Joint Power Allocation and Load Balancing Optimization for Energy-Efficient Cell-Free Massive MIMO Networks

I. INTRODUCTION

• Distributed coherent APs address cell-edge weakness, but AP activation in cell-free networks with fast fading and imperfect CSI had not been studied.The goal is to serve required SE while reducing power as traffic load changes.Source: §I · pp. 1–2

A. Main Contributions

• The non-convex joint power-allocation and AP-topology problem admits a global mixed-integer SOC solution, alongside two lower-complexity algorithms.The first heuristic uses optimized transmit powers and sparsity; the second uses optimized powers to select APs to turn off.Source: A. Main Contributions · p. 2

II. SYSTEM MODEL

• The network model assigns each of M APs to serve K users with required SE targets, allowing unneeded APs to sleep if the targets remain satisfied.A central processing unit connects to all APs through unlimited fronthaul links.Source: §II · p. 2

A. Uplink Pilot Transmission

• Users transmit assigned uplink pilots, which APs use to estimate channels; a sleeping AP is represented by setting its large-scale fading coefficients to zero.The paper uses MMSE channel estimation and assumes channel statistics are known at the APs.Source: A. Uplink Pilot Transmission · p. 3

B. Downlink Performance Analysis

• Each active AP constructs user-specific precoders from locally estimated channels and jointly transmits users’ data symbols.The user’s received signal combines contributions from the active APs.Source: B. Downlink Performance Analysis · p. 3
• The downlink ergodic SE lower bound separates desired signal, beamforming uncertainty, and inter-user interference contributions.Its effective SINR includes coherent pilot-contamination interference, non-coherent interference, and noise.Source: B. Downlink Performance Analysis · pp. 3–4
• Closed-form SE expressions are provided for MRT and F-ZF, and depend on transmit powers and the selected active AP set.F-ZF requires N > τp and uses τp spatial degrees of freedom to cancel interference between users with different pilots.Source: B. Downlink Performance Analysis · pp. 3–4

III. TOTAL POWER MINIMIZATION PROBLEM

• The objective includes transmit power and active-AP hardware dissipation, and the optimization variables are the active AP set and transmit powers.Activating an AP increases hardware dissipation but may reduce transmit power through coherent signal combination.Source: §III · p. 4, A. Problem Formulation · p. 4

A. Problem Formulation

• The formulation converts the non-convex problem into a mixed-integer SOC program solved by branch-and-bound.The hardware power term is simplified using the fact that SINR constraints are met with equality at an optimum.Source: §III · p. 4, A. Problem Formulation · p. 5
• The problem enforces AP maximum downlink powers and each user’s SE requirement, with transmit powers coupled through mutual interference.The SE expressions are based on fast fading and imperfect CSI, including channel-estimation errors and pilot contamination.Source: A. Problem Formulation · pp. 4–5
• For a fixed active-AP set, the problem reduces to an SOC program, but exhaustive search must consider 2^M − 1 nonempty AP selections.This exhaustive approach is computationally costly even for relatively small networks.Source: A. Problem Formulation · p. 5

B. Globally Optimal Solution to the Total Power Minimization Problem

• Setting an AP’s activity variable to zero forces its transmit powers to zero; setting it to one restores its total transmit-power constraint.The resulting binary solution identifies the active AP set in the equivalent formulation.Source: B. Globally Optimal Solution to the… · p. 6
• Binary activity variables link hardware and transmit power while remaining outside the SINR constraints, preserving the SOC structure.The implicit link is through zero maximum transmit power for inactive APs.Source: B. Globally Optimal Solution to the… · pp. 5–6
• Branch-and-bound can prune candidate AP sets, but its worst-case complexity matches exhaustive search and grows exponentially with M.The authors report that it finds a global optimum for a moderate-size network with 20 APs.Source: B. Globally Optimal Solution to the… · pp. 5–6

• The proposed algorithms seek good suboptimal solutions to the total power minimization problem with tolerable computational complexity.Source: §IV · p. 7
• The algorithms are designed to enable implementation in large Cell-free Massive MIMO networks.Source: §IV · p. 7
• Their motivation is the high computational complexity of solving the total power minimization problem using Lemma 3.Source: §IV · p. 7

• Group sparsity reformulates AP selection through transmit-power vectors, replacing the ℓ0 term with an ℓp relaxation that retains sparsity.An AP is implicitly active when its transmit-power vector has nonzero norm, so separate active-set variables are unnecessary.Source: A. Utilizing Sparsity to Turn Off APs · p. 7
• An iteratively reweighted least-squares procedure exploits group sparsity by solving an SOC program and updating weights across iterations.The objective decreases to a fixed point, and an AP with zero transmit power at one iteration remains at zero thereafter.Source: A. Utilizing Sparsity to Turn Off APs · pp. 7–8
• The stationary point may not be globally optimal, so its transmit powers are used to order APs for further turnoff rather than as the final solution.Source: A. Utilizing Sparsity to Turn Off APs · p. 8
• The algorithm tests candidate active-AP set sizes with SOC programs and updates the search bounds, retaining the best feasible solution.It bisects the range of AP counts and stops when the upper and lower bounds differ by one.Source: A. Utilizing Sparsity to Turn Off APs · p. 8
• The AP-turnoff search requires at most ⌈log2(M + 1)⌉ iterations, and the authors expect fewer iterations than branch-and-bound for large networks.Each iteration solves one SOC program.Source: A. Utilizing Sparsity to Turn Off APs · pp. 8–9

