4.1.1 Motivations and Background

Our proposed NAN design in the previous chapter was based on fixed data rate requirements in the smart grid. In practice, the data rate requirement may not be that demanding, since the smart grid will be rolled out in phases. Therefore, a NAN does not need to operate at it full designed capacity in early phases. In our proposed wireless NAN design, the networks are enabled by Wi‐Fi and WiMAX. Normal data aggregate points (DAPs) are enabled with Wi‐Fi for internal communications in a NAN. Gateway DAPs are also enabled with WiMAX for bridging a NAN and the utility backbone network.

WiMAX is chosen for gateway DAPs because it may be deployed in in the unlicensed 5.8 GHz unlicensed spectrum. Avoiding licensing fees could significantly lower costs. However, for systems operating in unlicensed bands, the maximum effective isotropic radiated power (EIRP) is limited to 30 dBm (or 1000 mW) by the Federal Communication Commission (FCC) [61]. If the equipment transmits in a fixed point‐to‐point link, the maximum EIRP can be slightly higher. Thus in our proposed network model, gateway DAPs are equipped with point‐to‐point WiMAX technology operating in 5.8 GHz bands to take advantage of the free spectrum and to achieve better performance. Compared to a normal Wi‐Fi based DAP, which operates at 100 mW, the power consumption for each gateway is a more important issue. If a gateway DAP can operate for a longer life cycle, so can a normal Wi‐Fi‐based DAP.

Increasing the energy efficiency of a NAN benefits its longevity as well as self‐sustainability. One needs to realize that one purpose of a NAN in the smart grid is to deliver information such as metering data. The advanced metering infrastructure (AMI) requires reliable data delivery to enable functions in the smart grid. In order to analyze the performance of a NAN transmission, we define data rate reliability as the ratio of the total effective uplink transmission data rate to the data rate generated by all the smart meters in the NAN. Data rate reliability is a metric to measure if a NAN successfully uploads sufficient smart meter data to the concentrator.

In this chapter, we focus our study on the uplink transmission, because the downlink transmission contains only a relatively small amount of information from the metering data management system (MDMS). Unlike the downlink transmission, achieving a very high data rate reliability (e.g. ) for the uplink transmission is challenging because of the increasing real‐time transmission data rate requirement as the smart grid rolls out. With more pervasive deployment of the smart grid in near future, smart meter data will be generated much more frequently than the current 15‐minute‐based sampling period. For example, if we adopt currently used 12 KB as one sampling smart meter data, and the minimum sampling period goes down from 15 minutes to 1 second, the data rate required by a smart meter would increase almost 100 times, from 100 bps to 96,000 bps. Moreover, it is estimated that a yearly amount data as massive as 100 PBytes will be generated by AMI within 10 years [78]. This will quickly cause storage and processing problems for the MDMS. Therefore, if a smart meter is smart enough to adaptively change its sampling frequency based on the household situation, the peak volume of smart meter data can be lowered dramatically, and this in turn reduces the burden of the MDMS. For example, fewer people stay at home during daytime on working days lower metering frequency; more appliances are used from evening to midnight when people are at home thus higher metering frequency.

4.1.2 Related Work

The uplink transmission from the gateway DAPs to the concentrator occurs over a multiaccess wireless network. There has been much research about the power efficiency for this topic [72–75]. However, that research was based on the noncooperative game theoretical approach, which achieves individual optimality. As an alternative, our proposed scheme is intended to achieve global optimality. Moreover, fairness to the customers is considered in our NAN design, so even the furthest customer in the neighborhood can receive the same quality of service provided by the smart grid communication infrastructure. As discussed earlier in the previous chapter, we consider geographical fairness when deploying gateway DAPs so that each gateway would roughly cover the same number of customers. Moreover, gateway DAPs operate at different power levels so that the receiving power at the receiver side is uniform. Besides fairness, it is also a requirement to consider the geographical deployment of gateway DAPs, since it is the first priority of a NAN to upload sensing/measuring status data from even the most distant customer to the concentrator within the latency requirement.

A wireless mesh network based on the IEEE 802.11s protocol [79] has been widely used in the literature as one of the best candidates for deploying a NAN [58, 63, 80]. A Wi‐Fi‐enabled NAN can handle high data rate transmission within the network, owing to the robust and still improving Wi‐Fi technology. Some uplink transmission issues have been studied, such as the uplink transmission of AMI in terms of latency and throughput based on multigate structure [63, 80]. However, energy efficiency was not mentioned in these works as they did not address green energy, which plays an important role for the power supply of the AMI in the smart grid. The deployment of an AMI with green energy (e.g. solar panel and battery) was studied in [10] without a specific power control scheme to further enhance the performance of a NAN. Energy efficiency for a multiple‐access system has been proposed in [72, 74, 75] by achieving maximum bits/joule utility. Since AMI is distinguished from those general multiple‐access systems by its demanding data rate and low latency requirement, the results from those studies cannot be directly applied to AMI.

