## 5 Simulation Results  
In this section, a series of numerical simulations are carried out to verify the accuracy of our derived EE equation and the effectiveness of our proposed optimization algorithm. Simulation parameters are listed in Table 1. For the power consumption model, we refer to the statistics in [41]. Figure 2 shows the impact of picocell BS density on EE. It can be seen that the simulation results are in line with the theoretical results, which validates the correctness of EE expression in Equation (13). We can see that if the MUE density, the macrocell BS density, and the power consumption of BS are given, there must exist an optimal picocell BS density that can maximize EE. It can be seen that the EE increases first and then decreases with the increase of the density of the picocell BS. The reason for the increase is that with the increase of the density of the picocell BS, since the distance from the picocell BS to the MUE is obviously smaller than that from the macrocell BS to MUE, more users will access the picocell BS. Additionally, the power consumption of the picocell BS is small, so EE will be improved at this time. However, when the density of the picocell BS is greater than a given value, the number of access users will be saturated, while a continuous increase of the density of the picocell BS will lead to an increase of power consumption, meaning the EE will gradually decrease. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters Name** & **Values** \\ \hline Bandwidth & \(10^{7}\) Hz \\ HetNet area & \(10\times 10\) km\({}^{2}\) \\ Density of MUEs, \(\lambda_{\text{u}}\) & 0.025 m\({}^{-2}\) \\ Density of macrocell BS, \(\lambda_{1}\) & \(10^{-5}\) m\({}^{-2}\) \\ Path-loss exponent, \(\alpha\) & \(\alpha=4\) \\ Circuit power consumption of MBS, \(\text{P}_{\text{C1}}\) & 130 W \\ Transmit power of MBS, \(\text{P}_{\text{T1}}\) & 20 W \\ Number of transceivers of MBS, \(\text{N}_{\text{TR1}}\) & 6 \\ Slope of MBS power consumption, \(\theta_{1}\) & 4.7 \\ Circuit power consumption of PBS, \(\text{P}_{\text{C2}}\) & 6.8 W \\ Transmit power of PBS, \(\text{P}_{\text{T2}}\) & 0.13 W \\ Number of transceivers of PBS, \(\text{N}_{\text{TR2}}\) & 2 \\ Slope of PBS power consumption, \(\theta_{2}\) & 4.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 2: Network energy efficiency (EE) versus picocell base station (BS) density. In addition, both biased mode and unbiased mode are compared in this simulation. Unbiased mode means that the SIR threshold of a microcell BS is the same as that of a picocell BS, while biased mode means that the SIR threshold of a microcell BS is higher than that of a picocell BS. The target SIR for the picocell BSs is set to 10 dB, while the target SIRs for picocell BSs are set to 10 dB in unbiased mode and 5 dB in biased mode, respectively. It can be seen that when \(\lambda_{2}\) is not very low, bias technology helps to improve EE. This is because, in both unbiased mode and biased mode, more MUEs choose to access macrocell BSs when the distribution of picocell BSs is sparse, since macrocell BSs can provide higher SIR compared to picocell BSs. In this case, the EEs of the two modes are close to each other. With the increase of the picocell BSs density, in biased mode, increasingly more MUEs choose to access picocell BSs. Because the power consumption of a picocell BS is much lower than that of a macrocell BS, the EE of the biased mode is much higher. Figure 3 gives the average EE against the transmit power of picocell BSs. Figure 3 shows the impact of picocell BS transmit power on EE. The target SIR for the macrocell BSs and picocell BSs are set to 10 dB and 5 dB, respectively. It can be seen that the EE increases first and then decreases with the increase of the transmission power of the picocell BS.