Game theory has proven to be a valuable tool to study strategic electricity consumers enrolled in a demand response (DR) program. Among the different billing mechanisms proposed, the hourly billing ...model is of special interest as an intuitive and fair mechanism. We focus on this model and answer to several theoretical and practical questions. First, we prove the uniqueness of the consumption profile corresponding to the Nash equilibrium and we analyze its efficiency by providing a bound on the price of anarchy. Next, we address the computational issue of this equilibrium profile by providing results on the convergence rates of two decentralized algorithms to compute the equilibrium: the cycling best response dynamics and a projected gradient descent method. Last, we simulate this DR framework in a stochastic environment where the parameters depend on forecasts. Numerically, we show the relevance of an online DR procedure which reduces the impact of inaccurate forecasts in comparison to a standard offline procedure.
Matrix versions of the Hellinger distance Bhatia, Rajendra; Gaubert, Stephane; Jain, Tanvi
Letters in mathematical physics,
08/2019, Letnik:
109, Številka:
8
Journal Article
Recenzirano
Odprti dostop
On the space of positive definite matrices, we consider distance functions of the form
d
(
A
,
B
)
=
tr
A
(
A
,
B
)
-
tr
G
(
A
,
B
)
1
/
2
,
where
A
(
A
,
B
)
is the arithmetic mean and
G
(
A
,
B
)
...is one of the different versions of the geometric mean. When
G
(
A
,
B
)
=
A
1
/
2
B
1
/
2
this distance is
‖
A
1
/
2
-
B
1
/
2
‖
2
,
and when
G
(
A
,
B
)
=
(
A
1
/
2
B
A
1
/
2
)
1
/
2
it is the Bures–Wasserstein metric. We study two other cases:
G
(
A
,
B
)
=
A
1
/
2
(
A
-
1
/
2
B
A
-
1
/
2
)
1
/
2
A
1
/
2
,
the Pusz–Woronowicz geometric mean, and
G
(
A
,
B
)
=
exp
(
log
A
+
log
B
2
)
,
the log Euclidean mean. With these choices,
d
(
A
,
B
) is no longer a metric, but it turns out that
d
2
(
A
,
B
)
is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of
m
positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.
We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation ...function in the output node, referred to as log-sum-exp (LSE) network, is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of network maps to a family of subtraction-free ratios of generalized posynomials (GPOS), which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for a design that possesses a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing an effective optimization-based design. We illustrate the proposed approach by applying it to the data-driven design of a diet for a patient with type-2 diabetes and to a nonconvex optimization problem.
•The design of a menu of contracts must consider uncertainty in the customers decision.•The formulation with a quadratically regularized lower-level leads to robust prices.•The pivoting heuristic, ...based on geometrical properties, solves realistic instances.•Study on French Electricity Retail market shows effectiveness of the quadratic model.
We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers’ choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer’s response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer’s response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.
•Multi-stage stochastic OPF models electricity network operations under uncertainty.•Multi-stage stochastic AC OPF is a large-scale non-convex optimization problem.•Some conditions ensure no ...relaxation gap for stochastic AC OPF problem.•Bounds on the relaxation gap can be computed using convex optimization only.•A scenario-tree generator is built based on a stochastic model for solar irradiance.
We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality. We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems.Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm.
In this paper, we show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is a universal approximator ...of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named log-sum-exp (<inline-formula> <tex-math notation="LaTeX">\mathrm {LSE}_{T} </tex-math></inline-formula>). Under a suitable exponential transformation, the class of <inline-formula> <tex-math notation="LaTeX">\mathrm {LSE}_{T} </tex-math></inline-formula> functions maps to a family of generalized posynomials <inline-formula> <tex-math notation="LaTeX">\mathrm {GPOS}_{T} </tex-math></inline-formula>, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an <inline-formula> <tex-math notation="LaTeX">\mathrm {LSE}_{T} </tex-math></inline-formula> network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a <inline-formula> <tex-math notation="LaTeX">\mathrm {GPOS}_{T} </tex-math></inline-formula> model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.
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