Optimal electricity tariff design with demand-side investments

Abstract

This paper proposes a method for evaluating tariffs based on mathematical programming. In contrast to previous approaches, the technique allows comparisons between portfolios of rates while capturing complexities emerging in modern electricity sectors. Welfare analyses conducted with the method can account for interactions between intermittent renewable generation, distributed energy resources and tariff structures. We explore the theoretical and practical implications of the model that underlies the technique. Our analysis shows that a regulator may induce the welfare maximizing configuration of the demand by properly updating portfolios of tariffs. We exploit the structure of the model to construct a simple algorithm to find globally optimal solutions of the associated nonlinear optimization problem; a computational experiment suggests that the specialized procedure can outperform standard nonlinear programming techniques. To illustrate the practical relevance of the rate analysis method, we compare portfolios of tariffs with data from two electricity systems. Although portfolios with sophisticated rates create value in both, these improvements differ enough to advise different portfolios. This conclusion is beyond the reach of previous techniques to analyze rates, illustrating the importance of using model-based data-driven approaches in the design of rates in modern electricity sectors.

Appendices

Proofs of results in Sect. 3

1.1 Proof of Lemma 1

Proof

Willig [54] shows that when focusing on goods with an associated expenditure relatively small with respect to the customer’s budget, such as the consumption of electricity, consumer surplus and compensating variation are equivalent. Thus, in the context of the present study, we can write

$$\begin{aligned} \varDelta Y = CS(l_2,{\bar{p}}^2) - CS(l_1,{\bar{p}}^1). \end{aligned}$$

(18)

By definition, the change in producer surplus is

$$\begin{aligned} \varDelta \varPi = \varPi (l_2,{\bar{p}}^2) - \varPi (l_1,{\bar{p}}^1). \end{aligned}$$

(19)

$\varPi (l,{\bar{p}}) = 0$ for the optimal solution of (6). Combing this condition with (1) and (5), and letting ${\mathcal {P}}_h = \left\{ {\bar{p}}^h\right\}$ for $h \in \left\{ 1,2\right\}$, we conclude that $v_h = E\left[ {\bar{S}}^I({\bar{p}}^h) \right] - C({\bar{D}}^I({\bar{p}}^h)) - \varPi _0$. Using this fact and adding (18) and (19), the expression (7) follows. $\square$

1.2 Proof of Proposition 1

Proof

First we show that C is a convex function. Consider ${\bar{d}}^1$, ${\bar{d}}^2$ non-negative, $\psi \in [0,1]$ and define $\psi ^1 :=\psi$, $\psi ^2 :=1 - \psi$. Let $w^j$ be the optimal solution of $M({\bar{d}}^j)$ for $j\in \left\{ 1,2\right\}$. Note that ${\hat{w}}:=\psi ^1 w^1 + \psi ^2 w^2$ is feasible for $M(\psi ^1{\bar{d}}^1 + \psi ^2{\bar{d}}^2)$. Indeed, since $Q_1$ is convex and $Q_2$ is a linear transformation we have that

$$\begin{aligned} Q_1(\psi ^1{\bar{d}}^1 + \psi ^2{\bar{d}}^2) \le \psi ^1 Q_1({\bar{d}}^1 ) + \psi ^2 Q_1({\bar{d}}^2) \le Q_2({\hat{w}}). \end{aligned}$$

(20)

Thus it holds that

$$\begin{aligned} C( \psi ^1{\bar{d}}^1 + \psi ^2{\bar{d}}^2) \le Q_0({\hat{w}}), \end{aligned}$$

(21)

Since $Q_0$ is a convex function we can write

$$\begin{aligned} Q_0({\hat{w}}) \le \psi ^1 C({\bar{d}}^1) + \psi ^2 C({\bar{d}}^2). \end{aligned}$$

(22)