In the rest of this chapter, we propose an uplink transmission power control scheme that aims to guarantee data rate reliability while maintaining a high level of energy efficiency in a NAN. Specifically, the proposed scheme for uplink transmission power control is based on a two‐level Stackelberg game theoretical approach.

4.3.1 Mathematical Formulation

Table 4.1 lists the key notations and terminology used throughout the rest of the section. We denote the DGD set as and the BGD set as . The numbers of DGDs and BGDs are and respectively. Note that if , then all gateway DAPs support two‐way communication.

Sets
	set of DGDs
	set of BGDs
	set of transmitting power of ,
	set of transmitting power of ,
Variables
	incoming data rate to
	incoming data rate to
	total generated uplink data rate
	total transmitted uplink data rate
	transmitting power of
	transmitting power of
	output SINR for
	output SINR for

Let be the incoming data rate to DGD , and be the incoming data rate to BGD . The total data rate required by the smart meters is calculated as follows:

(4.2)

Unlike the packet loss rate that only considers the data rate of the transmitters, the data rate reliability is a metric to measure whether or not the NAN successfully uploads sufficient smart meter data to the concentrator. For example, indicates that all the transmitters together should have an effective uplink transmission data rate over of the incoming data rate. However, it does not require the total packet loss rate to be less than when each transmitter has a relatively high transmission rate compared to its incoming data rate.

4.3.2 Energy Efficiency Utility Function

It is known that additional energy consumption changes the fundamental tradeoff between energy efficiency and the data rate [81]. Intuitively, achieving a higher signal‐to‐interference‐plus‐noise ratio (SINR) level requires the user terminal to transmit at higher power, which would result in lower energy efficiency. This tradeoff is well known and can be quantified [72–74]. The pure energy efficiency utility function of a user is defined as the ratio of its throughput to its transmit power (bits/joule) as follows:

(4.3)

where is the receiving data rate calculated as follows:

(4.4)

Function is the efficiency function, which is assumed to be increasing, continuous, and S‐shaped (sigmodial; more specifically, there is a point above which the function is concave, and below which the function is convex [76]) with and . This efficiency function is commonly adopted by many researchers [73–75] as follows:

(4.5)

where is the block length and is the output SINR for ‐th gateway. Assuming random spreading sequences, the SINR is calculated as follows:

(4.6)

where is the processing gain, and and are the transmit power and path gain of the ‐th gateway respectively. Eq. (4.6) can be rewritten to calculate the transmit power as follows:

(4.7)

According to Eq. (4.4), the total transmitted uplink data rate is calculated as follows:

(4.8)

With a given SINR, the energy utility function is quasiconcave. An illustration of the characteristics of a quasiconcave function is given in Figure 4.2, as indicated by the solid line. As it shows, there is a maximizer that can achieve the optimal pure energy efficiency Eq. (4.3) with a given SINR . However, the maximizer may not guarantee a required data transmission rate. According to the efficiency function Eq. (4.5), the higher the transmit power, the higher the data rate. Therefore, when the total generated uplink data rate is high, transmitting with is more likely to miss the data rate reliability requirement.

In order to meet the data rate reliability requirement in all scenarios while maintaining high energy efficiency, we introduce a weighted penalty function, denoted as . The penalty serves as a measure of the reliability gap caused by insufficient transmitting power. For practical purposes, we require to satisfy three properties with respect to with given noise:

1. It is a decreasing function.
2. It is convex.
3. It approaches 0 at infinity.

With , we then define the utility of the ‐th gateway DAP as follows:

(4.9)

For example, when a gateway DAP (e.g. ) transmits at a higher power than the maximizer of the pure energy efficiency, the penalty is less, because transmitting at higher power level would be more likely to meet the reliability requirement. The convexity of ensures that the weight of the penalty as the power consumption increases. In the defined energy efficiency, the marginal increase in data rate decreases as transmit power increases higher than the maximizer; thus the penalty should follow this characteristic. The penalty approaches zero when consuming infinite power.

The impacts of the penalty function are also illustrated in Figure 4.2. The maximizer calculated based on the utility function Eq. (4.9) is used as a benchmark point. The penalty in the rewritten utility Eq. (4.9) shifts the maximizer value to the right of the benchmark value. With a higher weighted penalty, the maximizer would be shifted more to the right. As shown in Figure 4.2, the maximizer of the dotted line is shifted higher than the maximizer of the dashed line, which means the penalty is weighted more in the dotted line. The exact penalty function will be presented with the transmission power control scheme in the next section.

4.4.1 DGD Level Game

The DGD level game is defined as . In this level of game, are the players, that is DGDs; is the action set (i.e. power consumption of each DGD), and is the set of individual utilities (i.e. power efficiency with defined penalty).

