A downlink power control algorithm for long-term energy efficiency of small cell network

Abstract

Recently, small cells have been densely deployed in hot spots and overlaid with macro cells in order to guarantee the quality of service for voluminous multimedia data. For a small cell network (SCN), the long-term energy efficiency is one of the most challenging issues for conserving energy while achieving link capacity over the air-interface. In this paper, we present a novel downlink power control scheme to maximize the long-term energy efficiency over a time division multiple access system, which includes multiple small cells and a central controller. Toward this goal, we formulate a long-term energy efficiency optimization problem by defining a long-term energy efficiency metric and performing an iterative algorithm to obtain optimal solutions in real-time via fast convergence. The simulation results show that the proposed metric is very effective at improving the long-term energy efficiency and conserving 8.1 % of the energy on average when compared to a conventional metric. In addition, it is verified that the proposed metric is more effective for an SCN, in particular, when the link capacity is severely degraded due to inter-cell interference.

Appendices

Appendix: Proof of the optimality of the solutions (14) and (17)

Here, we prove the optimality of the solutions (14) and (17) by referring to the numerical proof used in [21]. For brevity, we shall omit the time index $t$ from the notations. In addition, the two problems (8) and (9) can be rewritten as

$$\begin{aligned}&(\text {I}) \quad \max \{U_N({\mathbf P} ) / U_D({\mathbf P} ) \} \\&(\text {II}) \quad \max \{U_N({\mathbf P} ) - q\cdot U_D({\mathbf P} ) \}. \end{aligned}$$

To prove the optimality of the solutions, we first prove the convexity of the problem (II).

Corollary 2

$F(q)=\max \{U_N({\mathbf {P}}) - q\cdot U_D({\mathbf {P}})\}$ is convex.

Proof

Let ${\mathbf {P}}_{m}$ be the downlink power to maximize $F(\theta \cdot q_1+ (1-\theta )\cdot q_2)$ with $q_1 \ne q_2$ and $0 \le \theta \le 1$:

$$\begin{aligned}&F(\theta \cdot q_1+ (1-\theta )\cdot q_2) = U_N({\mathbf {P}}_{m}) - (\theta \cdot q_1 + (1-\theta )\cdot q_2) \cdot U_D({{\mathbf {P}}_{m}})\\&\quad = \theta \cdot \{ U_N({\mathbf {P}}_{m}) - q_1\cdot U_D({\mathbf {P}}_{m})\} + (1-\theta )\cdot \{ U_N({\mathbf {P}}_{m}) - q_2\cdot U_D({\mathbf {P}}_{m})\} \\&\quad \le \theta \cdot \max \{U_N({\mathbf {P}}) - q_1\cdot U_D({\mathbf {P}})\} + (1-\theta )\cdot \max \{U_N({\mathbf {P}}) - q_2\cdot U_D({\mathbf (P)})\} \\&\quad = \theta \cdot F(q_1) + (1-\theta )\cdot F(q_2). \end{aligned}$$

$\square$

Corollary 3

$F(q)=0$ has a unique solution of $q_0$.

Proof

See Appendix 9. $\square$

Corollary 4

$q_0 = U_N({\mathbf {P}}) / U_D({\mathbf {P}}) = \max \{U_N({\mathbf P} _0) / U_D({\mathbf {P}}_0) \}$ if and only if $F(q_0) = F(q_0, {\mathbf P} _0) = \max \{U_N({\mathbf P} ) - q_0\cdot U_D({\mathbf P} )\} = 0$.

Proof

• (a) Let ${\mathbf P} _0$ be a solution of problem (I). Therefore,

  $$\begin{aligned} q_0 = U_N({\mathbf P} _0)/U_D({\mathbf P} _0)\ge U_N({\mathbf P} )/U_D({\mathbf P} ). \end{aligned}$$

  From equation and inequation above,

  $$\begin{aligned}&U_N({\mathbf P} )-q_0\cdot U_D({\mathbf P} )\le 0, \end{aligned}$$

  (22)
  $$\begin{aligned}&U_N({\mathbf P} _0)-q_0\cdot U_D({\mathbf P} _0) = 0. \end{aligned}$$

  (23)

  From (22), $F(q_0)=\max \{U_N({\mathbf P} )-q_0\cdot U_D({\mathbf P} )\}=0$. In addition, from (23), $F(q_0) = F(q_0, {\mathbf P} _0) = \max \{U_N({\mathbf P} )\,-\,q_0\cdot U_D({\mathbf P} )\} = 0$.