• Algorithm 2 solves for transmit powers once with all APs active, then uses the solution to order APs for turnoff.Source: B. Total Transmit Power Minimization… · p. 9
• After ordering APs, Algorithm 2 follows the same procedure as Algorithm 1 to determine how many remain active.Source: B. Total Transmit Power Minimization… · p. 9
• Algorithm 2 has lower complexity than Algorithm 1 and complexity roughly O(M^3.5K^3.5).Source: B. Total Transmit Power Minimization… · p. 9

V. NUMERICAL RESULTS

• The simulations use randomly distributed APs and users, 20 antennas per AP, and a 2 b/s/Hz SE requirement per user in the main setup.The setup uses a 1 km2 square, 50 m minimum AP spacing, and wrap-around to remove edge effects.Source: §V · p. 9
• With MRT, the mixed-integer SOC method needs about 6.4 W average transmit power, versus 1.8 W when all APs are active; disjoint sparsity uses about 11.8 W.For F-ZF, transmit power is up to 12% lower than with MRT, while the average minimum is only 3% lower.Source: Figure 1 · p. 10, Figure 2 · p. 10, §V · pp. 10–11
• 49% with MRT and 55% with F-ZF: the mixed-integer SOC solution reduces average total power consumption versus the transmit-power-only baseline.The two proposed sparsity-based algorithms also use less total power than disjoint sparsity, with an improvement of up to 59% for MRT.Source: Figure 3 · p. 10, Figure 4 · p. 10, §V · p. 11
• The group-sparsity approach converges to a stationary point in fewer than 15 iterations for networks with 20 or 40 APs serving 20 users.The objective decreases monotonically; relative to the initial point, its value is 20% lower with 20 APs and about 32% lower with 40 APs.Source: Figure 7 · p. 11, §V · p. 11
• The mixed-integer SOC solution activates 9.1 APs on average with MRT and 7.8 with F-ZF, while satisfying the requested SEs.With MRT, more than 55% of APs are in sleep mode; the network needs only a subset of APs to meet the SE requirements.Source: §V · p. 11, Figure 5 · p. 11, Figure 6 · p. 11
• 2.3× with MRT and 2.5× with F-ZF: Algorithm 1 reduces power consumption versus transmit-power-only optimization in a 50-AP, 40-user network.The mixed-integer SOC benchmark is excluded in this large-network case because of its extremely high complexity; group sparsity provides up to 10% lower consumption than using optimized transmit powers as side information.Source: §V · pp. 11–12, Figure 8 · p. 12
• With heterogeneous user SE requirements from 1 to 2 b/s/Hz, Algorithm 1 uses 3× lower average total power than transmit-power-only optimization.Algorithm 2 substantially reduces complexity compared with Algorithm 1 while requiring only 2% more power.Source: §V · p. 12

VI. CONCLUSION

• The mixed-integer SOC formulation with branch-and-bound finds a globally optimal joint choice of transmit powers and active APs.The optimization satisfies the SE requirements of all users.Source: §VI · p. 12
• Joint optimization reduces total power consumption by more than 50% compared with minimizing transmit power alone.Source: §VI · p. 12
• The two low-complexity algorithms consume roughly 20% more power than the global optimum and are applicable to large cell-free networks.Source: §VI · p. 12

A. Proof of Lemma 4

• A lower bound for the global optimum is computed by solving an SOC program for the new box.The proof describes the program's optimization variables and SOC-constraint dimensions.Source: A. Proof of Lemma 4 · p. 12
• An upper bound is obtained by rounding the optimized binary-variable values and solving an SOC program on a new box.Source: A. Proof of Lemma 4 · p. 13
• The branch-and-bound complexity is obtained by summing the computational costs across N1 iterations.The per-iteration cost and iteration order for ε-accuracy are derived from the SOC programs.Source: A. Proof of Lemma 4 · pp. 12–13

B. Proof of Theorem 1

• The iterative process in Theorem 1 produces a non-increasing objective function for problem (39).Source: B. Proof of Theorem 1 · p. 13
• The iteratively weighted least-squares approach converges to a fixed point.Source: B. Proof of Theorem 1 · p. 13
• At the fixed point, the solution is a stationary point of problem (39), and the proof establishes a local minimum of f.Source: B. Proof of Theorem 1 · p. 13