Let be the maximum transmitting power for all the players. For an arbitrary player , given its action , the corresponding utility is calculated as follows:

(4.10)

The penalty function is defined as follows:

(4.11)

where is the weight factor, which depends on three factors: 1) 's incoming data rate , 2) the total incoming data rate , and 3) the maximum incoming data rate . As discussed earlier, a more weighted penalty leads to a higher power consumption for maximum utility. Therefore, the weight factor increases based on the increase of two conditions.

• The ratio of the incoming data rate of DAP to the total uplink date rate, that is, .
• The ratio of the total uplink data rate to the maximum uplink data rate, that is, .

The two conditions can be combined together as follows:

Without loss of generality, the weight factor for DGD is defined as follows:

(4.12)

where is a scaling factor so that the final results can be normalized into a reasonable range. Note that for a given , is a constant.

In the previous section, the penalty function was defined to have three properties. We then check if the penalty function can meet all properties. Taking the first derivative with respect to as follows:

(4.13)

Therefore, is a monotonically decreasing function with respect to . We then take the second derivative with respect to as follows:

(4.14)

Therefore, is convex. And finally, the penalty function has the following property:

(4.15)

In summary, in Eq. (4.11) satisfies all three properties required for a penalty function.

In the DGD level game, for an arbitrary and selfish player , the goal is to find the optimal power to achieve its maximum power efficiency when counting in the penalty. That can be achieved by solving the following optimization problem:

(4.16)

with given for all and power consumption of all BGDs.

4.4.2 BGD Level Game

The BGD level game is defined as . In this level of game, are the players (i.e. BGDs), is the action set (i.e. power consumption of each BGD), and is the set of individual utilities (i.e. power efficiency with defined penalty). Similarly, we assume that the maximum transmitting power is identical to all the players. Given the DGDs power profile generated in the DGD level game, an arbitrary player calculates its utility in the BGD level game as follows:

(4.17)

where the penalty function is defined as follows:

(4.18)

where is the weight factor of this penalty function calculated as

(4.19)

Similarly, this selfish player aims to maximize its utility with respect to the penalty function by solving the following optimization problem:

(4.20)

with given for all and power consumption of all DGDs.

4.5.1 Estimation of and

Before discussing the hierarchical uplink transmission power control scheme, we need to decide the actual number of players in each level of the Stackelberg game. Without loss of generality, when estimating , we assume that the DGDs are identical to each other. With this assumption, the power consumption levels of DGDs are the same, that is, , . The uplink data rate is also identical at all DGDs, i.e. . Note that these simplifications apply only to the estimations of and in this subsection. To clarify the illustration, we temporarily rewrite Eq. (3.28) as follows:

(4.21)

With a fixed noise and taking the first derivative with respect to as follows:

(4.22)

Therefore is monotonically increasing with respect to . This will help to estimate and . Recall that DGDs will transmit portion of the total uplink data rate, as follows:

(4.23)

where is the transmission rate of and is the maximum SINRs achieved by at its maximum power . Note that the maximum SINRs are also identical at each DGD:

The maximum SINR is achieved when all DGDs transmit at the maximum power. Applying to Eq. (4.23) as follows:

(4.24)

Eq. (4.24) may have two solutions, for example and . Assuming , the number of DGD is estimated as follows:

(4.25)

We then estimate , the number of BGDs. Similarly, all the BGDs are assumed to be identical to each other. With the interference from DGDs added, the estimated SINR for is calculated as follows:

(4.26)

Without loss of generality, we further assume that the DGDs are also identical to BGDs in the estimation, where and , . Thus Eq. (4.26) can be represented as follows:

(4.27)

Obviously, is monotonically increasing with respect to . Note that DGDs and BGDs together support all uplink transmissions:

(4.28)

If the transmission rates of BGDs and DGDs are identical to each other, where , and , the maximum SINRs are also identical, that is, , and . Eq. (4.28) can be represented as follows:

(4.29)

Eq. (4.29) may have two solutions, and . Assuming , the number of BGD is estimated as follows:

(4.30)

For to make the illustration clearer, the estimation of DGDs and BGDs will be represented as and in the rest of the chapter.

4.5.2 Analysis of the Proposed Stackelberg Game

We apply the backward induction method to approach the proposed Stackelberg game. In a two‐level Stackelberg game, since the follower's strategies will affect the leader's strategies, we first study the BGD level game . The goal is to find the Nash equilibrium (NE) for this game with a given action set of DGDs, that is, .

With a fixed noise, SINR is a function of transmit power . Therefore, the efficiency function can be represented as a function of . Given a weight factor, the penalty function is also a function of . Moreover, all BGDs are assumed to have the same physical layer transmission rate, since they are equipped with the same hardware, that is, . Without loss of generality, the transmission rate is normalized to 1. With a fixed noise and weight factor , the utility function Eq. (4.17) can be represented as follows:

(4.31)

where and .

