Stay or Switch: Competitive Online Algorithms for Energy Plan Selection in Energy Markets with Retail Choice

TL;DR

This paper develops and analyzes competitive online algorithms for energy plan selection in retail energy markets, addressing consumer decision complexity and demonstrating near-optimal cost savings through empirical evaluation.

Contribution

It introduces the first competitive online algorithms for energy plan selection with provable optimal ratios, extending to variable cancellation fees and validating with real-world data.

Findings

Deterministic algorithm achieves 14.6% cost saving versus offline optimal

Randomized algorithm improves savings up to 16.2%

Algorithms are proven optimal among their classes

Abstract

Energy markets with retail choice enable customers to switch energy plans among competitive retail suppliers. Despite the promising benefits of more affordable prices and better savings to customers, there appears subsided participation in energy retail markets from residential customers. One major reason is the complex online decision-making process for selecting the best energy plan from a multitude of options that hinders average consumers. In this paper, we shed light on the online energy plan selection problem by providing effective competitive online algorithms. We first formulate the online energy plan selection problem as a metrical task system problem with temporally dependent switching costs. For the case of constant cancellation fee, we present a 3-competitive deterministic online algorithm and a 2-competitive randomized online algorithm for solving the energy plan selection problem. We show that the two competitive ratios are the best possible among deterministic and randomized online algorithms, respectively. We further extend our online algorithms to the case where the cancellation fee is linearly proportional to the residual contract duration. Through empirical evaluations using real-world household and energy plan data, we show that our deterministic online algorithm can produce on average 14.6% cost saving, as compared to 16.2% by the offline optimal algorithm, while our randomized online algorithm can further improve cost saving by up to 0.5%.

Tables

Table 1. Table 1. Key notations.

Notation	Definition
$T$	The total number of time intervals (unit: month)
${\beta }_t$	The cancellation fee at time $t$ ($)
${\sigma }_t$	The joint input at time $t$
$s_t$	The selected plan at time $t$ (variable-rate plan denoted by ‘1’ and fixed-rate plan denoted by ‘0’)
$e_t$	The electricity demand of customer at time $t$ (kWh)
$p_t^0$	The price per unit of electricity usage for fixed-rate plan ($ / kWh)
$p_t^1$	The price per unit of electricity usage for variable-rate plan ($ / kWh)
$B_t$	The base load for fixed-rate plan (kWh)
$H$	The underusage charging rate for fixed-rate plan ($ / kWh)
R	The minimum contract length for fixed-rate plan (month)
L	The total length of contract for fixed-rate plan (month)
$\alpha$	The cancellation fee for the residual time in the contract ($ / month)

Table 2. Table 2. Acronyms for problems and algorithms.

Acronym	Meaning
EPSP	Energy Plan Selection Problem
SP	Simplified version of EPSP
dSP	EPSP with linearly decreasing switching costs
OFAs	The optimal offline algorithm for SP
gCHASEs	Generalized version of deterministic online
	algorithm CHASEs for SP
gCHASE${}_{}{}^r{}_s{}^{}$	Randomized version of gCHASEs for SP

Equations

$$g_t(s_t)\triangleq$$

$$\displaystyle\textbf{(EPSP)\ \ }\min\$$

$$s_{\tau }\le 1_{\{s_t\ge s_{(t-1)}\}},t+1\le \tau \le t+\text{R}-1,$$

$$\widehat{{\beta }_t}=1_{\{{\sum }_{t-\text{L}+1}^t{s}_{\tau }>0\}}\cdot {\beta }_t,$$

$$s_t\in \mathcal{N}\triangleq \{0,1\},t\in [1,T],$$

$$\displaystyle\textbf{(SP)\qquad}\min\ \$$

$$s_t\in \mathcal{N},$$

$$\displaystyle\textup{{(P1)\qquad}}\min\ \$$

$$s_t\in \mathcal{S}\triangleq \{0,1,\cdots ,n\}$$

$$\displaystyle\textup{{(P2)\qquad}}\min\ \$$

$$s_t\in \mathcal{S}\triangleq \{0,1,\cdots ,n\}$$

$$\delta (t)\triangleq g_t(0)-g_t(1).$$

$$\Delta(t)\triangleq\Big{(}\Delta(t-1)+\delta(t)\Big{)}_{-\beta}^{0}\ ,$$

$${\rm Cost}_{\mathcal{A}}(s_{0},\bm{\sigma})\leq c\cdot{\rm Opt}(s_{0},\bm{\sigma})+\gamma(s_{0}),\forall\mbox{\boldmath$\sigma$},$$

$${\mathbb{E}}[{\rm Cost}_{\mathcal{A}}(s_{0},\bm{\sigma})]\leq c\cdot{\rm Opt}(s_{0},\bm{\sigma})+\gamma(s_{0}),\forall\mbox{\boldmath$\sigma$},$$

$$[T_0,T_1-1],\cdots ,[T_n,T_{n+1}-1],\cdots ,[T_{2n},T_{2n+1}-1],$$

$$s_t=$$

$$\text{dSP:}\min \mathrm{Cost}(\mathbf{s})\triangleq$$

$$T_{2i+1}-T_{2i}\le \text{L},\forall i\in [0,n],$$

$$s_t\in \mathcal{N},2n\in [0,T],$$

$$\hat{\Delta}(t)\triangleq\Big{(}\hat{\Delta}(t-1)+\delta(t)-\alpha\Big{)}_{-\beta}^{0}\ ,$$

$$\sum _{t=1}^{T+1}\beta (s_t-s_{t-1}{)}^{+}=\sum _{t=1}^{T+1}\frac{\beta }{2}\left|s_t-s_{t-1}\right|$$

$$\sum _{t=1}^{T+1}(s_t-s_{t-1})=0$$

$$s_t-s_{t-1}$$

$$\left|s_t-s_{t-1}\right|$$

$$0=\sum _{t=1}^{T+1}(s_t-s_{t-1})=$$

$$\displaystyle-\sum^{T+1}_{t=1}(s_{t-1}-s_{t})^{+}\Big{|}_{s_{t}-s_{t-1}<0}$$

$$\sum^{T+1}_{t=1}\left|s_{t}-s_{t-1}\right|\Big{|}_{s_{t}-s_{t-1}\geq 0}=\sum^{T+1}_{t=1}\left|s_{t}-s_{t-1}\right|\Big{|}_{s_{t}-s_{t-1}<0}$$

$$\sum _{t=1}^{T+1}\left|s_t-s_{t-1}\right|=$$

$$=$$

$$[T_0,T_1],\cdots ,[T_i+1,T_{i+1}],\cdots ,[T_{2k}+1,T_{2k+1}],$$

$$z_t=$$

$$\displaystyle{\rm Cost}(\bm{s_{[\tau_{l},\tau_{r}]}})\triangleq\sum_{t=\tau_{l}}^{\tau_{r}}\Big{(}g_{t}(s_{t})+\beta\cdot(s_{t}-s_{t-1})^{+}\Big{)}+\beta\cdot(s_{\tau_{r}+1}-s_{\tau_{r}})^{+}$$

$$\Delta (t)$$

Full text

retail choice, energy markets, energy plans, competitive analysis

††journalyear: 2019††copyright: acmcopyright††conference: Proceedings of the Tenth ACM International Conference on Future Energy Systems; June 25–28, 2019; Phoenix, AZ, USA††booktitle: Proceedings of the Tenth ACM International Conference on Future Energy Systems (e-Energy ’19), June 25–28, 2019, Phoenix, AZ, USA††price: 15.00††doi: 10.1145/3307772.3328287††isbn: 978-1-4503-6671-7/19/06††ccs: Theory of computation Online algorithms