• (b) Let ${\mathbf P} _0$ be a solution of problem (II). From the definition of (II)

  $$\begin{aligned} U_N({\mathbf P} ) - q_0\cdot U_D({\mathbf P} ) \le U_N({\mathbf P} _0) - q_0\cdot U_D({\mathbf P} _0) = 0. \end{aligned}$$

  Therefore,

  $$\begin{aligned}&U_N({\mathbf P} ) - q_0\cdot U_D({\mathbf P} ) \le 0, \end{aligned}$$

  (24)
  $$\begin{aligned}&U_N({\mathbf P} _0) - q_0\cdot U_D({\mathbf P} _0) = 0. \end{aligned}$$

  (25)

  From (24), $q_0 \ge U_N({\mathbf P} )/U_D({\mathbf P} )$, which means that $q_0$ is the maximum of problem (I). In addition, from (25), $q_0 = U_N({\mathbf P} _0)/U_D({\mathbf P} _0)$, which means that ${\mathbf P} _0$ is a solution vector of problem (I).$\square$

From Corollaries 2 and 3, it is proved that the problem (II) is convex and has a unique solution. The unique solution is an optimal solution of the problem (II). In addition, from Corollary 4, the optimal solution of the problem (II) is equivalent to the optimal solution of the problem (I).

The objective functions of the primal problems (7) and (15) become concave functions by Corollary 1. In addition, the constraints in the primal problems (7) and (15) are convex sets. In this case, the optimal duality gap is zero, i.e., strong duality holds [28] (see Chapter 5.2 The Lagrange dual problem). Therefore, the optimality of the solutions (14) and (17) are guaranteed.

Appendix: Proof of the convergence of the iterative algorithm

Referring to the numerical proof used in [21], we prove the convergence of the iterative algorithm in Sect. 4.3. For ease of understanding, we use the notations ${\mathbf {P}}_n$ and $q_n$ instead of using ${\mathbf {P}}(t)$ and $q(t)$, respectively, for the $n$th iteration parameters. First, it is necessary to prove that $q_{n + 1} > q_n$ for $n>0$ when ${\mathbf {P}}_n$ and $q_n$ satisfy the inequality $F(q_n) = U_N({\mathbf {P}}_n) - {q_n}\cdot {U_D}({\mathbf {P}}_n) > 0$. According to the definition of energy efficiency $q(t)$, the equality $U_N({\mathbf {P}}_n) = q_{n + 1} \cdot {U_D}({\mathbf {P}}_n)$ holds. Then $F(q_n)$ becomes

$$\begin{aligned} F(q_n)&= {U_N}({\mathbf {P}}_n) - {q_n}\cdot {U_D}({\mathbf {P}}_n)\\&= {q_{n + 1}}\cdot {U_D}({\mathbf {P}}_n) - {q_n}\cdot {U_D}({\mathbf {P}}_n)\\&= \left( q_{n + 1} - q_n\right) {U_D}({\mathbf {P}}_n) > 0. \end{aligned}$$

Since ${U_D}({\mathbf {P}}_n) > 0$, $q_{n + 1} > q_n$. $F(q_n)$ is a strictly monotonic decreasing function and is continuous for $q_n$. The solution of $F(q_n)=0$ is unique because of the monotonicity property of $\lim \limits _{q_n \rightarrow - \infty } F(q_n) = \infty$ and $\lim \limits _{q_n \rightarrow \infty } F(q_n) = - \infty$. Let $q_0$ be the unique solution. To prove the convergence of the iterative algorithm, we prove the following assertion: $\lim \limits _{n \rightarrow \infty } q_n = q_0$. If this is not true, the following inequality holds: $\lim \limits _{n \rightarrow \infty } q_k = q^* < q_0$. Since $F(q_n)$ is a strictly monotonic decreasing function, $F\left( q^*\right) > F\left( q_0\right)$. However, according to (9), $F\left( q^*\right) = 0$. From the definition of the unique solution $q_n$, $F\left( q_n\right) = 0$.

