Ferkans — Interactive Telecom Tutor

The Energy Cost of Distributed Infrastructure

Distributing $L$ access points across a coverage area improves spectral efficiency and fairness, but at what energy cost? Each AP requires power for RF chains, baseband processing, and the fronthaul link — even when the AP carries no traffic. As $L$ grows, the total hardware power consumption can dominate the transmit power, and the energy efficiency (bits per Joule) can actually decrease. This section develops a realistic power consumption model and identifies the optimal AP density that maximizes energy efficiency.

Definition:
Total Power Consumption Model

The total power consumed by a cell-free massive MIMO system with $L$ APs is

$P_{\text{total}} = \sum_{l=1}^{L} \left(\frac{1}{\zeta} \sum_{k=1}^{K} \eta_{lk} \gamma_{lk} N + P_{\text{AP}} + P_{\text{fh},l}\right) + P_{\text{CPU}}$

where:

$\zeta \in (0, 1]$ is the power amplifier efficiency
$P_{\text{AP}}$ is the per-AP circuit power (RF chains, oscillators, DACs/ADCs, cooling)
$P_{\text{fh},l}$ is the per-AP fronthaul link power (optical transceiver, switching)
$P_{\text{CPU}}$ is the CPU processing power (baseband computation, fronthaul aggregation)

The fronthaul power scales with the data rate: $P_{\text{fh},l} = P_{\text{fh},0} + \Delta_{\text{fh}} \cdot C_{\text{fh},l}$ where $C_{\text{fh},l}$ is the fronthaul rate from AP $l$ and $\Delta_{\text{fh}}$ is the incremental power per bit/s.

For single-antenna APs ( $N = 1$ ), $P_{\text{AP}} \approx 0.2$ -- $1$ W and $P_{\text{fh},l} \approx 0.5$ -- $2$ W per AP. With $L = 100$ APs, the fixed hardware power alone is $P_{\text{hw}} = L(P_{\text{AP}} + P_{\text{fh}}) \approx 70$ -- $300$ W, which can exceed the total transmit power of $L \cdot P_t \approx 10$ -- $50$ W.

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Definition:
Energy Efficiency

The energy efficiency (EE) of the cell-free system is defined as the ratio of total throughput to total power consumption:

$\text{EE} = \frac{B \cdot \sum_{k=1}^{K} \text{SE}_k}{P_{\text{total}}} \quad [\text{bits/Joule}]$

where $B$ is the system bandwidth in Hz and $\text{SE}_k$ is the spectral efficiency of user $k$ in bits/s/Hz. The EE captures the fundamental tradeoff: adding more APs increases the sum-SE (numerator) but also increases the total power (denominator). The optimal operating point balances these competing effects.

Theorem: Optimal AP Density for Energy Efficiency

For a cell-free system with fixed total coverage area $|\mathcal{A}|$ , user density $\lambda_u = K / |\mathcal{A}|$ , and per-AP power $P_{\text{AP}} + P_{\text{fh}}$ , the energy efficiency as a function of AP density $\lambda_{\text{AP}} = L / |\mathcal{A}|$ is

$\text{EE}(\lambda_{\text{AP}}) = \frac{B \cdot K \cdot \overline{\text{SE}}(\lambda_{\text{AP}})}{\lambda_{\text{AP}} |\mathcal{A}| (P_{\text{AP}} + P_{\text{fh}} + P_t/\zeta) + P_{\text{CPU}}}$

where $\overline{\text{SE}}(\lambda_{\text{AP}})$ is the average per-user SE. Since $\overline{\text{SE}}$ is a concave, saturating function of $\lambda_{\text{AP}}$ (due to diminishing returns from adding more APs), and the denominator is linear in $\lambda_{\text{AP}}$ , $\text{EE}(\lambda_{\text{AP}})$ is quasi-concave with a unique maximum at

$\lambda_{\text{AP}}^* = \arg\max_{\lambda_{\text{AP}}} \text{EE}(\lambda_{\text{AP}})$

Beyond $\lambda_{\text{AP}}^*$ , adding more APs decreases energy efficiency.

Initially, adding APs provides large SE gains (macro-diversity from uncovered areas). Eventually, diminishing returns set in: the $L$ -th AP adds little SE but still costs $P_{\text{AP}} + P_{\text{fh}}$ watts. The optimal density is where the marginal SE gain per AP equals the marginal power cost — a classical economic efficiency argument.