Now we show the concavity of $g(\cdot )$. Let $\alpha ^1,\alpha ^2 \in {\mathcal {A}}$ and ${\hat{\alpha }}=\psi ^1 \alpha ^1+\psi ^2 \alpha ^2$. Consider the following point,

$$\begin{aligned} \hat{{\bar{p}}}^{h}&:={\left\{ \begin{array}{ll} \frac{\psi ^1 \alpha _h^1}{{\hat{\alpha }}_h}{\bar{p}}^{h1} + \frac{\psi ^2 \alpha _h^2}{{\hat{\alpha }}_h}{\bar{p}}^{h2} &{}\text{ if } \alpha ^1_h + \alpha ^2_h > 0 \text{, }\\ \psi ^1 {\bar{p}}^{h1} + \psi ^2 {\bar{p}}^{h2} &{}\text{ otherwise, } \end{array}\right. } \end{aligned}$$

(23)

where $p^j$ is an optimal solution for the problem $P_{\alpha ^j}$, $j \in \left\{ 1,2 \right\}$. Observe that by the convexity of ${\bar{D}}^h(\cdot )$ it holds that

$$\begin{aligned} \underbrace{\sum _{h=1}^n {\hat{\alpha }}_h {\bar{D}}(\hat{{\bar{p}}}^h)}_{d({\hat{\alpha }})} \le \psi ^1 \underbrace{\sum _{h=1}^n \alpha _h^1 {\bar{D}}({\bar{p}}^{h1})}_{d(\alpha ^1)} + \psi ^2 \underbrace{\sum _{h=1}^n \alpha _h^2 {\bar{D}}({\bar{p}}^{h2})}_{d(\alpha ^2)}, \end{aligned}$$

(24)

which implies that the optimal solution of $M(\psi ^1d(\alpha ^1) + \psi ^2d(\alpha ^2))$ is feasible for $M(d({\hat{\alpha }}))$. Then we can write

$$\begin{aligned} C(d({\hat{\alpha }})) \le C(\psi ^1d(\alpha ^1) + \psi ^2d(\alpha ^2)) \le \psi ^1C(d(\alpha ^1)) + \psi ^2C(d(\alpha ^2)), \end{aligned}$$

(25)

where the last inequality follows from the convexity of $C(\cdot )$. Because ${\mathcal {P}}$ is convex, $\hat{{\bar{p}}}$ is feasible for $P_{{\hat{\alpha }}}$. Thus, it holds that

$$\begin{aligned} g({\hat{\alpha }})&\ge \sum _{h = 1}^n{\hat{\alpha }}_h E \left[ {\bar{S}}^h (\hat{{\bar{p}}}^h) - r_h \right] - C\left( d({\hat{\alpha }})\right) \end{aligned}$$

(26)

$$\begin{aligned}&\ge \sum _{j = 1}^2\psi ^j \sum _{h = 1}^n{\hat{\alpha }}_h^j E \left[ {\bar{S}}^h ({\bar{p}}^{hj}) - r_h \right] - \psi ^1 C\left( d(\alpha ^1)\right) - \psi ^2 C\left( d(\alpha ^2)\right) , \end{aligned}$$

(27)

where (27) follows from (25) and the fact that ${\bar{S}}^h(\cdot )$ is concave. $\square$

1.3 Proof of Proposition 2

Proof

The Lagrangian of (12)–(14) is

$$\begin{aligned} \begin{aligned} L&= \sum _{h = 1}^n\alpha _h E \left[ {\bar{S}}^h ({\bar{p}}^h) - r_h \right] - C\left( \sum _{h = 1}^n \alpha _h {\bar{D}}^h({\bar{p}}^h)\right) \\&\quad + \eta ^\top (\nu - \Gamma \alpha ) + \zeta ^\top \alpha + \kappa ^\top {\bar{h}}({\bar{p}}), \end{aligned} \end{aligned}$$

