We model end-users’ electricity consumption in presence of tariff pricing. That means that end-users are not directly subject to wholesale prices. Instead, *suppliers* offer tariffs to them. As opposed to flat tariffs, time-dependent-pricing (TDP) tariffs show varying prices for different times, e.g., hours during one day. Suppliers are passing on the volatility of the wholesale market to the downstream retail market to set the right incentives for customer behavior. A novelty to the approach is that pricing horizons do not have a fixed length, e.g., 24 hours (day-ahead pricing). Instead, we explore different pricing horizons’ effect on resulting prices and suppliers’ profits. Table 1 gives an overview of notation.

*H* time-slots of equal length constitute a pricing horizon. The supplier maintains a rolling pricing horizon of length *H*, i.e., after every time-slot, she announces a new price. The supplier optimizes the risk-adjusted profit for a specific period of consideration, e.g., one day. We denote it as *T*. Other periods of consideration are equally possible, suppliers could optimize profits for a month or a year.

End-users in the model have a known time-dependent baseline demand *x*_{t,base} under flat pricing *p*_{flat}. If end-users have advanced metering infrastructure (AMI), the supplier can estimate the baseline demand from historical data, otherwise, she may use standard load profiles. Users might be able and willing to shift some demand from a time-slot to another. This shifting does not necessarily require actual interventions by the users. Much rather software agents such as home energy management systems (HEMSs) would act on their behalf. In the past, residential electricity consumption has mostly been assumed to be inelastic as the inconvenience of behavioral change outweighs potential savings for most conceivable pricing regimes. However, with the growing availability of technological flexibility, e.g., battery storage or electric vehicles, combined with the increasing availability of HEMSs, it is conceivable that demand elasticity will grow. Given these technologies, the potential to shift demand can be substantial. Note, however, that we are not assuming that consumers at any point would shift all of their demand. Partial inflexibility of demand is within the scope of our model.

### Baseline demand

*x*_{t,base} denotes the baseline demand of a population of end-users for electricity at any given time *t*=1,2,…,*T*. The baseline is the demand of the users at time *t* under flat pricing *p*_{flat}. While *x*_{t,base} would be a random variable with substantial variance for single users, it can be treated as a deterministic variable for a sufficiently large population of customers. As most tariffs for residential or small-scale industrial and commercial customers in the electricity sector still follow a flat pricing scheme, it can be precisely estimated based on historical data, e.g., through the use of standard load profiles for residential customers (Meier et al. 1999) or using statistical learning methods. For the remainder of the work, we will assume that a sufficiently large population of customers has subscribed to the proposed TDP tariff, and treat *x*_{t,base} deterministically. This is a reasonable assumption as suppliers in tariff markets typically serve a vast number customers.

### Demand shifting

Users can shift demand through technological flexibility. This flexibility may be due to actual storage technology such as batteries, as well as *abstract* storage technologies such as electric vehicles or thermal storage. Previous work in the field of smart grids lays the foundation for the model of demand shifting (Joe-Wong et al. 2012). The model can easily be adapted to incorporate a larger group of heterogeneous users by aggregating the demand of individual users or user groups.

The *waiting function* *w*(*d*,*s*) denotes the share of consumption the user population shifts by an amount of time *s*, given a relative amount of money saved *d*. Even with 100% of payments saved, it does not necessarily have to equal 1, as some share of the baseline demand might be inflexible. Thus, \(w\left (\frac {p_{t} - p_{s}}{p_{{flat}}}, \left |s - t\right |\right)\) denotes the shifted share of demand from time *t* to time *s*, where |*s*−*t*| is the amount of time between periods *s* and *t*, and \(\frac {p_{t} - p_{s}}{p_{{flat}}}\) is the amount of money saved, relative to the known flat pricing scheme from before (*p*_{flat} serves as a baseline for shifting decisions). Note that this shifting can be bidirectional; users can shift their normal electricity usage to a later time, or they can use more electricity at the current time and less at a later time. Again, the model can be extended to a heterogeneous group of users by defining different waiting functions, or different parameters for individual users or user groups. Users are less likely to shift their usage from a time-slot to another as more time elapses between the two. We can mathematically capture this behavior by imposing \(\frac {\partial w(d,s)}{\partial s}<0\); we also impose \(\frac {\partial w(d,s)}{\partial d}>0\), since users would shift more usage if they can save more money by doing so. To incorporate these requirements, we choose