1. Introduction

Retail choice in energy markets aims to provide diverse options to residential, industrial and commercial customers by enabling selectable purchases of electricity and natural gas from multiple competitive retail suppliers (Zhou, 2017). Traditionally, the energy sector is a notoriously monopolized industry with vertically integrated providers spanning energy generation, transmission and distribution. Limited options of energy suppliers and tariff schemes have been available in most of the world. However, the energy sector is being restructured by deregulation, and new legislation has been launched worldwide to promote more competitive energy retail markets, giving customers higher transparency and more options. Liberating the energy retail markets to competitive suppliers not only allows more affordable tariff schemes and better savings to customers, but it also encourages more customer-oriented and flexible services from suppliers. With more bargaining leverage, customers can also influence the energy suppliers to be more socially conscious and sustainable toward a low-carbon society. Furthermore, the emergence of virtual power plants (Ward et al., 2012), where household PVs and batteries are aggregated as an alternate supplier, can become a new form of energy suppliers in competitive retail markets.

In competitive energy retail markets (including electricity and natural gas), there is a separation between utility providers (who are responsible for the management of energy transmission and distribution infrastructure, as well as the maintenance for ensuring its reliability) and energy suppliers (who generate and deliver energy to utility providers). Energy suppliers are supposed to compete in an open marketplace by providing a range of services and tariff schemes. With retail choice, customers can shop around and compare different energy plans from multiple suppliers, and then determine the best energy plans that suit their needs. The switching from one supplier to another can be attained conveniently via an online platform, or through third-party assistance services.

Since the deregulation in the energy sector in several countries, there has been a blossom of energy retail markets. As of 2018, there are 23 countries in the world offering energy retail choice, including the US, the UK, Australia, New Zealand, Denmark, Finland, Germany, Italy, the Netherlands and Norway (Morey and Kirsch, 2016). In the US, there are over 13 states offering electricity retail choice (Zhou, 2017; Littlechild, 2018). In particular, there are over 109 retail electric providers in Texas offering more than 440 energy plans, including 97 of which generated from all renewable energy sources (Energy, 2017). In the UK, there are over 73 electricity and natural gas suppliers (of Gas and (Ofgem), 2019). In Australia, there are over 33 electricity and natural gas retailers and brands (AEMC, 2018).

Despite the promising goals of retail choice in energy markets, there however appears subsided participation particularly from residential customers. Declining residential participation rates and diminishing market shares of competitive retailers have been reported in the recent years (Administration, 2018; Graves et al., 2018). While there are several reasons behind the subpar customer reactions, one identified major reason is the confusion and complication of the available energy plans in energy retail markets (AEMC, 2018). First, there are rather complex and confusing tariff structures by energy retailers. It is not straightforward for average consumers to comprehend the details of consumption charges and different tariff schemes. Second, savings and incentives among energy retailers are not easy to compare. Some discounts are conditional on ambiguous contract terms. Without discerning the expected benefits of switching their energy plans, most customers are reluctant to participate in energy retail markets. Third, there lack proper evaluation tools for customers to keep track of their energy usage and expenditure. Last, the increasing market complexity with a growing number of retailers and agents obscures the benefits of retail choice. For example, using real-world official datasets, we found 165 energy plans available in Buffalo NY (zip code: 14201), and 434 electricity plans available in Sydney (postal code: 2000). See the screenshots in Fig 1. A large number of available energy plans cause considerable confusion and complexity to average customers.

To bolster customers’ participation, several government authorities and regulators have launched websites and programs to educate customers the benefits of retail choice in energy markets (of Texas, 2018; of Public Service New York State, 2018; Regulator, 2018; of Gas and (Ofgem), 2018). Recently, a number of start-up companies emerged to capitalize the opportunities of retail choice in energy markets by providing assistance services and online tools to automatically determine the best energy plans for customers as well as offering personalized selection advice. These assistance services and online tools are integral to the success of retail choice in energy markets by automating the confusing and complex decision-making processes of energy plan selections. In the future, household PVs, batteries, and smart appliance will optimize their usage and performance in conjunction with energy plan selection. Therefore, we anticipate the importance of proper decision-making processes for energy plan selection in energy markets with retail choice.

There are several challenging research questions arisen in the decision-making processes for energy plan selection:

1. (1)

Complex Tariff Structures: There are diverse tariff structures and properties in practical energy plans. For example, the tariff schemes may have different contract periods (e.g., 6 months, one year, or two years). There may be different time-of-use and peak tariffs as well as dynamic pricing depending on renewable energy sources. Also, it is common to have various administrative fees, such as connection, disconnection, setup and cancellation fees. When there are rooftop PVs or home batteries, there will be feed-in tariffs for injecting electricity to the grid. In some countries, different appliances can be charged by different rates (e.g., boilers and heaters are charged differently). 2. (2)

Uncertain Future Information: To decide the best energy plan that may last for a long period, one has to estimate the future usage and fluctuation of energy tariffs. It is difficult to predict the uncertain information accurately to make the best decisions. For example, energy tariffs may depend on global energy markets. Unexpected guests or travel plans will largely effect energy demand of consumers. If there are rooftop PVs, their performance is conditional on the unpredictable weather. The various sources of uncertainty complicate the decision-making process of energy plan selection. 3. (3)

Assurance of Online Decision-Making: The energy plan selection decision-making process is determined over time when new information is gradually revealed (e.g., the present demand and updated tariffs). A sequential decision-making process with a sequence of gradually revealed events is called online decision-making. The average customers are reluctant to switch to a new energy plan, unless there is certain assurance provided to their selected decisions. The online decision-making process should incorporate suitable bounds on the optimality of decisions as a metric of confidence for customers.

In this paper, we shed light on the online energy plan selection problem with practical tariff structures by offering effective algorithms. Our algorithms will enable automatic systems that monitor customers’ energy consumption and newly available energy plans from an energy market with retail choice, and automatically recommend customers the best plans to maximize their savings. We note that a similar idea has been explored recently in practice (Guardian, 2018).

Our results are based on competitive online algorithms. Decision-making processes with incomplete knowledge of future events are common in daily life. For decades, computer scientists and operations researchers have been studying sequential decision-making processes with a sequence of gradually revealed events, and analyzing the best possible strategies regarding the incomplete knowledge of future events. For example, see (Borodin and El-Yaniv, 2005) and the references therein. Such strategies are commonly known as online algorithms. The analysis of online algorithms considering the worst-case impacts of incomplete knowledge of future events is called competitive analysis. Competitive online algorithms can provide assurance to online decision-making processes with uncertain future information.

We summarize our contributions in the following.