(28)

where $\eta$ and $\kappa$ are Lagrange multipliers, and ${\bar{h}}$ is the mapping associated to the constraint set ${\mathcal {P}}$, such that ${\bar{p}}\in {\mathcal {P}}\Leftrightarrow {\bar{h}}({\bar{p}})\ge 0$. The optimal solution of the Ramsey problem satisfies the following first order necessary condition for $\alpha _h$

$$\begin{aligned} E \left[ {\bar{S}}^h ({\bar{p}}^h) - r_h \right] - \frac{d}{d\alpha _h} C\left( \sum _{h = 1}^n \alpha _h {\bar{D}}^h({\bar{p}}^h)\right) - \eta _i\Gamma _{i h} + \zeta _h = 0 \end{aligned}$$

(29)

The second term can be derived using a result from sensitivity analysis for nonlinear programming problems. Let

$$\begin{aligned} L' = Q_0(w) + \lambda ^\top \left( Q_1({\bar{d}}) - Q_2(w) \right) \end{aligned}$$

(30)

be the Lagrangean of $M({\bar{d}})$. We can use the Theorem 6.67 in [37] to write

$$\begin{aligned} \frac{d}{d\alpha _h} C\left( \sum _{h = 1}^n \alpha _h {\bar{D}}^h({\bar{p}}^h)\right) = E\left[ {\bar{D}}^h({\bar{p}}^h)^\top \nabla _{{\bar{d}}} L' \right] = E\left[ {\bar{D}}^h({\bar{p}}^h)^\top \mathbb {J}_{{\bar{d}}}Q_1 \lambda \right] . \end{aligned}$$

(31)

Then, we can write

$$\begin{aligned} E \left[ {\bar{S}}^h ({\bar{p}}^h) - {\bar{D}}^h({\bar{p}}^h)^\top \mathbb {J}_{{\bar{d}}}Q_1 \lambda - r_h \right] = \eta _i - \zeta _h. \end{aligned}$$

(32)

Given our definition of fixed charges, the left hand side is the consumer surplus of the customers in h, whereas the right hand side is lower or equal to $\eta _i$. Thus, as long as there are customer in h, they do not have incentives to switch. $\square$

1.4 Proof of Proposition 3

Proof

For simplicity, we denote the consumer surplus of a customer in rate h (the left hand side of (32)) as $cs_h(\alpha )$, when the population distribution is $\alpha$. Let $\alpha (t)$ be a curve such that

$$\begin{aligned} \alpha _h(t) :={\left\{ \begin{array}{ll} \alpha '_{h1} - t &{}\text{ if } h = h_1 \text{, }\\ \alpha '_{h2} + t &{}\text{ if } h = h_2 \text{, }\\ \alpha '_{h} &{}\text{ otherwise, } \end{array}\right. } \end{aligned}$$

(33)

for $t \in [0,\alpha '_{h1}]$. A switch from h1 to h2 is a movement along this curve from t to $t+\varDelta t$. Let $f(t):=g(\alpha (t))$, and note that this is a concave function since $g(\cdot )$ is concave and $\alpha (t)$ is linear.

In the steps that follow, we compute the derivative of f(t) applying Theorem 6.67 in [37] to the Lagrangian of $P_\alpha$

$$\begin{aligned} L = \sum _{h = 1}^n\alpha _h E \left[ {\bar{S}}^h ({\bar{p}}^h) - r_h \right] - C\left( \sum _{h = 1}^n \alpha _h {\bar{D}}^h({\bar{p}}^h)\right) + \kappa ^\top {\bar{h}}({\bar{p}}), \end{aligned}$$

(34)

where we define $\kappa$ and ${\bar{h}}({\bar{p}})$ as in Appendix A.3.

We show first (iii). For $\varDelta t$ small enough we have that the sign of $f(0+\varDelta t) - f(0)$ is equal to the sign of

$$\begin{aligned} \frac{d}{dt}f(0)&= \nabla g(\alpha ')^\top \frac{d}{dt}\alpha (0)\end{aligned}$$

(35)

$$\begin{aligned}&= cs_{h2}(\alpha ') - cs_{h1}(\alpha ') \end{aligned}$$

(36)

$$\begin{aligned}&> 0, \end{aligned}$$

(37)

where (36) follows from the fact that $\nabla g(\alpha ')_h$ is equal to $cs_h$ (as we showed in the proof of Proposition 2), and (37) holds because otherwise customers in h1 would not have incentives to switch.