This is a contradiction to $\lim \limits _{n \rightarrow \infty } q_n \ne q_0$, so $\lim \limits _{n \rightarrow \infty } q_n = q_0$. Thus, $\lim \limits _{n \rightarrow \infty } F\left( q_n\right) = F\left( q_0\right)$, since $F(q_n)$ is a continuous function for $q_n$. Therefore, we prove the convergence of the iterative algorithm.

Appendix: Discussion of practical issues of the proposed algorithm

For practicality of the iterative algorithm, an orthogonal frequency division multiple access (OFDMA) scheme used for 3GPP Long Term Evolution (LTE) downlink is considered.

For brevity, we shall hereafter insert the subcarrier index $s\in \{1,...,S\}$ to the notations, where $S$ is the total number of subcarriers. The received symbol at user $k$ in BS $i$ on subcarrier $s$ at the time slot $t$ can be expressed as

$$\begin{aligned} y_{i,k,s}(t)&= \sqrt{P_{i,s}(t)a_{i,k}(t)l_{i,k}(t)}h_{i,k,s}(t)x_{i,s}(t)+z_{i,k,s}(t) \\&\quad+ \underbrace{\sum \limits _{j\ne i} \sqrt{P_{j,s}(t)a_{j,k}(t)l_{j,k}(t)}h_{j,k,s}(t)x_{j,s}(t)}_{\rm Inter-cell \, Interference} \end{aligned}$$

(26)

where $P_{i,s}(t)$ and $x_{i,s}(t)$ are the donwlink power and transmitted symbol of BS $i$ on subcarrier $s$ at time slot $t$.

In 3GPP LTE downlink system, the scrambling which is one of the interference randomization schemes is performed to make the ICI perform like the AWGN, and to average the interference in time domain by scrambling the signal from different BSs with different codes which have low correlation with each other [29, 30]. However, it is difficult to reflect the scrambling and other modulation schemes to Eq. (26) from the beginning. In terms of long-term energy efficiency, the signal $x_{i,s}(t)$ in Eq. (26) could be assumed to be a scrambled symbol. In such a case, it is expected to have similar tendency of energy efficiency even if those schemes are involved. Since the scrambling does not reduce the average interference level, the performance is not affected by the scrambling [31, 32].

Therefore, the interference power at user $k$ in BS $i$ on subcarrier $s$ at time slot $t$ can be expressed as $I_{i,k,s}(t) = \sum \limits _{j\ne i} G_{j,k,s}(t)P_{j,s}(t)$, where $G_{j,k,s}(t) = a_{i,k}(t)l_{i,k}(t)|h_{j,k}(t)|^{2}$ is the channel gain from BS $i$ to user $k$ on subcarrier $s$. In addition, the moving average interference ${\tilde{I}}_{i,k,s}(t)$ can be expressed as ${\tilde{I}}_{i,k,s}(t) = \left( 1 - \frac{1}{t_e}\right) {\tilde{I}}_{i,k,s}(t-1) + \frac{1}{t_e}I_{i,k,s}(t-1)$.

Therefore, the channel capacity at user $k$ in BS $i$ on subcarrier $s$ can be expressed as ${\tilde{C}}_{i,k,s}(t) = \log _{2}\{1+\tilde{\varGamma }_{i,k,s}(t)\}$, where $\tilde{\varGamma }_{i,k,s}(t) = \frac{G_{i,k,s}(t){P_{i,s}}(t)}{N_0 + {\tilde{I}}_{i,k,s}(t)}$ is the estimated SINR at user $k$ in BS $i$ on subcarrier $s$.