1. $\rhd$

We present a case study in Section 2 to evaluate the potential saving of selecting an appropriate energy plan using real-world data of household demands and energy plan prices. 2. $\rhd$

We formulate the online energy plan selection problem as a metrical task system problem with temporally dependent switching cost in Section 3. 3. $\rhd$

For constant cancellation fee, the problem is reduced to a discrete online convex optimization problem. In Section 4, we present an offline optimal algorithm, a 3-competitive deterministic online algorithm, and a 2-competitive randomized online algorithm for solving the energy plan selection problem. These algorithms maintain great consistency while achieving their optimality. 4. $\rhd$

We extend our study to consider cancellation fee to be temporally dependent on the residual contract period by incorporating additional constraints in Section 5. We conjecture that our proposed algorithms should preserve their competitiveness under proper modification. 5. $\rhd$

Through empirical evaluations using real-world data in Section 6, we show that our deterministic online algorithm can produce on average 14.6% cost saving, as compared to 16.2% by the offline optimal algorithm. Our randomized online algorithm can further improve the cost saving.

Due to space limit, all proofs are presented in the technical report (Zhai et al., 2019), and we refer interested readers to it for details.

2. Case Study

To motivate our results, we first present a case study to evaluate the potential saving of selecting an appropriate energy plan. We use real-world data of household demands and retailers’ energy plan prices to estimate the costs and savings of selecting different energy plans.

2.1. Dataset

2.1.1. Household Electricity Consumption

We use the electricity consumption data from Smart∗ project (Barker et al., 2012), which consists of 114 single-family apartments recorded in 2015-2016. A typical family’s consumption profile in two years is plotted in Figure 2 (a). By comparing the consumption pattern in two consecutive years, we observe dramatic fluctuations from 2015 to 2016. Hence, the consumption rate of the previous year may not be a reliable indicator for selecting the next year’s energy plan.

2.1.2. Energy Plan Prices

We consider the historic data of energy plan prices from the real-world official dataset in New York State. From Figure 2 (b)-(c), we observe that variable-rate energy plans have a large price range, whereas fixed-rate energy plans have a stable price range. Note that fixed-rate energy plans are associated with different kinds of fees when customers switch their selections. For example, cancellation fee applies in the US when cancelling a long-term plan, whereas connection/disconnection fee applies in Australia when switching to other retailers. At the same time, fixed-rate energy plans have additional fee when the usage differs greatly from previous year.

2.2. Potential Savings

2.2.1. Setting

We consider the case of a household switching between a fixed-rate energy plan and a variable-rate energy plan in one year. Our evaluation is based on the consumption data and energy plan prices in Figure 2 (a)-(c). Suppose that a household having the same consumption as is shown in 2016, with previous years data as in 2015, and selects Public Power as the retailer, which provides either a variable-rate energy plan or a 12-month fixed-rate energy plan with the rate in 2017. We consider four representative choices as follows:

1. (1)

remaining in a variable-rate plan, 2. (2)

remaining in a fixed-rate plan, 3. (3)

first staying in a fixed-rate plan, and then cancelling with $100 cancellation fee to switch to a variable-rate plan, or 4. (4)

first staying in a variable-rate plan, and then switching to a fixed-rate plan without switching cost.

2.2.2. Observations

The energy costs under different choices are plotted in Figure 2 (d). Since the household consumption varies considerably from 2015 to 2016, remaining in a variable-rate plan (Choice (1)) will cost more than other energy plans. On the other hand, remaining in a fixed-rate plan (Choice (2)) can save 15.68% cost. However, staying in a fixed-rate plan in the beginning and cancelling before the end of contract (Choice (3)) only saves 11.73% cost due to the cancellation fee. The best choice is Choice (4) - staying in a variable-rate plan at first and then switching to a fixed-rate plan, which saves up to 22.88% cost. The reason is that the consumption was much lower in January and February 2016 than 2015. Therefore, it is not economical to sign up a fixed-rate plan too early. The high consumption in September and onwards makes a fixed-rate plan more economical.

Note that we assume a single retailer without switching to another retailer. It can be shown later that an even bigger saving can be achieved by switching to another retailer.

In this section, we showed the substantial potential savings by selecting a proper energy plan. In practice, there may be many more retailers with a large number of energy plans. Hence, it requires a systematic solution for the energy plan selection problem.

3. Problem Formulation

To solve the energy plan selection problem, we first present a formal model of it. We consider a typical setting where a consumer can select his/her energy plan offered by various energy retailers. We formulate the energy plan selection problem in EPSP, which considers arbitrary demands, time-varying prices, and cancellation fees. Since energy plans differ considerably from one country to another, we consider a typical energy plan model in the US. Some key notations are defined in Table 1 and acronyms are listed in Table 2.

3.1. Model

3.1.1. Uncertain Electricity Demands

We consider arbitrary consumers demand throughout the whole period. Let the electricity demand at time $t$ be $e(t)$. Note that we do not rely on historic data for any prediction, nor any stochastic model of $e(t)$.

3.1.2. Pricing Schemes

We describe the pricing schemes of different energy plans in the US. We first consider two major energy plans to be selected by a customer. Let $s_{t}$ be the selected plan, where 0 represents a fixed-rate plan and 1 represents a variable-rate plan.

• •

Variable-Rate Plan: Switching between different variable-rate plans will not incur any cost. As a result, we can always assume the least-cost variable-rate plan when selecting a variable-rate plan. Hence, there is only a single variable-rate plan to be considered at each time. Denote by $p^{1}_{t}$ the price per electricity unit for the variable-rate plan, and the consumption charge at time $t$ is $p^{1}_{t}\cdot e_{t}$.

• •

Fixed-Rate Plan: Fixed-rate plans also vary with market prices, but do not fluctuate as much as variable-rate plans and are characterized by a stable ranking for a long period of time. Moreover, there is always a relatively high cancellation fee when cancelling a fixed-rate plan. With this understanding, we assume that a consumer will not switch between fixed-rate plans.

Note that a fixed-rate plan will not offer the same electricity price for arbitrary demand, but a tiered-pricing scheme up for a certain level of demand (Gas, 2018). Let $B_{t}$ be a base load for a consumer for each time $t$ in the previous year, and $p^{0}_{t}$ be the price per electricity unit for a fixed-rate plan. A consumer will pay $B_{t}\cdot p^{0}_{t}$ if his/her demand is between $0.9B_{t}$ and $1.1B_{t}$. Otherwise, an underusage fee will be charged at rate $H$ when $e_{t}$ is less than $0.9B_{t}$, or an overusage fee will be charged at the same rate of a variable-rate plan when $e_{t}$ is more than $1.1B_{t}$.

We define the cost function $g_{t}(s_{t})$ based on the input $\sigma_{t}\triangleq(e_{t},p_{t}^{0},p_{t}^{1},B_{t})$ and the selected plan $s_{t}$ at time $t$ as:

[TABLE]

The model can be extended to describe other advanced settings, including the one in Australia (Regulator, 2018). We only focus on the one in (1) in this paper.

3.1.3. Cancellation Fee

When a customer cancels a fixed-rate plan, there will be a cancellation fee of the following types:

1. (1)

No Fee: The customer does not need to pay any cancellation fee but s/he needs to inform the retailer in advance (e.g., 30 days ahead). 2. (2)

Constant Fee: If the residual number of months in the contract is within a certain level, then a fixed amount of cancel fee is required (e.g., $100 if less than 12 months left in the contract, or$200 if between 12 to 24 months left in the contract). 3. (3)

Temporally Linear-dependent Fee: a pre-specified charge times the residual number of months in the contract (e.g., $10 per remaining month in the contract).

Let $\beta_{t}$ be the cancellation fee at time $t$. We note that within each period of a fixed-rate plan, it is either a constant, or a linearly decreasing value proportional to the residual contract period.