Now we focus on (ii) and (iv). Note that, in view of (36), both claims are equivalent. By the concavity of f, we have

$$\begin{aligned} \left( \frac{d}{dt}f(0+\varDelta t)-\frac{d}{dt}f(0)\right) (0 + \varDelta t - 0) \le 0, \end{aligned}$$

(38)

from where the result follows.

To show (i) it is enough to observe that $\nabla g$ is a continuous mapping, because g is concave and differentiable [23, pp. 282–284]. Then, for small enough $\varDelta t$ we have

$$\begin{aligned} cs_{h2}(\alpha (0)) - cs_{h1}(\alpha (0))> 0 \Rightarrow cs_{h2}(\alpha (\varDelta t)) - cs_{h1}(\alpha (0)) > 0. \end{aligned}$$

(39)

$\square$

1.5 Proof of Proposition 4

Before proving the proposition, we introduce notation and make some observations. Define

$$\begin{aligned} {\mathcal {A}}_i = \left\{ \alpha \in \mathbb {R}^{n_i}_+: \nu _i \ge \sum _{h \in H_i}\alpha _h \right\} , \end{aligned}$$

(40)

where $H_i = \left\{ h_1^i,\ldots ,h_{ni}^i\right\}$ is the set of rates associated to types i. The set ${\mathcal {A}}$ is the Cartesian product of these sets. Based on ${\mathcal {A}}_i$, we define the simplex

$$\begin{aligned} {\mathcal {A}}_i^0 = \left\{ v^i \in \mathbb {R}^{n_i+1}_+: \nu _i = \sum _{h \in H^0_i} v_h \right\} , \end{aligned}$$

(41)

were $H_i^0 = H_i\cup \left\{ h_0^i\right\}$, and $v^i = (v_{h_0^i},\ldots ,v_{h_{ni}^i})$. In addition, define

$$\begin{aligned} {\mathcal {A}}^0 :={\mathcal {A}}_1^0 \times \cdots \times {\mathcal {A}}_{|I|}^0 \end{aligned}$$

(42)

and the problem

$$\begin{aligned} \min \left\{ f(v) :v \in {\mathcal {A}}^0 \right\} , \end{aligned}$$

(43)

where v is a block vector that has $v^i$ in its i-th component, $f(v) :=g(v_{-0})$ and $v_{-0}$ results after removing $v_{h_0^i}$ from v, for all $i\in I$.

Note that (43) is concave. Moreover, this problem is equivalent to (15). The optimal values of both problem coincide, and for $\alpha \in {\mathcal {A}}$ there is a unique $v \in {\mathcal {A}}^0$ and vice versa. Further, applying Theorem 6.67 in [37] to (43), we have that for $h \ne h_0^i$, $\nabla f(v)_{h} = cs_h$ (the consumer surplus of customers enrolled in h). We define $cs_{h_0^i}:=0$ for all $i\in I$. Given this, $\nabla f(v) = cs$.

We also make explicit the way in which customer react to the incentives as well as how the utility updates rates as customers switch. Given the previous definitions, one can interpret an update in $\alpha$ as a switch between two groups, say from h1 to h2. We distinguish the following three cases:

• If $h1 = h_0^i$, new customers enroll in rate h2.

• If $h2 = h_0^i$, existing customers end their utility services.

• Otherwise, existing customers replace h1 with h2.

We define a mapping s(v) such that if t customers switch the new distribution is $v + t \cdot s(v)$. If customers switch from h1 to h2 then

$$\begin{aligned} s(v)_h = {\left\{ \begin{array}{ll} 1 &{}\text{ if } h = h2 \text{, }\\ -1 &{}\text{ if } h = h1 \text{, }\\ 0 &{}\text{ otherwise. } \end{array}\right. } \end{aligned}$$

(44)

We consider that customers switch rationally but not strategically. That is, they consider current differences in consumer surpluses among rates but do not internalize the impacts of other customers’ decisions on their after-switching surpluses. In terms of timing, rationality implies that the first to switch are those customers with the highest opportunity for increasing their surpluses.

