Background

The increasing penetration of RESs has been evidently increasing the risk of reliable electric-ity grid operation. In recent years, the increasing pressure on the power grid has also been caused by the rising peak load demand, which is partly due to the rapid growth in the use of air conditioners and other appliances, as well as the introduction of some new types of demand to the grid, e.g. plug-in electric vehicles (EVs). Moreover, the increasing amount of renewable generations is not always capable of providing support to reduce the peak load.

This problem was observed back in 2013, when California’s grid operator CAISO released a report on how the growth of solar power was going to change the state’s energy balance over the coming years. In this report, the famous ’duck curve’, shown in figure 5.1, drew enormous attention. This graph illustrates a deep drop in the net load around noon, which is driven by a large amount of PV power injected into the grid, followed by a sharp ramp-up from the late afternoon until the early night time, when PV power fades as residents start to come home from work and to consume electricity at the same time. It should also be noted that peak demand occurs only a limited number of times per year. Thus, many new assets are in fact idle and operated at a mere fraction of their capacity. For example, in the US, 10%–20% of electricity costs are determined by peak demand summing up to only 100 hours

annually. In the Australian National Electricity Market (NEM), around 20%–30% of the $60 billion of electricity network capacity is operated for no more than 90 hours per year.

Fig. 5.1 The duck chart, showing a deep midday drop in net load and a steep ramp-up starting in the late afternoon and extending into evening peak load; the observation of oversupply risks is attributable to variable generation resources[7]

To tackle aforementioned issue, DSM has been developed and employed. DSM includes manifold programs depending on the different electricity markets and policy. Time-of-use pricing (TOU) [94] is a popular scheme. In gneral, under this scheme, electricity is priced with different rates for different time periods, normally 2 to 5 periods, such as peak, shoulder and off-peak hours. The prices are predetermined and typically adapted on a monthly or seasonal basis, so as to capture consumers’ expected demand. However, the main drawback of the TOU scheme is that the prices are set well in advance, and cannot be adjusted to capture an actual situation if the demand changes widely from the expected values. It is known a large number of users tend to favour consumption when the price is cheap, consequently resulting in a new peak. This phenomenon is called the ’rebound effect’.

An alternative to TOU is the RTP. In this scheme, prices are not predetermined and are adapted to each time slot, typically an hour. The price can be increased when the demand rises, and lowered when the demand drops. In other words, if an increasing number of users begin tend to consume electricity (the rebounding effect is about to start), the price

will rise accordingly and thus mitigate the ’rebounding effect’. This scheme can reflect the current conditions and effectively capture the actual demand. Therefore, the RTP scheme can overcome the disadvantage of TOU. Mohsenian-Rad et al. [92, 95, 96] proposed innovative models for RTP. Their models have received much attention and citations in the last several years. In general, these models aim to maximise the aggregated welfare for users while minimising the cost for the energy provider through an optimal RTP strategy. Furthermore, based on Mohsenian-Rad et al’ models, some extended models have been proposed for different applications. For sintance, Chai et al. extended this work to a two-level game model given for multiple utility companies [97]. The authors argue that the equilibrium points can be achieved in games formulated between users and utility companies through distributed algorithms. A load scheduling strategy is proposed in [98], where the RESs and storages are installed. The problem is again solved in a distributed fashion. Study [99] extends the work by considering load uncertainty in the optimisation constraints. Similar schemes can also be found in [100, 101].

In general, these models [92, 95–101] are concave maximisation problem and can be solved by using the IPM in a centralised fashion to obtain global optimal solutions if all parameters are known to the energy provider. However, when these models are applied in practice such as the optimal RTP problems, it is difficult to collect the parameters of all households due to privacy concerns. Moreover, when the user number is large, centralised techniques may not be feasible. Instead, a distributed sub-gradient algorithm is widely applied [92, 95–101]. This algorithm simulates the dynamic behaviours between users and the energy provider, and finally converges to an equilibrium point. Furthermore, the smart grid environment with two-way communication networks provides such a platform for the bi-directional information exchange between customers and the energy provider. In general, a one-day operation can be divided into several time slots. For example, a one-hour-based time slot is often adopted. The price for each time slot is computed in real-time and then has to be announced at the beginning of each time slot. However, there are still some problems that require further discussion. For example, it is unknow that whether the equilibrium point is unique or the same as the solution from the centralised algorithm (IPM). On the other hand, during the optimising process, although the distributed sub-gradient algorithm [102] is widely used in many RTP models to achieve an optimal solution, this algorithm in fact suffers from a slow convergence and numeric sensitivity to the step size, or even from non-convergence.

More specifically, the step size parameter in the distributed algorithm has a strong impact on the convergence rate. In practice, it is difficult to adapt the step size efficiently because oscillation occurs when the step size is big, while convergence becomes too slow when the step size is small.