Proof

SE saturation

The per-user SE under MRC scales as $\log_2(1 + c \cdot \lambda_{\text{AP}})$ for a constant $c$ (proportional to the SINR, which grows linearly in $L$ when interference is held fixed). This is concave in $\lambda_{\text{AP}}$ .

Linear power growth

$P_{\text{total}} = \lambda_{\text{AP}} |\mathcal{A}| (P_{\text{AP}} + P_{\text{fh}} + P_t/\zeta) + P_{\text{CPU}}$ is affine (hence linear) in $\lambda_{\text{AP}}$ .

Quasi-concavity

The ratio of a concave function to a positive affine function is quasi-concave. A quasi-concave function on a convex set has a unique maximum (up to a plateau). The optimal density satisfies the first-order condition $\overline{\text{SE}}'(\lambda^*) / (P_{\text{AP}} + P_{\text{fh}} + P_t/\zeta) = \overline{\text{SE}}(\lambda^*) / P_{\text{total}}(\lambda^*)$ . $\blacksquare$

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Energy Efficiency vs Number of APs

Explore how energy efficiency varies with AP density for different hardware power parameters. Observe the optimal AP count that maximizes EE and how it depends on the per-AP hardware power consumption.

Parameters

K

(users)10

L_{\max}

(max APs)160

P_{\text{AP}}

(W)0.5

P_{\text{fh}}

(W)1

SNR (dB)10

PA efficiency

\zeta

0.4

Example: Energy Efficiency Comparison

Compare the energy efficiency of cell-free ( $L = 64$ APs, $N = 1$ ) versus co-located ( $L = 1$ BS, $M = 64$ antennas) for $K = 10$ users over a 1 km $^2$ area with bandwidth $B = 20$ MHz. Use $P_{\text{AP}} = 0.5$ W, $P_{\text{fh}} = 1.0$ W, $P_t = 0.2$ W per AP, PA efficiency $\zeta = 0.4$ , and $P_{\text{CPU}} = 10$ W.

Solution

Cell-free power consumption

Transmit: $64 \times 0.2 / 0.4 = 32$ W. Hardware: $64 \times (0.5 + 1.0) = 96$ W. CPU: 10 W. Total: $P_{\text{total}}^{\text{cf}} = 138$ W.

Co-located power consumption

Transmit: $12.8 / 0.4 = 32$ W (same total $P_t$ ). Hardware: $1 \times (P_{\text{BS}} + P_{\text{cooling}}) \approx 50$ W (one BS is more power-hungry but no fronthaul). Total: $P_{\text{total}}^{\text{col}} \approx 92$ W.

Spectral efficiency

From Section 15.2 analysis: cell-free average SE $\approx 3.5$ bits/s/Hz/user, co-located average SE $\approx 2.8$ bits/s/Hz/user.

Energy efficiency

Cell-free: $\text{EE}^{\text{cf}} = \frac{20 \times 10^6 \times 10 \times 3.5}{138} = 5.07$ Mbits/J. Co-located: $\text{EE}^{\text{col}} = \frac{20 \times 10^6 \times 10 \times 2.8}{92} = 6.09$ Mbits/J. The co-located system is $\sim 20\%$ more energy-efficient due to lower hardware overhead, despite lower SE. Cell-free's higher SE does not fully compensate for the 64 fronthaul links.

When cell-free wins in EE

Reducing $P_{\text{fh}}$ to 0.3 W (e.g., with PoE and passive optical network) gives $P_{\text{total}}^{\text{cf}} = 83.2$ W and $\text{EE}^{\text{cf}} = 8.41$ Mbits/J — now 38% better than co-located. The fronthaul power cost is the key lever for cell-free EE.

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Power Consumption Parameters by Deployment Scenario

Parameter	Indoor (PoE AP)	Urban Outdoor	Rural Macro
$P_{\text{AP}}$ (W)	0.2-0.5	0.5-2.0	5-20
$P_{\text{fh}}$ per AP (W)	0.1-0.5	0.5-2.0	2-10
$P_t$ per AP (W)	0.05-0.2	0.2-1.0	5-40
PA efficiency $\zeta$	0.2-0.3	0.3-0.4	0.3-0.5
Typical AP count	50-200	20-100	5-20
Total power (W)	20-100	50-400	200-1000
Fronthaul type	Ethernet/PoE	Fiber/eCPRI	Fiber/CPRI

Key Takeaway

Energy efficiency in cell-free massive MIMO is not a monotonically increasing function of the number of APs. There exists an optimal AP density $\lambda_{\text{AP}}^*$ that balances the SE gain from macro-diversity against the hardware power cost of each AP and its fronthaul link. Reducing per-AP hardware power — through simpler AP designs and efficient fronthaul (PoE, passive optics) — shifts $\lambda_{\text{AP}}^*$ higher and makes denser deployment viable.