The energy efficiency metric over a long-term duration becomes

$$\begin{aligned} {\rm EE}_{\rm long}(t) = \frac{\sum \nolimits _{i = 1}^M \sum \nolimits _{s = 1}^{S} b_{i,k,s}(t) {\tilde{C}}_{i,k,s}(t) + \sum \nolimits _{t'=t-t_h}^{t-1} {\sum \nolimits _{i = 1}^M \sum \nolimits _{s = 1}^{S} b_{i,k,s}(t')C_{i,k,s}(t')}}{\sum \nolimits _{i = 1}^M \sum \nolimits _{s = 1}^{S} b_{i,k,s}(t) P_{i,s,{\rm tot}}(t) + \sum \nolimits _{t'=t-t_h}^{t-1} \sum \nolimits _{i = 1}^M \sum \nolimits _{s = 1}^{S} b_{i,k,s}(t') P_{i,s,{\rm tot}}(t')} \end{aligned}$$

with $P_{i,s,{\rm tot}}(t) = P_{i,s}(t) + P_{i,c}$. The variable $b_{i,k,s}(t)\in \{0,1\}$ represents the assignment status of subcarrier $s$ at user $k$ in BS $i$ at time slot $t$, i.e., $b_{i,k,s}(t)=1$ for the assignment, or $b_{i,k,s}(t)=0$ otherwise.

Therefore, the energy efficiency optimization problem becomes

$$\begin{aligned} \max \limits _{{\mathbf {P}}(t),{\mathbf {B}}(t)}& \quad {\rm EE}_{\rm long}(t)\\ \text{ subject } \text{ to }& \quad ({\rm C1}) \sum \limits _{s=1}^{S} b_{i,k,s}(t){\tilde{C}}_{i,k,s}(t)\ge R_{k,0}, k\in K_i,\forall i\in M, \\& \quad ({\rm C2}) \sum \limits _{s=1}^{S} b_{i,k,s}(t)P_{i,s}(t) \le P_{\max }, \forall i\in M, \\& \quad ({\rm C3}) \sum \limits _{k=1}^{K_i} b_{i,k,s}(t) \le 1, \forall s\in S,\forall i\in M, \\& \quad ({\rm C4})\,b_{i,k,s}(t) \in \{0,1\}, \forall {k}\in K_i,\forall s\in S,\forall i\in M, \\& \quad ({\rm C5})\,0 \le P_{i,s}(t), \forall s\in S,\forall i \in M. \end{aligned}$$

(27)

where ${\mathbf {P}}(t) = [P_{1,1}(t),\ldots , P_{1,S}(t), \cdot \cdot \cdot , P_{M,1}(t),\ldots , P_{M,S}(t)]$, and ${\mathbf {B}}(t) = [b_{i,1,1}(t),\ldots , b_{i,1,S}(t),\ldots , b_{i,M,1}(t),\ldots , b_{i,M,S}(t)]$ for $i=1,...,M$. Here, (C1) is the target data rate constraint for each user. (C2) and (C5) are the downlink power constraints for each BS. In addition, (C3) and (C4) mean that each subcarrier can be assigned to only one user.

Compared to (15), the energy efficiency optimization problem in (27) includes the subcarrier assignment problem for the multi-cell multi-carrier system. In addition, the objective function of (27) is a fractional form including binary variables. The constraints in (27) are non-convex sets. Therefore, the optimization techniques and approaches in Sect. 4 cannot be used to find solutions of (27) anymore. Even though it is difficult to find ${\mathbf {P}}^{*}(t)$ and ${\mathbf {B}}^{*}(t)$ at the same time, the local solution of ${\mathbf {P}}(t)$ can be found by using the proposed algorithm under the assumption that the subcarriers are already assigned to each user. In this case, the subcarrier assignment can be achieved based on the CSI feedback. For instance, a subcarrier can be assigned to each user whose channel gain is largest. Under this assumption, the proposed method in Sect. 4 is directly applied to each subcarrier as a single carrier system.