$$ w(d,s) = \frac{\max(d,0)}{C_{w}(\beta)(s + 1)^{\beta}}, $$

(1)

where *β*>0 parameterizes users’ *willingness to shift*. A large *β*, for instance, would indicate that the users’ probability of shifting decays rapidly as *s* increases, indicating impatience. Here *C*_{w}(*β*) is a parameter-specific normalization constant. Note that the waiting functions may also be time-dependent without changing the basic structure of the model. If the end-user price has significant fixed components (such as grid fees), the relative savings \(\frac {p_{t} - p_{s}}{p_{{flat}}}\) will be substantially smaller than under more flexible pricing schemes. As the resulting lower amount of shifting can also be modeled with *C*_{w}(*β*). For brevity, we are assuming a fully flexible end-user price.

### Pricing horizon

The supplier maintains a window of *H* future prices by posting one new price after each time-slot; we denote this a *continuous pricing horizon*. Figure 1 shows an explanatory case with *H*=8. The end-user makes a demand decision at *t*=11. Cells colored light blue are time-slots to which or from which users can shift energy demand, cells colored light red are time-slots to which or from which they could have shifted. Besides the *continuous pricing horizon*, there are other possibilities for variable price announcements, e.g., blocks of fixed length (simultaneous announcement of all prices for a certain time frame). However, we do not model such pricing schemes, as we assume the case of a rolling window to be the most realistic and most applicable.

Since the provider announces prices so that she maintains a continuous price horizon of *H* time-slots, users can always shift their demand to the full extent of *H*. We find the end-user demand at time *t* is

$$\begin{array}{*{20}l} x_{t} = x_{t,base} &- \sum_{s\neq t, s = t - H+1}^{t + H -1} x_{t,base}w\left(\frac{p_{t} - p_{s}}{p_{{flat}}}, \left|s - t\right|\right) \\ &+ \sum_{s\neq t, s = t - H+1}^{t + H-1} x_{s,base}w\left(\frac{p_{s} - p_{t}}{p_{{flat}}}, \left|t - s\right|\right). \end{array} $$

(2)

Note that shifting from or to periods outside the period of consideration is possible, i.e., even when we are maximizing daily profits, we have to account for the fact that shifting can occur across days given sufficiently large pricing horizons. We ignore the potential start and end effects, as we can safely assume that tariff contracts persist substantially longer than both the periods and horizons we explore.

### Objective

The supplier’s objective is to find the prices *p*_{t} and time horizon *H* that optimize the (risk-adjusted) profit in the period of consideration, e.g., one day. For a continuous pricing horizon, the profit *Π*_{c} for the period of consideration is

$$ \Pi_{c} = \sum_{t = 1}^{T} \Pi_{t} = \sum_{t = 1}^{T} \left(p_{t}x_{t}-C_{t}x_{t}\right), $$

(3)

where *C*_{t} is a random variable that represents the per-unit cost at time *t*. It could represent the supplier’s production or her purchases on the wholesale market. Even if *C*_{t} denotes the cost of the supplier’s production, it is a random variable, as generation quantity (for variable renewable energy) and operational disruptions can occur randomly, resulting in the necessity to procure electricity on the wholesale market. Note that *T*=24 in this work, however, other periods of consideration are equally possible. Since we choose a short period of consideration, we do not discount future profits.