We now make precise what we mean by the utility updating rates relatively fast. Suppose that the first event after the last rate update is a switch from h1 to h2, i.e., h1 and h2 produce the lowest and highest surpluses respectively. The maximum number of customers that can switch before the next update is $t(v) = \theta ^p\epsilon$, for some $\epsilon > 0$ such that $x_{h1} \ge \epsilon$, $\theta \in (0,1)$, and p being the smallest number in ${\mathbf {Z}}_+$ satisfying

$$\begin{aligned} f(v + \theta ^p\epsilon \cdot s(v)) \ge f(v) + \beta \theta ^p\epsilon \cdot s(v)^\top \nabla f(v), \end{aligned}$$

(45)

with $\beta \in (0,1)$. Further, we consider that the utility decreases $\epsilon$ as the difference between the maximum and minimum consumer surpluses decreases across rate updates.

Proof

Given our previous considerations, the evolution of v across time follows the iterations of the Selective Bi-coordinate Method described in [32]. The algorithm converges to the optimum for concave problems with compact feasible regions, and with objective functions differentiable, with continuous gradient [32]. The problem (43) is concave, f is differentiable because $P_\alpha$ meets the conditions of Theorem 6.67 in [37], and $\nabla f$ is a continuous mapping because f is concave and differentiable [23, pp. 282–284]. Thus, v converges to the Ramsey optimum. $\square$

Implementation details

1.1 Demand model calibration

We consider customers with linear demands. These functions differ in their price responsiveness but have the same intercept. Let ${\bar{d}}_0$ and ${\bar{p}}_0$ be the demand and price intercepts, and ${\bar{B}}^h\prec 0$ the Jacobian of the demand of customers in h. The statement ${\bar{B}}^h\prec 0$ indicates that ${\bar{B}}$ is negative definite. This assumption is implied by the strict concavity of the utility function assumed in Peak-Load Pricing. A system of demand equation that satisfies these conditions is

$$\begin{aligned} {\bar{D}}^h ({\bar{p}}) = {\bar{B}}^h ({\bar{p}} - {\bar{p}}_0) + {\bar{d}}_0, \end{aligned}$$

(46)

with an associated gross surplus function

$$\begin{aligned} {\bar{S}}^h ({\bar{p}}) = \frac{1}{2}({\bar{p}}^\top {\bar{B}}^h {\bar{p}} - {\bar{p}}_0^\top {\bar{B}}^h {\bar{p}}_0). \end{aligned}$$

(47)

The latter expression results from assuming that the optimum of the utility maximization problem is interior, a standard assumption in applied microeconomics analysis. This implies that $\nabla {\bar{U}}({\bar{D}}({\bar{p}})) = {\bar{p}}$, which also provides an expression for the Hessian of ${\bar{U}}(\cdot )$. Normalizing ${\bar{U}}(0) = 0$, researchers can derive an expression for the Taylor expansion of ${\bar{U}}(\cdot )$ about ${\bar{d}}_0$.

It follows that ${\bar{d}}_0$, ${\bar{p}}_0$ and ${\bar{B}}^h$ fully determine the demand and gross surplus functions. The demand intercept ${\bar{d}}_0$ corresponds to the consumption baseline that we described in the previous subsection. We assume that the price intercept is a flat rate, i.e., ${\bar{p}}_0 = e \cdot \tau _0$ (with $e:=(1,\ldots ,1)^\top$). Following a procedure similar to [15], we set $\tau _0$ simply as the average long-run marginal cost of electricity for a system with aggregated demand equal to $\sum _i \nu _i\cdot d_0$ (recall that $\sum _i \nu _i$ is the total number of customers in the population).