Common Mistake: Ignoring Hardware Power in EE Comparisons

Mistake:

Computing energy efficiency as $\text{EE} = B \sum_k \text{SE}_k / (L \cdot P_t)$ , using only transmit power in the denominator and ignoring circuit, fronthaul, and processing power.

Correction:

In practical cell-free deployments, hardware power dominates transmit power by a factor of 3-10x. The correct formula uses $P_{\text{total}}$ including all components. Ignoring hardware power makes denser deployment look artificially attractive and hides the diminishing-returns behavior that determines the optimal AP density.

🔧Engineering Note

AP Sleep Modes for Energy Savings

When traffic load is low (e.g., nighttime), a significant fraction of APs can be put into sleep mode to save the fixed power $P_{\text{AP}} + P_{\text{fh}}$ . The challenge is to select which APs to deactivate while maintaining coverage for the remaining active users. This is a binary optimization problem (NP-hard in general) but admits efficient greedy approximations. With sleep modes, the effective $P_{\text{AP}}$ is load-dependent: $P_{\text{AP}}^{\text{eff}} = P_{\text{AP}} \cdot L_{\text{active}} / L$ , which can reduce total power by 50-70% during off-peak hours.

Practical Constraints

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Active AP set must provide minimum SINR to all users
•
Transition time from sleep to active: 10-100 ms (impacts latency-sensitive traffic)
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O-RAN supports AP sleep through the non-RT RIC energy-saving use case

Quick Check

In a cell-free system with $L = 100$ single-antenna APs, each with $P_{\text{AP}} = 0.5$ W, $P_{\text{fh}} = 1$ W, and transmit power $P_t = 0.2$ W with PA efficiency $\zeta = 0.4$ , what fraction of total power (excluding CPU) is consumed by hardware (non-transmit)?

About 25%

About 50%

About 75%

About 90%

Correction:

About 75%

Hardware = $100 \times 1.5 = 150$ W. Transmit = $100 \times 0.2/0.4 = 50$ W. Total = 200 W. Hardware fraction = 75%. This confirms that hardware power dominates in cell-free systems.

Energy Efficiency (EE)

The ratio of total throughput (bits/s) to total power consumption (Watts), measured in bits/Joule. Captures the tradeoff between spectral efficiency gains from more APs and the hardware power cost of each additional AP and its fronthaul link.

Power Amplifier Efficiency

The ratio $\zeta = P_{\text{out}} / P_{\text{in}}$ of RF output power to DC input power. Typical values: 20-40% for linear operation. Lower efficiency means more DC power is wasted as heat, increasing total power consumption.

Related: Energy Efficiency, Total Power

Historical Note: The Green Communications Movement

2010-present

Energy efficiency in wireless networks became a major research topic around 2010, driven by the realization that ICT was responsible for 2-3% of global CO $_2$ emissions. The GreenTouch consortium (2010-2015) set a goal of 1000x improvement in network energy efficiency. Bjornson, Hoydis, and Sanguinetti (2017) showed that massive MIMO could achieve 100x EE improvement over single-antenna systems by serving many users simultaneously with linear processing. Cell-free massive MIMO extends this by additionally averaging path loss through AP proximity, potentially reducing the required transmit power by another order of magnitude. The challenge is that the hardware power of distributed APs can offset these gains if not carefully managed.

The EE-SE Tradeoff

Energy efficiency and spectral efficiency are not simultaneously maximizable. Operating at maximum SE (all APs active, maximum power) typically does not maximize EE, and vice versa. The EE-SE tradeoff curve is convex: starting from the EE-optimal point, increasing SE requires disproportionately more power. For cell-free systems, the EE-optimal operating point typically uses 40-70% of the APs that the SE-optimal point would use. System designers must choose where on this tradeoff curve to operate based on the deployment objective.

Energy Efficiency