Under monopoly assumption, the supplier could charge infinitely high prices as the short-term price elasticity of consumers is zero; users might shift their demand, but do not reduce their total demand in the short run. However, the supplier cannot charge prices that will increase total costs for the end-users, as suppliers in electricity (and other liberalized) markets face competition. We assume that users would only subscribe to TDP if their daily cost would not increase (even without shifting). Under this assumption, subscribing to TDP is a *no-regret* measure for users. Additionally, suppliers might have to offer a horizon-specific discount to incentivize users to subscribe to TDP. As TDP increases inconvenience and transaction costs, this discount is positive and decreasing in *H*.

$$ \sum_{t = 1}^{T}\left(p_{t}x_{t,base}\right) \leq \left(\sum_{t = 1}^{T} \left(p_{{flat}} x_{t,base}\right)\right)\left(1-\frac{\delta}{H}\right). $$

(4)

We assume a *mean-variance* (aka *μ*−*σ*) utility function of the supplier (cf. (Oum et al. 2006)). While such a utility function is not the only option to model risk aversion, it is a standard choice. With a deterministic demand pattern, the distribution of profits only depends on the distribution of the cost. The supplier forecasts cost up to a normally distributed error term, the profits are therefore also distributed normally; subtracting a risk term from the expected profits captures risk aversion. *λ* is the coefficient of risk aversion. The objective of the supplier is then to maximize risk-adjusted profits

$$\begin{array}{*{20}l} \max_{p_{1},...,p_{T},H} z_{c}\left(p_{1},...,p_{T},H\right) &\equiv \mathbf{E}[\Pi_{c}] - \frac{\lambda}{2} \mathbf{Var}[\Pi_{c}]. \end{array} $$

(5)

The user population’s baseline demand is deterministic, **V****a****r**[*x*_{t,base}]=0. Cost is predicted by an unbiased estimator *c*_{t}, i.e., \(\mathbf {Var}[C_{t}]=\mathbf {Var}\left [\hat {c}_{t}+\epsilon _{t}\right ]=\mathbf {Var}[\epsilon _{t}]\), where *ε*_{t} is the randomly distributed error term of the prediction at time *t*. The error terms are independently distributed between time-slots, therefore, **C****o****v**[*Π*_{t},*Π*_{s}]=*x*_{s}*x*_{t}**C****o****v**[*C*_{t},*C*_{s}]=*x*_{s}*x*_{t}**C****o****v**[*ε*_{t},*ε*_{s}]=0 ∀*t*≠*s*. With this, we get

$$\begin{array}{*{20}l} z_{c}(\cdot) &= \mathbf{E}\left[\sum_{t = 1}^{T} \Pi_{t}\right] - \frac{\lambda}{2} \mathbf{Var}\left[\sum_{t = 1}^{T} \Pi_{t}\right] \end{array} $$

(6)

$$\begin{array}{*{20}l} &= \sum_{t = 1}^{T} \left(\mathbf{E}[\Pi_{t}] - \frac{\lambda}{2} x_{t}^{2} \mathbf{Var}[C_{t}]\right) \end{array} $$

(7)

$$\begin{array}{*{20}l} &= \sum_{t = 1}^{T} \left((p_{t}-c_{t})x_{t}- \frac{\lambda}{2} x_{t}^{2} \mathbf{Var}[\epsilon_{t}]\right). \end{array} $$

(8)

**V****a****r**[*ε*_{t}], the variance of the cost forecasting error term at time *t*, equals the mean square error of the prediction. The functional form of **V****a****r**[*ε*_{t}] depends highly on the used forecasting algorithm, as algorithms are performing best for different forecasting horizons (Aggarwal et al. 2009). We model **V****a****r**[*ε*_{t}] based on three assumptions. First, we assume that the supplier’s forecasting error will be smaller for shorter forecasting horizons, as more relevant information for the price formation (e.g., power plant availability, weather) is available shortly before market clearing. Second, we assume the forecasting error converges to a final value for very long forecasting horizons. In particular, we suppose that it does not grow infinitely large. This is reasonable as even without precise short-term information there is a prior on cost distribution. Both these characteristics correspond to forecasting with autoregressive models (cf. (Hamilton 1994, Chapter 4)). Third, we assume that the forecasting error is proportional to the squared expected cost, i.e., there is a constant coefficient of variation. This is the strongest assumption, especially as it implies zero variance for an expected cost of zero. However, such a functional relationship between the error term and the expected value seems particularly reasonable for the electricity sector. As there is a convex supply curve, variance due to both supply and demand variation is bigger for higher price levels. Note, however, that other functional forms of **V****a****r**[*ε*_{t}] can easily be incorporated in our model. Under the given assumptions we can model **V****a****r**[*ε*_{t}] by

$$ \mathbf{Var}[\epsilon_{t}] = \varphi_{1}\left(1-\varphi_{2}^{-H}\right) c_{t}^{2}, \hspace{0.1in} 0 < \varphi_{1}, \hspace{0.1in} 1 < \varphi_{2}, $$