3.1.4. Contract Period

The retailers often offer several contracts with various contract periods and different restrictions. Since variable-rate plans have higher per-unit rates than those of fixed-rate plans, it is uncommon that variable-rate plans are limited by a contract period. However, retailers may want to retain consumers in a fixed-rate plan by stipulating a certain fixed period of committing to the contract. Denote by R the minimum contract length for a fixed-rate plan, by which a consumer shall not switch to a variable-rate plan during this period. Let L be the total contract length, so that consumers will not need to pay any cancellation fee when terminate by the end of contract.

3.2. Problem Definition

The total time period $T$ is divided into integer slots $\mathcal{T}\triangleq\{1,\cdots,T\}$, each is assumed to last for one month, corresponding to the available minimum contract period. Our goal is to find a solution $\bm{s}\triangleq(s_{1},s_{2},\cdots,s_{T})$ to the following energy plan selection problem:

[TABLE]

where $(x)^{+}=\max\{x,0\}$ and $\bm{1}_{\{\cdot\}}$ is an indicator function. Function $g_{t}(s_{t})$ is defined in (1). Without loss of generality, the initial state $s_{0}$ is set to be 0. The total cost function in (2a) is consisted of operational cost $g_{t}(s_{t})$ and switching cost when cancelling a fixed-rate plan. The constraints in (2b) capture the minimum contract length for the fixed-rate plan. The constraints in (2c) capture zero cancellation fee when terminating a fixed-rate plan exactly by the end of contract.

Remarks: Our problem formulation bears similarity with the online optimization problems in LCP (Lin et al., 2011), CHASE (Lu et al., 2013) or their extended version (Bansal et al., 2015; Albers and Quedenfeld, 2018a; Hajiesmaili et al., 2016). However, there is a fundamental difference that makes our problem more challenging. In the constraints (2c), we note that the cancellation fee depends on the last L states, which cannot be reduced to a sub-problem as in the prior studies. Thus, our problem is harder and requires non-trivial treatments.

4. Constant Switching Cost

In this section, we consider a simplified version of this problem by restricting our attention to the essential part. First, by a survey of the existing energy plans in (of Public Service New York State, 2018), we observe that it is common for retailers to offer fixed-rate plans without minimum contract period. Further, if there is a contract period, the longer the contract period, the lower price it has. Under such setting, we can drop the constraints in (2b) and (2c) by assuming R is 0 and L approaches infinity with respect to $T$. Moreover, we focus on the setting with constant cancellation fee, which is also common in many fixed-rate plans.

This basic problem can be reformulated as follows:

[TABLE]

Note that this problem involves only two states as the decision variables, and the switching cost $\beta$ depends on the current consecutive time slots. Proposition 1 shows that problem SP (or equivalently, P1) is equivalent to P2, which is a classical online decision problem known as the Metrical Task System problem (Borodin and El-Yaniv, 2005).

Proposition 1**.**

The following two problems are equivalent under the boundary condition of $s_{0}=s_{T+1}=0$, $g_{T+1}(0)=0$:

[TABLE]

[TABLE]

See Appendix A for the full proof.

4.1. Offline Optimal Algorithm

In this section, we provide an offline optimal solution to the problem SP, where input $\bm{\sigma}$ is given in advance. Our offline optimal solution (OFAs) does not incur high space and computational complexity, and hence can be implemented efficiently. The offline optimal solution will motivate our design of proper online algorithm in the next section. The basic idea is based on the theoretical framework in (Lu et al., 2013), from which we adopt similar notations.

First, we note that the cost function $g_{t}(\cdot)$ is non-negative. Hence, we can focus on the cost difference between two states.

Definition 1**.**

Define the one-timeslot cost difference by

[TABLE]

Positive $\delta(t)$ suggests changing to state 1, or otherwise, state 0. Next, we define the cost difference for consequent time slots.

Definition 2**.**

Define the cumulative cost difference by

[TABLE]

where $(x)_{a}^{b}\triangleq\min\{b,\max\{x,a\}\}$. The initial condition is $\Delta(0)=-\beta$.

If $\Delta(t)$ increases to 0, it means that staying at state 0 will cost more than changing to state 1, and hence, it is more desirable to change to state 1 afterwards. Otherwise, it is more desirable to change to state 0 if $\Delta(t)$ decreases to $-\beta$.

Theorem 1**.**

$\text{OFA}_{s}$* (Algorithm 1) is an offline optimal algorithm for problem SP.*

Proof.

(Sketch) At first, we need to show the behavior of $\text{OFA}_{s}$ is the same as the optimal solution in (Lu et al., 2013), which is evident followed by its definition. Meanwhile, we note that $\text{OFA}_{s}$ is also similar to the one in (Lin et al., 2011), in which the property of the solution vector is related by a backward recurrence relation.

See Appendix B for the full proof. ∎

Theorem 2**.**

Both the running time and space requirement of $\text{OFA}_{s}$ for problem SP are $\mathcal{O}(T)$.

Proof.

At each time slot $t$, $\Delta(t)$ can be computed in $\mathcal{O}(1)$ time. Thus, the computation and storage of $\Delta(t)$ for all $t$ in $\mathcal{T}$ require at most $\mathcal{O}(T)$ in time and space. Based on the structure of Algorithm 1, to run this algorithm only needs $\mathcal{O}(T)$ time, and takes additionally $\mathcal{O}(T)$ space. In total, the running time is no larger than $\mathcal{O}(T)$ and requires only $\mathcal{O}(T)$ in space. ∎

Furthermore, we claim that OFAs is still an offline optimal algorithm when taking time and space complexity into consideration. It is evident that no algorithm can run faster than $\mathcal{O}(T)$ because of the length of the input sequence. Meanwhile, to store the solution vector, a minimum space of $\mathcal{O}(T)$ is necessary.

4.2. Competitive Online Algorithm

This section is divided into two parts. First, a deterministic online algorithm is presented to output a deterministic solution over time. Second, a randomized online algorithm is presented to generate a probabilistic ensemble of solutions over time.

4.2.1. Deterministic Online Algorithm

We formally define online algorithm and competitive ratio (Borodin and El-Yaniv, 2005) first. Let the input to the problem be $\bm{\sigma}=(\sigma_{t})_{t=1}^{T}$. Given $\bm{\sigma}$ in advance, the problem can be solved offline optimally. Let ${\rm Opt}(\bm{\sigma})$ be the offline optimal cost for input $\bm{\sigma}$, which is given in advance. A deterministic online algorithm ${\mathcal{A}}$ decides each output $s_{t}$ deterministically only based on $(\sigma_{\tau})_{\tau=1}^{t}$. We say the online algorithm $\mathcal{A}$ is $c$-competitive if

[TABLE]

where $s_{0}$ is the initial state at time [math] defined by the problem formulation, and $\gamma(s_{0})$ is a constant value only depended on $s_{0}$. The smaller $c$ is, the better online algorithm ${\mathcal{A}}$ is. The smallest $c$ satisfying (8) is also named as the competitive ratio of ${\mathcal{A}}$. We will devise a competitive online algorithm for problem SP.