To compute ${\bar{B}}^h$ we use the definition of the price elasticity matrix as follows

$$\begin{aligned}{}[E^h]_{tl} :=\frac{\partial {\bar{D}}_t^h({\bar{p}})}{\partial {\bar{p}}_l} \cdot \frac{{\bar{p}}_{l0}}{{\bar{d}}_{t0}} \Leftrightarrow [{\bar{B}}^h]_{tl} = [E^h]_{tl} \frac{{\bar{d}}_{t0}}{\tau _0}, \end{aligned}$$

(48)

where $[E^h]_{tl}$ is the element tl of the price-elasticity matrix of the corresponding segment. To calibrate the elasticity matrix with the information we have, we use the following procedure.

Let $\varepsilon _o^h$ and $\varepsilon _c^h$ be the pair of own- and cross-price elasticities of customers in h. First, we define ${\hat{E}}^h$, which is not necessarily symmetric or negative definite,

$$\begin{aligned}{}[{\hat{E}}^h]_{tl} :={\left\{ \begin{array}{ll} \varepsilon _o^h &{}\text{ if } t = l \text{, }\\ \varepsilon _c^h/10 &{}\text{ if } t \in \left\{ l - 5,\ldots , l + 5 \right\} \setminus \left\{ l\right\} , \\ 0 &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

(49)

To ensure concave utility functions we then compute $E^h$ as the closest elasticity matrix to ${\hat{E}}^h$ consistent with a negative definite ${\bar{B}}^h$ via the following conic program:

$$\begin{aligned} \min _{({\bar{B}}^h,E^h)} \left\{ \frac{\Vert E^h - {\hat{E}}^h\Vert }{\Vert {\hat{E}}^h\Vert } : \,\, [{\bar{B}}^h]_{tl} = [E^h]_{tl} \frac{{\bar{d}}_{t0}}{\tau _0}\, \forall tl, \,\, {\bar{B}}^h \prec 0 \right\} , \end{aligned}$$

(50)

where $\Vert \cdot \Vert$ is the Frobenius norm.

1.2 Modeling a Renewable Portfolio Standard

A renewable portfolio standard is a policy in which the regulator mandates the utility to produce a minimum fraction $\zeta$ of its total energy from renewable sources. In order to include the effect of this policy in our analysis, we modify the model of the cost function (2)–(4) by including a new constraint. Let $e :=[1,\ldots ,1]^\top \in \mathbb {R}^{T}$, the associated constraint is

$$\begin{aligned} e^\top E \left[ \zeta \cdot {\bar{d}} - \sum _{k\in K_R} {\bar{y}}_k \right] \le 0, \end{aligned}$$

(51)

where $K_R$ is the set of qualified renewable technologies.

An improved model of the supply-side

In this appendix, we show how to modify (2)–(4) to produce an improved model of the supply-side. Specifically, we add a model of the transmission network and include a modified version of the RPS equation of (51). Next, we show that this more complex model is a special case of $M({\bar{d}})$. We finally discuss further improvements to the transmission network model.

1.1 The model

Here we consider the representation of the transmission network described in [46]. It correspond to the dc approximation of the power flow equations, which is commonly used in planning studies [51]. In this model, there are vectors ${\bar{u}}_j$ of net imports at each bus $j\in \left\{ 1,\ldots ,J\right\}$. A matrix of power transfer distribution factors, ${\bar{F}}$, describes the incremental change in power that occur in the transmission lines due to power transfers between two buses. For a given vector ${\bar{u}} = \left( {\bar{u}}_1,\ldots ,{\bar{u}}_J\right) ^\top$, a component of ${\bar{F}}{\bar{u}}$ is the total flow across a line at a given time period. The thermal limits of the network are included in the components of the vector ${\bar{f}}$.