(9)

where *φ*_{1} and *φ*_{2} are forecasting parametrs. A higher value for *φ*_{1} indicates a generally higher error term variance, whereas a higher value for *φ*_{2} indicates faster convergence to the residual error variance of an infinitely long forecasting horizon; it denotes the myopia of the forecast. Strictly speaking, \(\frac {ln(2)}{ln(\varphi _{2})}\) is the half-life of the error variance reduction through forecasting. Note that *φ*_{1} is multiplied with *λ* in (8), rendering a distinction between them redundant. Therefore, *φ*_{1}≡1 for the remainder of this paper. Note that adjustment of *λ* in the simulations accounts for the dropped *φ*_{1}. For brevity, we define *φ*_{2}≡*φ*. It follows

$$ z_{c}(\cdot) = \sum_{t = 1}^{T} \left(\underbrace{\vphantom{\frac{\lambda}{2}}\left(p_{t}-c_{t}\right)x_{t}}_{\text{Gross Profit}} - \underbrace{\frac{\lambda}{2} x_{t}^{2} \left(1-\varphi^{-H}\right) c_{t}^{2}}_{\text{Risk}}\right). $$

(10)

The intuition behind the non-discriminability of *λ* and *φ*_{1} is straightforward: given a better forecasting quality, the supplier’s constant coefficient of risk aversion has to be higher to observe the same optimal decision.

Since we admit that suppliers can both increase and decrease prices compared to flat pricing, suppliers will exhaust their opportunity to increase prices to a degree where they are just competitive. However, we are restricting prices to 0≤*p*_{t}≤2*p*_{flat} ∀*t* to be able to calculate *C*_{w}. Note that this restriction is never binding in our simulations. Under these assumptions, constraint (4) binds so that the maximization of (10) resembles a cost minimization problem.

The previously used concept of risk-adjustment is ill-defined for flat pricing, as there is no specific decision horizon. To compare the results of TDP and flat pricing, we adjust gross profits under flat pricing with a risk term corresponding to an infinitely long forecasting horizon, assuming that the supplier sets the flat pricing scheme well in advance of the period of consideration.

$$\begin{array}{*{20}l} z_{{flat}}(\cdot) = \sum_{t= 1}^{T} \left(\underbrace{\vphantom{\frac{\lambda}{2}}\left(p_{{flat}}-c_{t}\right)x_{t,base}}_{\text{Gross Profit}}-\underbrace{\frac{\lambda}{2} x_{t,base}^{2} c_{t}^{2}}_{\text{Risk}}\right) \end{array} $$

(11)

Note that *z*_{flat} is not a function of the pricing horizons, as is intuitive. For non-zero usage and cost, the risk under flat pricing is strictly higher than under TDP, because the supplier has less freedom to manage its risk. Comparing (10) and (11) one recovers the supplier’s three-part gain from offering a TDP tariff.

$$\begin{array}{*{20}l} \Delta z &= z_{c}(\cdot) - z_{{flat}}(\cdot) \\ &= \underbrace{\sum_{t = 1}^{T} \left(p_{t}x_{t}-p_{{flat}}x_{t,base}\right)}_{\text{\(\leq0\), due to discount and shifting}} + \underbrace{\sum_{t = 1}^{T} \left(c_{t}\left(x_{t,base}-x_{t}\right)\right)}_{\text{\(\geq 0\), procurement savings}} \\ &+ \underbrace{\sum_{t = 1}^{T} \left(\frac{\lambda}{2} c_{t}^{2} \left(x_{t,base}^{2}-x_{t}^{2}\left(1-\varphi^{-H}\right)\right)\right)}_{\text{\(\geq 0\), risk avoided}} \end{array} $$

The magnitude of the terms determines the profitability of TDP for the supplier.