From Algorithm 1, we observe that the optimal solution will certainly stay in state 1 when $\Delta(t)$ is 0, and stay in state 0 when $\Delta(t)$ is $-\beta$. Hence, we replicate such behavior in an online fashion as an online algorithm in $\text{gCHASE}_{s}$ (Algorithm 2), which appears to “chase” $\text{OFA}_{s}$ in a literal sense.

Theorem 3**.**

The competitive ratio of $\text{gCHASE}_{s}$ (Algorithm 2) for problem SP is 3.

Proof.

(Sketch) We classify the time intervals into several critical segments, as in (Lu et al., 2013). Then we compare the costs for each type of these segments between $\text{gCHASE}_{s}$ and $\text{OFA}_{s}$. By combining all segments together and upper bounding their cost ratio, we can obtain the overall competitive ratio as 3.

See Appendix C for the full proof. ∎

Theorem 4**.**

The lower bound on competitive ratio of any deterministic online algorithms for problem SP is 3.

Proof.

(Sketch) We adopt the similar ideas from (Lu et al., 2013; Albers and Quedenfeld, 2018a) and (Bansal et al., 2015). A specific input sequence is constructed progressively depending on the behavior of a given online algorithm $\mathcal{A}$. Then, we bound the cost of $\mathcal{A}$ and the minimum cost for an offline optimal algorithm by 3. ∎

4.2.2. Randomized Online Algorithm

If an online algorithm ${\mathcal{A}}$ is a randomized algorithm (namely, making decisions probabilistically), we define the expected competitive ratio of ${\mathcal{A}}$ by the smallest constant $c$ satisfying

[TABLE]

where ${\mathbb{E}}[\cdot]$ is the expectation over all random decisions. We next devise a competitive randomized online algorithm for problem SP.

Instead of changing the state only at the moments of observing the cumulative cost difference $\Delta(t)$ reaching 0 or $-\beta$, we introduce randomization to change the state at an earlier random moment. The basic idea is that for increasing $\Delta(t)$, the faster it increases, the higher probability of changing the state to 1. On the other hand, for decreasing $\Delta(t)$, the faster it decreases, the higher probability of changing the state to 0. This idea was first proposed in (Albers and Quedenfeld, 2018a) and (Bansal et al., 2015).

We remark that randomized algorithms do not necessarily entail random decisions of a single customer. When we consider an ensemble of a large number of customers using an automatic energy plan recommendation system, each customer can be given a deterministic decision rule drawn from a probabilistic ensemble of decision rules. In the end, the expected cost of a customer can be computed by the expected cost of a randomized algorithm.

Theorem 5**.**

The expected competitive ratio of $\text{gCHASE}_{s}^{r}$ (Algorithm 3) for problem SP is 2.

Proof.

(Sketch) By relaxing the discrete states to a continuous setting, we show that the expected cost of $\text{gCHASE}_{s}^{r}$ is equal to a continuous version of $\text{gCHASE}_{s}^{r}$. Then we show that the continuous version of $\text{gCHASE}_{s}^{r}$ has a competitive ratio of 2 in the continuous setting. Lastly, it can be verified that the optimal cost in the discrete setting is an upper bound to the continuous one.

See Appendix D for the full proof. ∎

Theorem 6**.**

The lower bound on expected competitive ratio of any randomized online algorithm for problem SP is 2.

Proof.

(Sketch) The proof is similar to the ones in (Albers and Quedenfeld, 2018b) and (Antoniadis and Schewior, 2018). First, the expected cost of an arbitrary algorithm $\mathcal{A}$ is not smaller than a converted deterministic algorithm $\mathcal{A^{*}}$ in the continuous setting. Then a 2-competitive deterministic algorithm $\mathcal{B}$ is constructed in the continuous setting to provide the lower bound on the cost of any $\mathcal{A^{*}}$. Lastly, use the discrete setting to provide an upper bound on the cost of an offline optimal algorithm. ∎

5. Linearly Decreasing Cancellation Fee

In this section, we consider the extension to the setting with linearly decreasing cancellation fee by adding back constraint (2c) in EPSP. Note that in our study, $\beta_{t}$ is proportional to the amount of residual time remaining in a fixed-rate plan, we consider that the original problem should be properly reformulated in order to taking it into account. We propose another problem setting in the following, and discuss its inherited connection with problem SP.

First, we divide the total time period $[0,T+1]$ into consecutive time segments, such that states remain unchanged within the same segment:

[TABLE]

where $T_{0}=0$, $T_{2n+1}-1=T+1$, and

[TABLE]

We assume that a consumer is not allowed to maintain a fixed-rate plan longer than its total length L. This is reasonable because after the contract is expired, the consumer will be automatically switched to a variable-rate plan, if s$\backslash$he has not explicitly renewed the contract. Based on the above assumption, we rewrite the problem objective and constraints in below.

[TABLE]

where $\alpha\cdot[\text{L}-(T_{2i+1}-T_{2i})]$ is the cancellation fee each time for cancelling a fixed-rate plan. Constraint (12b) captures the fact that staying in a fixed-rate plan cannot exceed its maximum length L.

Similar to problem SP, we use Definition 1 for $\delta(t)$, but the cumulative cost difference ${\Delta}(t)$ is replaced by $\hat{\Delta}(t)$ as follows.

Definition 3**.**

Define the cumulative cost difference for dSP by

[TABLE]

where $\beta=\alpha\cdot\text{L}$. Initially set $\hat{\Delta}(0)=-\beta$.

Intuitively, we can divide the total cancellation fee $\alpha\cdot\text{L}$ into L time slots. Then in each time slot, the fixed-rate plan suffers $\alpha$ more than the variable-rate plan due to its potential cancellation fee. Because of the similar structure between dSP and SP, we can now use $\text{gCHASE}_{s}$ as a heuristic online algorithm by replacing ${\Delta(t)}$ with $\hat{\Delta}(t)$ to solve dSP. In Section 6, we show by empirical evaluations that such a heuristic online algorithm can indeed produce a good approximation to an offline optimal solution. We further conjecture that the competitive ratio when applying heuristic online algorithm $\text{gCHASE}_{s}$ to dSP is $(3+\frac{1}{\text{L}-1})$. Namely, the longer the contract period L is, the more competitive the heuristic online algorithm is. See Appendix E for a further discussion.

Note that OFAs is not an optimal offline algorithm for dSP, because backward recurrence relation is not in compliance with the linearly decreasing switching cost scenario. Borrowing ideas on constructing offline algorithms from (Lu et al., 2013; Albers and Quedenfeld, 2018a), it is always possible to use dynamic programming to solve this problem or similar ones. Even though dynamic programming may lead to high time and space complexity, it is so far the best one in our mind which can guarantee the optimal results. Hence, we will use dynamic programming to obtain the empirical optimal solutions instead of OFAs in Section 6. Furthermore, we anticipate that the randomized online algorithm gCHASE${}_{s}^{r}$ should also be competitive after changing to the new $\hat{\Delta}(t)$.

6. Empirical Evaluations

In this section, we evaluate our proposed algorithms under different settings using real-world traces. Our objectives are threefold: (i) estimating the potential savings by switching energy plans as compared to staying in a variable-rate plan, (ii) comparing the performances of both deterministic and randomized online algorithms against the offline optimal algorithm, and (iii) analyzing the effect of changing the cancellation fee.

