kallitsis, G. Michailidis and M. Devetsikiotis.

By: Renyong WuHunan University

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(a) Schematic of a 4-bus network. (b) instantaneous optimal power flow.

All transmission lines have maximum capacity 50 KW.

3(c) Convergence of the proposed (decentralized) algorithm.

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1. Introduction

• Network externality (effect): is a economic concept, has been defined as a change in the benefit, or surplus, that an agent derives from a good when the number of other agents consuming the same kind of good changes.

• As fax machines increase in popularity, for example, your fax machine becomes increasingly valuable since you will have greater use for it.

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1. Introduction

• Efficient resource allocation is an important but traditional problem to both electric power grid and communication networks.

• Two-way data communication can serve as a feedback loop between users and power provider to bidirectionally transmit some important messages.

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1. Introduction

• The problem has the nature of a local public goods allocation problem, arising in networks with externalities, i.e., networks such that each user’s utility is directly affected by other users’ actions.

• A existing lower bound on the dimensionality of the message space required by any algorithm in order to achieve the global optimum is o(N2).

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1. Introduction

Contributions of the paper:• The proposed a social welfare maximization framework

aims to reduce the uncertainty (e.g., delay) on the communication of the plethora of the control messages exchanged.

• A decentralized algorithm is proposed to allow efficient resource allocation in large-scale networks.

• In the formulation, the data network of the smart grid introduces externalities to the power network of the smart grid via coupled utility functions.

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2. The social welfare optimization framework

• The provider operates G generating units. Unit j has power output Kj and marginal generating cost λj.

• There are Np power users. Each user has to take an action pi (for example, an action is the requested power pi).

• So the energy balance constraint is satisfied:

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2. The social welfare optimization framework

• The power network topology consists of B nodes and L transmission lines. A node may produce power (a generator), withdraw power (a user), or perform both direction.

• Line admitances are given by the diagonal matrix Ω[L ×L].

• Network interconnection matrix A [L ×(B-1) ], the elements of A, Alb= 0, 1 or -1.

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2. The social welfare optimization framework

• The transfer admittance matrix H is calculated as:

• Matrix H takes values in [-1, 1] and Hlb represents the power that is distributed on line l when 1W in injected into bus b and withdraw at the reference bus.

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2. The social welfare optimization framework

• We seek to maximize the sum of users’ utility functions minus the production cost (optimization problem 3.1):

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2. The social welfare optimization framework

Dealing with communication uncertainty• Property1: the user with the highest demand is allocated

more bandwidth.• Property 2: high delay leads to allocate more

communication resources. Moreover, critical services should be allocated more bandwidth resources.

• Property 3: high delay leads to allocate more power

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2. The social welfare optimization framework

• Optimization problem 3.2 considering the cost function Ci(pi, Φl(i)) is:

We will obtain [15-16, 19]

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2. The social welfare optimization framework

A centralized mechanism for solving (5) is impractical because

• users’ utilities are considered private information and cannot be revealed.

• It would be infeasible to communicate all information to a centralized location.

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3. Mechanism design in a distributed manner

• Consider the set of users and each user i has to take an action ai ∈Ai where Ai is the action space and user i’s private information.

• Denote the set of users that affect user i by Ri={k∈N | ki} where ki indicates that user k affects user i.

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3. Mechanism design in a distributed manner

• A utility function quantifies the performance of a user as a result of the actions of users in its neighbor set Ri.

• User i selects an action from space Di, so the aim is to design a mechanism to determine the users’ action profile aN = (a1, a2, …, aN). The optimization problem 4.1 (here N is users’ number)

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3. Mechanism design in a distributed manner

• The information available to user i ∈ N is its utility function ui, the set of its feasible action Ai, the sets of its neig

hbors Ri and Ci, and an estimate of the set of feasible actions of its neighbors.

• Each user solves an individual optimization problem using solely the above information, and communicates the outcome to its neighbors.

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3. Mechanism design in a distributed manner

• Before starting the iterative algorithm, all users agree on a common initial action profile and consent to a sequence of modification parameters.

• As shown in [8], the convergence of the algorithm is guaranteed when the users have strictly concave utility functions.

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3. Performance evaluztion

• The cost function for user I is given by

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Thank you!