Let and ; the utility function Eq. (4.31) can be represented as follows:

(4.37)

The utility function for BGD level game is accordingly.

In summary, we know that the BGB level noncooperative game has a unique NE with a given , and it can be calculated by Eq. (4.38). Let be the set of NEs for BGDs when DGDs play strategy . In the leader's strategy, DGDs then play their noncooperative game and look for the NE based on . Mathematically, the DGD level game has the same structure as the BGD level game . Hence, with any results from , an NE also uniquely exists in . Specifically, given a calculated set from the follower's game and its weight factor , the 's best response to a given vector in the leader's game is unique and calculated as

(4.39)

where is the solution to . We then formally define the Stackelberg equilibrium for the two‐level Stackelberg game.

In order to find the SE, Eq. (4.38) and Eq. (4.39) are computed reciprocally to find the NE for both games in each level as follows:

(4.40)

4.5.3 Algorithms to Approach NE and SE

Based on the analysis above, we propose an algorithm to approach the NE in the two‐level games. The BGD level game is presented in Algorightm 4.1. Switch and , and the algorithm approaches the NE for the DGD level game .

To approach SE, we need to run Algorithm 4.1 reciprocally for both and . This process is summarized in Algorithm 4.2.

4.6.1 Simulation Settings

In the simulation set up, the NAN is in an area of 1 km  1 km, and it consists of 2000 smart meters, each having an adaptive sampling period 1‐. Each sample generates 12 KByte, or 48–Kbps real time data rate and is finally sent to a DGD or BGD by following the HWMP routing protocol. Therefore, when all smart meters have the same sampling period at 1 sec. Noise  Watt (−120 dBm). The BGDs and DGDs have the same transmission rate at =17 Mbps according to the IEEE 802.16 protocol. A concentrator is located 1 kilometer away from the border of the NAN. The DGDs and BGDs are supposed to be deployed as close to the concentrator as possible in order to save transmitting energy. Without loss of generality, let the DGDs be 1 kilometer away from the concentrator, and BGDs be roughly 100 m further. Considering , , the path gain of DGDs and BGDs are and respectively. To calculate transmission efficiency, let processing gain and packet length M=100 bits. Let so that the NAN can meet the maximum demand.

4.6.2 Estimate of and

With different values of , the numbers of DGDs and BGDs have different estimates, as shown in Figure 4.4. Note that when or , there is no hierarchical structure, and therefore the Stackelberg game approach is relaxed to a noncooperative game approach.

4.6.3 Data Rate Reliability Evaluation

Without loss of generality, assume , which returns and . We use the same Stackelberg approach to obtain the results targeting pure power efficiency utility (denoted as “no penalty” in simulation results). Figure 4.5 clearly shows that achieving pure power efficiency does not guarantee the reliability requirement.

We then discuss how the proposed utility performs with penalty. We first show the impact on the weight of the penalty function . With a higher weight, the proposed scheme achieves a higher reliability when the smart meters generate more data. When and , it always achieves reliability. However, the question is which to choose for deployment. Before answering this question, we need to show the power efficiency aspect of our proposed scheme. Specifically, we examine the impact of different incoming data rates on the total power usage with two sets of settings. One set is with ; in other words, the Stackelberg game is degraded to a noncooperative game only with . (We keep the path gain settings such that 5 BGDs are closer to the concentrator.) The other set is with , where and , and the results are obtained by the Stackelberg game. Each point is the average value of 300 simulation runs.

As shown in Figure 4.6, the pure power efficiency scheme returns the lowest power usage in both sets of the settings. Our proposed scheme returns the results in between those two schemes. And the total power usage increases when the generated data rate by smart meters increases. In comparison, our proposed scheme results in a significantly lower transmit power consumption than using the maximum transmit power. If we compare the two sets of settings ( and ), we can see that the Stackelberg game approach () returns a significantly lower power usage as compared with the noncooperative game approach (). We also see that a larger returns a higher power usage. To this end, we shall answer the question given before that with these settings, is a better choice than because its power usage is lower. The optimal decision of those parameters remain a future research topic in the smart grid.

4.6.4 Evaluation of the Proposed Algorithms to Achieve NE and SE

In this subsection, we show the performance of the two algorithms proposed to obtain the NE in the BGD level game and DGD level game and to obtain the SE in the two‐level Stackelberg game. The evaluation of the algorithms uses the same set of randomly generated incoming data rates applied to the previous analysis for consistency.

As shown in Figure 4.7, both BGD‐ and DGD‐level noncooperative games can converge after very few iterations using Algorithm 4.1. As shown in Figure 4.8, the SE can converge after very few iterations using Algorithm 4.2.