6.1. Dataset and Parameters

6.1.1. Electricity Demand

The demand traces are from Smart∗ project (Barker et al., 2012). We use the electrical dataset which involves 114 single-family apartments for the period 2015-2016. Their monthly average consumption is around 765 kWh, which matches with the monthly average consumption in the US (of Texas, 2018). As shown in Figure 2(a), the daily consumption pattern is rather sporadic with limited regularity.

6.1.2. Energy Plans

We consider the energy plans available in New York State (of Public Service New York State, 2018). Due to a large number of suppliers in this region, we select four representative energy retailers: Agera Energy, LLC, BlueRock Energy, InC, East Coast Power & Gas, LLC, and Public Power, LLC. These four retailers have constant cancellation fee $100 for a 12-month fixed-rate plan. We believe the results can hold similarly in other regions.

6.1.3. Electricity Prices

A seasonal data in 2017 is collected from New York state (Zip code: 10001). We use interpolation to obtain the exact price ($p^{0}_{t}$ and $p^{1}_{t}$) for each month. For the fixed-rate plan, we set $H$ be $0.1p^{0}_{t}$, which is about the difference between rates of a variable-rate plan and a fixed-rate plan.

6.1.4. Contract Period and Cancellation Fee

We set the length of a fixed-rate plan to be 12 months, with $100 constant cancellation fee when quitting before the term ends. As for the linearly decreasing cancellation fee plan, we charge$10 to every rest month remaining in the contract, which is consistent with many settings for energy retailers, for example Kiwi Energy NY, LLC and North America Power & Gas, LLC.

6.1.5. Cost Benchmark

As discussed in Section 1, we view current consumers as stationary plan users, who do not change their original plan throughout an entire year. Since the fixed-rate plan from East Coast Power & Gas is the lowest in 2017, we assume customers sticking to the variable-rate plan offered by this retailer. We evaluate their cost reduction by applying different algorithms.

6.1.6. Comparisons of Algorithms

We compare algorithm OFAs, gCHASEs and gCHASE${}_{s}^{r}$ for the same application scenario. The difference is that OFAs gets all input information before the beginning of time, but gCHASEs and gCHASE${}_{s}^{r}$ are not feed with the current input until time arrives. For each particular setting, gCHASE${}_{s}^{r}$ runs 100 times and is judged by its average outcome.

6.2. Constant Cancellation Fee

6.2.1. Purpose

In this setting, we aim to answer two questions. First, what is the maximum saving a consumer can benefit from by applying the optimal offline algorithm? From Figure 2(c), staying in a variable-rate plan of the same retailer for long is not economic. It is then interesting to evaluate the exact gap between switching and not. Second, how well can our proposed online algorithms behave comparing to the optimal one? Offline algorithm needs full time input before time starts, which is not practical for consumers to use. On the other hand, online algorithms are more practical in the sense that they do not require any predictions or stochastic models of future inputs, but their performances need to be validated.

6.2.2. Observations

From Figure 4, we observe that OFAs can save 16% - 18% for 50% of the families. We regard it as a large benefit, since on average it indicates that a family saves 2 months’ bill in a year. The results justify importance on switching energy plans properly. Further, by comparing it with our proposed deterministic and randomized online algorithms, we verify that OFAs is always the best. We claim that both gCHASEs and gCHASE${}_{s}^{r}$ are competitive as it is clearly shown in the figure that 14% - 16% saving can be guaranteed for 50% of the families. We find out that our two online algorithms have roughly the same behavior. Due to the inherited randomness in gCHASE${}_{s}^{r}$, a particular family may not be guaranteed to reap more savings by introducing the randomized online algorithm to this setting.

6.3. Linearly Decreasing Cancellation Fee

6.3.1. Purpose

For the case when cancellation fee is linearly decreasing with the time of enrollment in a fixed-rate plan, we implement the same online algorithms to cost data as in Section 6.2. However, the optimal offline solution is derived by applying dynamic programming technique, which is denoted by OFA in the legend.

In this experiment, we would like to identify the influence of changing settings, and find out whether the high percentage of savings can still be achieved or performances of algorithms will change greatly. Moreover, our conjecture in Section 5 is evaluated by checking behaviours of online algorithms.

6.3.2. Observations

We notice that the behaviour of OFA in Figure 4 is exactly the same as that of OFAs in Figure 4. Since the total decision space is only 12 months, not introducing cancellation fee should be an optimal scheduling for the consumer. As a result, in both scenarios, the optimal offline algorithm for every family is staying in a variable-rate plan for a period, then switching to the fixed-rate plan if needed. For the deterministic online algorithm gCHASEs, in equation (13) $\alpha$ is relatively small comparing to $\delta(t)$ in reality. Hence, for most families, we observe minor distinctions in their cost savings between Figure 4 and Figure 4, which means their decisions are almost unchanged. For the randomized online algorithm gCHASE${}_{s}^{r}$, an obvious shift can be detected. The cost saving percentage by applying two online algorithms vary largely for most of the families. Additionally, there are more than 60% of the families having less cost by utilizing gCHASE${}_{s}^{r}$ rather than gCHASEs, whereas the rest see the converse pattern. Overall, we conclude that randomization makes behaviour of the online algorithm in a larger range, thus its performance is less stable. However, it also has benefits of surpassing the deterministic one in some cases.

6.4. Effect of Cancellation Fee

6.4.1. Purpose

Previous experiments show that our proposed online algorithms are indeed competitive with large saving. This section is designed to investigate the impact of changing cancellation fee while keeping other parameters unchanged. Whether competitiveness of online algorithms can be preserved in various scenarios remains a question to be answered. Based on intuition, when cancellation fee is relatively high, consumers tend to be inactive without taking extra thoughts for choosing to enroll or quit a fixed-rate plan as they do not tend to suffer huge punishment. Our goal is to find out if consumers still benefit from switching, and the gap between online algorithms and the offline optimal one. Ultimately, we long for guiding consumers on properly choosing their energy plans.

To increase chances of switching between plans, we let cancellation fee $\beta$ vary from $1 to$100, with 1 increment between each two trials. To understand its different influences under both constant and linearly decreasing cancellation fee settings, we conduct two experiments for them separately. Figure [6](#S6.F6) plots the average cost saving percentage of all families under 100 distinct constant cancellation fee settings. As for the linearly decreasing cancellation fee case, we divide \beta$by length L to get$\alpha$ in each trial, and results are shown in Figure 6.

6.4.2. Observations

First of all, two figures generally reveal same trends for these algorithms. With the growth of cancellation fee, saving percentage decreases. The offline algorithm is relatively unchanged, due to the reason that for most of the times, it simply chooses to change from the variable-rate plan to the fixed-rate plan without incurring any additional cost. However, for gCHASEs and gCHASE${}_{s}^{r}$, when the cancellation fee takes large value, the impact of making a wrong decision incurs higher penalty, thus its competitiveness decreases.

Second, the plot indicates that by decreasing cancellation fee, online algorithms turn to be more competitive. Overall, the randomized online algorithm has better behavior than the deterministic one, where the advantage of making decision randomly is revealed. It seems that all algorithms reach ‘saturated’ saving percentage with the growth of cancellation fee. Moreover, by viewing from a larger angle, we notice that although the cancellation fee has high variation, the saving percentage only differs by around 1.5%. These results indicates that the most saving comes from choosing the cheapest plan for all the time, and online algorithms will not get too bad even when cancellation fee is high.

