Quality of service aware traffic scheduling in wireless smart grid communication

Abstract

The next generation electrical power grid, known as smart grid (SG), requires a communication infrastructure to gather generated data by smart sensors and household appliances. Depending on the quality of service (QoS) requirements, this data is classified into event-driven (ED) and fixed-scheduling (FS) traffics and is buffered in separated queues in smart meters. Due to the operational importance of ED traffic, it is time sensitive in which the packets should be transmitted within a given maximum latency. In this paper, considering QoS requirements of ED and FS traffics, we propose a two-stage wireless SG traffic scheduling model, which results in developing a SG traffic scheduling algorithm. In the first stage, delay requirements of ED traffic is satisfied by allocating the SG bandwidth to ED queues in smart meters. Then, in the second stage, the SG rest bandwidth is going to the FS traffic in smart meters considering maximizing a weighted utility measure. Numerical results demonstrate the effectiveness of the proposed model in terms of satisfying latency requirement and efficient bandwidth allocation.

Appendix

1.1 Convergence of the Stochastic iteration

Without loss of generality, consider the problem

$$\begin{aligned} \underset{x}{\min }~ \mathbb {E}_{r} \left[ f(x,r) \right] \end{aligned}$$

(17)

where r is a random variable and f(x, r) is a convex function in x. To find the optimal solution $x^{*}$ and optimal value $p^*=\mathbb {E}_{r} \left[ f(x^*,r)\right]$, the following gradient iteration is used.

$$\begin{aligned} x(t+1) = x(t)-\alpha g(t) \end{aligned}$$

(18)

where $\alpha$ is a step size and g(t) is the gradient of f(.) with respect to x(t), i.e., $g(t) \triangleq \nabla f_x (x(t),r(t))$. Taking norm-2 of $(x(t+1)-x^{*})$, we derive

$$\begin{aligned}&\left\| x(t+1) - x^{*} \right\| ^{2} = \left\| x(t) - \alpha g(t) - x^{*}\right\| ^{2} \nonumber \\&\quad = \left\| x(t)- x^{*}\right\| ^{2} - 2\alpha g(t)(x(t)- x^{*}) + \alpha ^{2} \left\| g(t)\right\| ^{2}.\qquad \end{aligned}$$

(19)

Due to the convexity of f(x(t), r(t)) in x(t), the following inequality holds [27].:

$$\begin{aligned} f(x^*, r(t)) \ge f(x(t),r(t))+g(t)(x^*-x(t)). \end{aligned}$$

(20)

Applying this inequality to (19), it is written as

$$\begin{aligned}&\left\| x(t+1) - x^{*} \right\| ^{2} \le \nonumber \\&\left\| x(t)- x^{*}\right\| ^{2} - 2\alpha \left\{ f(x(t), r(t)) - f(x^*,r(t))\right\} \nonumber \\&\quad + \alpha ^{2} \left\| g(t)\right\| ^{2}. \end{aligned}$$

(21)

Taking a similar recursive approach from x(t) to x(0) as an initial value, we derive

$$\begin{aligned} \left\| x(t+1) - x^{*} \right\| ^{2}&\le \left\| x(0) - x^{*}\right\| ^{2} + \alpha ^{2} \sum _{i=0}^{t} \left\| g(i)\right\| ^{2} \nonumber \\&- 2 \alpha \sum _{i=0}^{t}\left\{ f(x(i), r(i)) - f(x^*,r(i)) \right\} . \nonumber \\ \end{aligned}$$

(22)

Since the left-hand side is always non-negative, then

$$\begin{aligned}&2 \alpha \sum _{i=0}^{t}\left\{ f(x(i), r(i)) - f(x^*,r(i)) \right\} \nonumber \\&\quad \le \left\| x(0)- x^{*}\right\| ^{2} + \alpha ^{2} \sum _{i=0}^{t} \left\| g(i)\right\| ^{2}. \end{aligned}$$

(23)

Now consider the following two assumptions:

• $\left\| g(i)\right\| \le G$, for all i.

• $\left\| x(0)- x^{*}\right\| ^{2} \le R^{2}$.

With reference to the system model in Sect. 2, these assumptions are reasonable and can be provided in the model. Dividing both sides of (23) by $2\alpha t$, it is concluded that

$$\begin{aligned} \frac{1}{t} \sum _{i=0}^{t}\left\{ f(x(i),r(i)) - f(x^*,r(i))\right\} \le \frac{R^{2}}{2 \alpha t} + \frac{\alpha ^2 t G^{2}}{2\alpha t}. \end{aligned}$$

(24)

If $t \rightarrow \infty$, by the law of large numbers

$$\begin{aligned} \overline{f(x,r)} - p^* \le \frac{\alpha }{2} G^2. \end{aligned}$$

(25)

where $\overline{f(x,r)}=\frac{1}{t} \sum _{i=0}^{t}f(x(i),r(i))$ and $p^* = \mathbb {E}_{r} \left[ f(x^*,r)\right] = \frac{1}{t} \sum _{i=0}^{t} f(x^*,r(i))$.

Since f(.) is a convex function, by the Jensen’s inequality [24] we have $\overline{f(x,r)} \ge f(\bar{x},r)$, and consequently

$$\begin{aligned} f(\bar{x},r) - p^* \le \frac{\alpha }{2} G^2. \end{aligned}$$

(26)

Choosing step size $\alpha$ small enough, we conclude that the gradient iteration (18) converges statistically. In other words, as t goes to infinity, the solution derived from gradient iteration (18), i.e. $f(\bar{x},r)$, converges to the optimal value $p^*$.