The following constraint restricts flows according to the thermal limits

$$\begin{aligned} {\bar{f}} \ge {\bar{F}}{\bar{u}} \ge - {\bar{f}}. \end{aligned}$$

(52)

The next, establishes that the network does not create nor sink power

$$\begin{aligned} \sum _{j=1}^J {\bar{u}}_j = 0, \end{aligned}$$

(53)

and the following balances supply and demand at each bus

$$\begin{aligned} {\bar{d}}_j \le \sum _{k \in K} {\bar{y}}_{kj} + {\bar{u}}_j \,\,\, \forall j\in \left\{ 1,\ldots ,J\right\} . \end{aligned}$$

(54)

The last equation of the network imposes (4) at each bus

$$\begin{aligned} 0 \le {\bar{y}}_{kj} \le x_{kj} {\bar{\rho }}_{kj} \,\,\, \forall k\in K, \, j\in \left\{ 1,\ldots ,J\right\} . \end{aligned}$$

(55)

In order to include the RPS policy, we slightly modify (51)

$$\begin{aligned} e^\top E \left[ \zeta \cdot \sum _{j=1}^J \left( {\bar{d}}_j - \sum _{k\in K_R} {\bar{y}}_{kj} \right) \right] \le 0. \end{aligned}$$

(56)

Finally, in this model the objective function is

$$\begin{aligned} \sum _{j=1}^J \sum _{k \in K} E\left[ {\bar{y}}_{kj}^\top {\bar{c}}_{kj} + x_{kj} {\hat{r}}_{kj} \right] . \end{aligned}$$

(57)

1.2 A special case

To see that the model we presented in the previous subsection is a special case of $M({\bar{d}})$, consider the following definitions:

$$\begin{aligned} w&= (x,y,u)^\top , \end{aligned}$$

(58)

$$\begin{aligned} Q_1({\bar{d}})&= \begin{bmatrix} d_1 \\ \vdots \\ d_J \\ e^\top E \left[ \zeta \cdot \sum _{j=1}^J {\bar{d}}_j \right] \end{bmatrix}, \end{aligned}$$

(59)

$$\begin{aligned} Q_2(w)&= \begin{bmatrix} \sum _{k \in K} y_{k1} + u_1 \\ \vdots \\ \sum _{k \in K} y_{kJ} + u_J \\ e^\top E \left[ \sum _{j=1}^J \sum _{k \in K_R} {\bar{y}}_{kJ} \right] \end{bmatrix}, \end{aligned}$$

(60)

$$\begin{aligned} W_1&= \left\{ w : \, {\bar{f}} \ge {\bar{F}}{\bar{u}} \ge - {\bar{f}}\right\} , \end{aligned}$$

(61)

$$\begin{aligned} W_2&= \left\{ w : \, \sum _{j=1}^J {\bar{u}}_j = 0\right\} , \end{aligned}$$

(62)

$$\begin{aligned} W_3&= \left\{ w : \, 0 \le {\bar{y}}_{kj} \le x_{kj} {\bar{\rho }}_{kj} \,\,\, \forall k\in K, \, j\in \left\{ 1,\ldots ,J\right\} \right\} , \end{aligned}$$

(63)

$$\begin{aligned} W&= W1\cap W_2\cap W_3, \text{ and } \end{aligned}$$

(64)

$$\begin{aligned} Q_0(w)&= \sum _{j=1}^J \sum _{k \in K} E\left[ {\bar{y}}_{kj}^\top {\bar{c}}_{kj} + x_{kj} {\hat{r}}_{kj} \right] + \mathbb {I}_W (w). \end{aligned}$$

(65)

1.3 Further improvements

The model of the transmission system we present in Appendix C.1 is an approximation. An exact representation requires a set of quadratic power flow equations, constituting a non-convex set. This implies that $M({\bar{d}})$ does not capture the exact case, which is a limitation of the framework we propose in this paper. However, we observe that, with the exception of [47], all the applied work on rate analysis we reviewed omits completely the transmission system, and the majority does not even model a supply-side (see references in Sects. 3.1 and 3.2). In this dimension, our work does make important progress. It allows researchers to incorporate in their applied studies of tariffs considerably richer representations of the supply-side of the electricity sector.

In addition, we observe that using convex approximations or relaxations of an exact model of the transmission system may suffice. Indeed, [3, 42, 51] show that the dc approximation of the power flow equations works fairly well for market analysis of the power sector. Besides, recent work on the Optimal Power Flow problem—a mathematical program whose constraints are the power flow equations—shows that for certain network topologies convex relaxations of the problem can produce exact results [35, 36].