Third, by comparing Figure 6 and Figure 6, we observe that online algorithms achieve better behaviour in the linearly decreasing cancellation fee setting. It can be interpreted by considering that even if a wrong decision is made, for most circumstances, only partial of the maximum cancellation fee $\beta$ is needed to be charged. Further, randomization has more advantage in the linearly decreasing cancellation fee setting, although it is an average result for all households instead of each individual case.

6.5. To Stay or to Switch

Thus far, we have examined the performance of our algorithms in both settings of constant and linearly decreasing cancellation fee, and under various scenarios of cancellation fee. In practice, how and when should a household choose to stay in or switch to an energy plan depends on several critical factors as follows:

• •

Among a large number of variable-rate plans provided by retailers, consumers should always find the one who provides with the lowest rate for the current month.

• •

For all available fixed-rate plans, they should first be ranked from the lowest to the highest by average monthly rate throughout a year. Four criteria are to facilitate consumers’ decisions: (1) low rate rank, (2) low cancellation fee (average $ / month), and (3) linearly decreasing setting.

• •

If possible, try to implement a randomized online algorithm, especially for the linear decreasing cancellation fee scenario.

Following the above tips, a household can save up to 15% energy cost on average in our evaluation.

7. Related Work

The problem formulation studied in this paper is closely related to the subjects of Metrical Task System problem (Borodin and El-Yaniv, 2005) and online convex optimization problem with switching cost (Bansal et al., 2015).

When the state space belongs to a discrete set, such an online decision problem is known as Metrical Task System (MTS) problem, whose competitive ratio is known to be $2n-1$ (Borodin et al., 1992) in a general setting with $n$ states. Recently, (Lu et al., 2013) studied the energy generation scheduling in microgrids, which belongs to a subclass of MTS problems with convex objective function and linear switching cost. An online algorithm is developed called CHASE and can achieve a competitive ratio of 3, which is the lower bound for any deterministic online algorithms for this problem. Remarkably, restricting to convex operational cost functions and linear switching cost can reduce the optimal competitive ratio (termed as price of uncertainty) for MTS problems from $2n-1$ to a constant $3$.

On the other hand, an online convex optimization problem with switching cost is similar to an MTS problem, except that the state space is a continuous set. A 3-competitive algorithm called LCP is developed in (Lin et al., 2011) for it. Recently, (Albers and Quedenfeld, 2018a) shows that the same competitive ratio can be maintained in the discrete setting under proper rounding. Furthermore, it is proved that the competitive ratio of 3 is the lower bound of all deterministic online algorithms in the discrete setting.

Moreover, (Albers and Quedenfeld, 2018a) and (Albers and Quedenfeld, 2018b) show that by applying proper randomization, a 2-competitive online algorithm can be constructed based on (Antoniadis and Schewior, 2018). In this paper, we combine these two papers and construct an uniform version of algorithms, which is more practical and easier to be implemented.

Finally, a key difference from the previous studies that makes our problem harder is the presence of temporally dependent switching cost. To the best of our knowledge, this work appears to be the first study considering this class of problems.

8. conclusion and Future Work

This paper presents effective online decision algorithms to assist the energy plan selection in a competitive energy retail market to save energy cost. We devised offline optimal and competitive online algorithms. For the case of constant switching cost, we characterized the competitive ratios of our deterministic and randomized online algorithms and proved that they are the best possible in their classes. For a more general case with linearly decreasing switching cost, we developed a heuristic online algorithm and conjectured its performance. Empirical evaluations based on real-world data traces corroborated the effectiveness of our algorithms and demonstrated that opportunistic switching among energy plans can indeed bring considerable savings in energy cost to average customers.

As for the future work, we plan to evaluate the effectiveness of our algorithms with more diverse real-world data traces. Also, it is theoretically important and practically useful to develop more general online algorithms with improved competitive ratios for metrical task system problems with arbitrarily temporally dependent switching cost. Such an extension will be integral in solving a more generic class of energy plan selection problems, as well as online decision problems in broad applications.

Acknowledgements.

We thank the anonymous reviewers of e-Energy 2019 for giving insightful comments on the conference version of this article.

Appendix A Proof of Proposition 1

To show that (4) and (5) are equivalent, it suffices to show that

[TABLE]

Since $s_{0}=s_{T+1}=0$, we obtain

[TABLE]

By the definition of operation $(\cdot)^{+}$ and $\left|\cdot\right|$, we obtain

[TABLE]

By substituting (16) into (15), we obtain

[TABLE]

Further, we substitute (17) into (18) and obtain

[TABLE]

Therefore,

[TABLE]

By multiplying $\frac{\beta}{2}$ on both sides, we obtain (14), which completes the proof. ∎

Appendix B Proof of Theorem 1

Let $\bm{z}\triangleq(z_{1},\cdots,z_{T})$ be the solution vector of OFAs. Suppose $\bm{z^{*}}\triangleq(z^{*}_{1},\cdots,z^{*}_{T})$ is an optimal solution for SP and different from $\bm{z}$. We use contradiction to prove by showing that ${\rm Cost}(\bm{z^{*}})\geq{\rm Cost}(\bm{z})$ holds for any input sequences.

Divide $[0,T+1]$ into segments:

[TABLE]

where $T_{0}=0$, $T_{2k+1}=T+1$, and

[TABLE]

with $T_{2i+1}\geq T_{2i}+1$.

Define for any $0\leq\tau_{l}\leq\tau_{r}\leq T+1$,

[TABLE]

where $s_{-1}=s_{T+1}=0$.

Case-1: $z^{*}_{t}=1$, for any $t\in[\tau_{l},\tau_{r}]\subseteq[T_{2i}+1,T_{2i+1}]$

From Algorithm 1, we conclude that

[TABLE]

By (6) and (7), for any $t\in[T_{2i}+1,T_{2i+1}]$,

[TABLE]

Then we obtain

[TABLE]

Case-1a: $\tau_{l}=T_{2i}+1$ and $\tau_{r}=T_{2i+1}$

[TABLE]

Case-1b: $\tau_{l}>T_{2i}+1$ and $\tau_{r}=T_{2i+1}$

[TABLE]

Case-1c: $\tau_{l}=T_{2i}+1$ and $\tau_{r}<T_{2i+1}$

[TABLE]

Case-1d: $\tau_{l}>T_{2i}+1$ and $\tau_{r}<T_{2i+1}$

[TABLE]

Case-2: $z^{*}_{t}=0$, for any $t\in[\tau_{l},\tau_{r}]\subseteq[T_{2i+1}+1,T_{2i+2}]$

From Algorithm 1, we conclude that

[TABLE]

By (6) and (7), for any $t\in[T_{2i+1}+1,T_{2i+2}]$,

[TABLE]

Then we obtain

[TABLE]

Case-2a: $\tau_{l}=T_{2i+1}+1$ and $\tau_{r}=T_{2i+2}$

[TABLE]

Case-2b: $\tau_{l}>T_{2i+1}+1$ and $\tau_{r}=T_{2i+2}$

[TABLE]

Case-2c: $\tau_{l}=T_{2i+1}+1$ and $\tau_{r}<T_{2i+2}$

[TABLE]

Case-2d: $\tau_{l}>T_{2i+1}+1$ and $\tau_{r}<T_{2i+2}$

[TABLE]

In summation, as long as there exists $z_{t}^{*}\neq z_{t}$ for some $t$, ${\rm Cost}(\bm{z^{*}})\geq{\rm Cost}(\bm{z})$ always holds. Therefore $\bm{z_{t}}$ is an optimal offline solution. ∎

Appendix C Proof of Theorem 3

Let $\bm{z}\triangleq(z_{1},\cdots,z_{T})$ be the solution vector of OFAs and $\bm{x}\triangleq(x_{1},\cdots,x_{T})$ be the solution vector of gCHASEs. We prove that ${\rm Cost}(\bm{x})\leq 3{\rm Cost}(\bm{z})$ holds for any input sequences.

If ${\rm Cost}(\bm{z})<\beta$, both $\bm{z}$ and $\bm{x}$ must be zero vectors, then ${\rm Cost}(\bm{x})={\rm Cost}(\bm{z})$. In the following, we only consider the case where ${\rm Cost}(\bm{z})\geq\beta$.

From Algorithm 1 and Algorithm 2, we analysis time segments where $\bm{z}$ and $\bm{x}$ are different.

Define for any $1\leq\tau_{l}\leq\tau_{r}\leq T$,

[TABLE]

Case-1: $z_{t}=0$ and $x_{t}=1$, for any $t\in(\tau_{l},\tau_{r})$, where

[TABLE]

Let the union of such time segments $[\tau_{l},\tau_{r}]$ be $\mathcal{T}^{d}$.

By (6) and (7), for any $t\in(\tau_{l},\tau_{r})$,

[TABLE]

Then we obtain

[TABLE]

Case-2: $z_{t}=1$ and $x_{t}=0$, for any $t\in(\tau_{l},\tau_{r})$, where

[TABLE]

Let the union of such time segments $[\tau_{l},\tau_{r}]$ be $\mathcal{T}^{u}$.

Similarly, we obtain

[TABLE]

Let the rest time in which $\bm{z}$ and $\bm{x}$ are the same be $\mathcal{T}^{e}\triangleq\mathcal{T}\backslash\mathcal{T}^{d}\backslash\mathcal{T}^{u}$.

By summing over all time segments, we obtain

[TABLE]

∎

Appendix D Proof of Theorem 5

We extend problem SP to the continuous problem cSP defined as follows:

[TABLE]

where

[TABLE]

And we define an online algorithm for problem cSP as shown in Algorithm 4.

Lemma 1**.**

The expected cost of algorithm $gCHASE_{s}^{r}$ is equal to the cost of algorithm $cCHASE$ given the same input.

Proof.

For algorithm $gCHASE_{s}^{r}$, let $p_{t}\triangleq\text{Pr}[s_{t}=1]$. Therefore, $\text{Pr}[s_{t}=0]=1-p_{t}$.

Step-1: show that $p_{t}=x_{t}$. We prove by induction.

For $t=0$, $p_{t}=x_{t}=0$. If $p_{t-1}=x_{t-1}$ holds,

Case-1: $\Delta(t-1)\leq\Delta(t)$

[TABLE]

Case-2: $\Delta(t-1)>\Delta(t)$

[TABLE]

Step-2: show that $\mathbb{E}[g_{t}(\sigma_{t},s_{t})]=g_{t}(\sigma_{t},x_{t})$.

[TABLE]

Step-3: show that $\mathbb{E}[(s_{t}-s_{t-1})^{+}]=(x_{t}-x_{t-1})^{+}$.

[TABLE]

By summing the above over $t\ \in\mathcal{T}$, we obtain ${\rm Cost}_{gCHASE_{s}^{r}}(\bm{\sigma})={\rm Cost}_{cCHASE_{s}^{r}}(\bm{\sigma})$. ∎

Lemma 2**.**

Algorithm $cCHASE$ is 2-competitive for problem cSP.

Proof.

We denote cOPT as the optimal offline algorithm for problem cSP and the state it chooses is $z_{t}$ at time $t$. We make the following simplifications and assumptions: define

[TABLE]

and category them to two kinds of functions, namely

[TABLE]

and

[TABLE]

for any $t\in\mathcal{T}$.

Further, by combining definitions of $\Delta(t)$, $f_{t}(x)$ and $x_{t}$, we observe that

[TABLE]

Therefore,

[TABLE]

We use the potential function argument which needs to show the following

[TABLE]

where

[TABLE]

Step-1: show that (75b) holds.

[TABLE]

Step-2: show that (75a) holds.

We use the idea of interleaving moves (Borodin and El-Yaniv, 2005). Suppose at each time $t$, only cOPT or cCHASE changes its state. If both of them change their states, the input function can be decomposed into two stages, and feed them one after another so that two algorithms change states sequentially.

Case-1: $x_{t}=x_{t-1}$, $z_{t}\neq z_{t-1}$.

Case-1a: $f_{t}(x_{t})=f^{<}_{t}(x_{t})$ . By (73) and (74a), $x_{t}=0$.

[TABLE]

Case-1b: $f_{t}(x_{t})=f^{>}_{t}(x_{t})$. By (73) and (74b), $x_{t}=1$.

[TABLE]

Case-2: $z_{t}=z_{t-1}$, $x_{t}\neq x_{t-1}$.

Case-2a: $f_{t}(x_{t})=f^{<}_{t}(x_{t})$.

Case-2aI: $x_{t}\neq 0$. By (73), $x_{t}=x_{t-1}-\frac{f_{t}(1)}{\beta}$.

[TABLE]

Case-2aII: $x_{t}=0$. By (73), $x_{t-1}-\frac{f_{t}(1)}{\beta}\leq 0$.

[TABLE]

Case-2b: $f_{t}(x_{t})=f^{>}_{t}(x_{t})$.

Case-2bI: $x_{t}\neq 1$. By (73), $x_{t}=x_{t-1}+\frac{f_{t}(0)}{\beta}$.

[TABLE]

Case-2bII: $x_{t}=1$. By (73), $x_{t-1}+\frac{f_{t}(0)}{\beta}\geq 1$.

[TABLE]

To show the competitive ratio of cCHASE is not smaller than ($2-\epsilon$), for any $\epsilon>0$, we construct a special input sequence:

[TABLE]

where $\delta$ ia postive real constant smaller than 1 and $\delta\rightarrow 0$.

The state vector of cCHASE is:

[TABLE]

The cost of cCHASE is:

[TABLE]

We claim that the state vector of cOPT is $z_{t}=0$ for all $t\in\mathcal{T}$, and its cost is $\delta\beta$. In this case, the cost ratio between two algorithms is:

[TABLE]

∎

Last but not least, we need to verify that ${\rm Cost}_{cOPT}\leq{\rm Cost}_{OFA_{s}}$ always holds. As in the decision space in continuous version include the one in discrete space, therefore ${\rm Cost}_{OFA_{s}}$ must be a lower bound to the continuous setting.

In conclusion, we show that

[TABLE]

which completes the proof. ∎

Appendix E Discussion on Linearly Decreasing Cancellation Fee

We first show that problem SP can be reformulated in a similar way as dSP, then claim two problems only differs from a constant $\alpha$.

Define

[TABLE]

By using time segments defined by (10), we obtain

[TABLE]

Similarly, for problem dSP, we define

[TABLE]

[TABLE]

By regarding $\delta(\tau)-\alpha$ as a new input, we claim that SP differs from dSP by only a constant. Therefore, it is possible for the previous algorithms to apply in this case.