Alberto Benegiamo
email: alberto.benegiamo@gmail.com

Patrick Loiseau
email: patrick.loiseau@univ-grenoble-alpes.fr

Giovanni Neglia
email: giovanni.neglia@inria.fr

Dissecting Demand Response Mechanisms: the Role of Consumption Forecasts and Personalized Offers

Keywords: smart grid, demand-response, incentive mechanisms, energy network

Demand-Response (DR) programs, whereby users of an electricity network are encouraged by economic incentives to re-arrange their consumption in order to reduce production costs, are envisioned to be a key feature of the smart grid paradigm. Several recent works proposed DR mechanisms and used analytical models to derive optimal incentives. Most of these works, however, rely on a macroscopic description of the population that does not model individual choices of users.

In this paper, we conduct a detailed analysis of those models and we argue that the macroscopic descriptions hide important assumptions that can jeopardize the mechanisms' implementation (such as the ability to make personalized offers and to perfectly estimate the demand that is moved from a timeslot to another). Then, we start from a microscopic description that explicitly models each user's decision. We introduce four DR mechanisms with various assumptions on the provider's capabilities. Contrarily to previous studies, we find that the optimization problems that result from our mechanisms are complex and can be solved numerically only through a heuristic. We present numerical simulations that compare the different mechanisms and their sensitivity to forecast errors. At a high level, our results show that the performance of DR mechanisms under reasonable assumptions on the provider's capabilities are significantly lower than those suggested by previous studies, but that the gap reduces when the population's flexibility increases.

Introduction

Demand Response (DR hereinafter) programs are envisioned to be a key feature of the Smart Grid paradigm [START_REF] Albadi | A summary of demand response in electricity markets[END_REF]. By means of economic incentives (discounts or penalties), DR schemes encourage users to rearrange their consumption in response to the network state, thus mitigating the grid overload and driving wholesale prices down.

Several analytical models are available in the literature, which describe and quantify the effects of DR mechanisms. Whatever their specifics are, these schemes need to model how users react to the incentives. Ideally the models should capture the most realistic features of a practical DR mechanism while maintaining tractability.

Among these contributions, the authors of [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] study how an energy provider should select time-dependent discounts to minimize its production costs. They assume that the percentage of users who shift their consumption from slot i to slot j is a decreasing function of the temporal distance between slots i and j and a concave and increasing function of the discount offered in slot j (R j ), independent from discounts in other slots. The paper claims that, under these assumptions along with the requirement of piecewise linearity of energy production costs, the problem of finding the set of discounts that minimize the provider's cost is convex and therefore simple to solve. Under similar modeling assumptions, however, we find that the optimization problem can be non-convex even in such a simple scenario (see Sec. 4). The same user's model as in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] is adopted also in [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF], where the optimization problem is extended in order to account for battery storages and distributed renewable sources available into a specific microgrid. Authors of [START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] propose a day ahead pricing scheme which maximizes the provider's profitability and capacity utilization. Users are assumed to reschedule their consumption by comparing the utility v i they get by scheduling a task in each timeslot i; therefore they allocate their consumption proportionally to these utilities, i.e., they consume a fraction vi T j=1 vj of their total energy demand in timeslot i. The resulting optimization problem is non convex but some relaxation techniques are introduced, which allow one to calculate a solution within a reasonable amount of time. In [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF], a more realistic model is proposed where each user first calculates the welfare (defined as utility minus time-dependent cost) she gets from consuming electricity in each of the possible timeslots, and then allocates all the consumption to the slot returning the largest welfare. As we show below (see Sec. 4.4) this model can lead to a much more complex optimization problem than the one presented in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF]. Finally, the authors of [START_REF] Song | Demand Side Management in Smart Grids Using a Repeated Game Framework[END_REF] propose a full-fledged game theoretical model, but their results hold only if users experience a large number of interactions without any change in the system.

We claim that these studies rely on too strong assumptions, which jeopardize their usability for practical purposes. Interestingly, we observe that the assumptions are sometimes hidden in the macroscopic models the papers start from. In particular in this paper we focus on [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and show that its model requires personalized offers and a very precise forecast of the baseline consumption of each user.

The implementation of these features may require potentially significant costs in terms of communication, measurement and computation infrastructure. Besides highlighting these implicit requirements in the analytical framework in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] (and then also in [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF]), we explore their potentials considering four DR mechanisms with different levels of complexity:

1. the base mechanism corresponds to an optimization problem similar to the one considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF], it requires personalized offers and individual consumption forecasts; the energy production cost is optimized over the discount values, each of which is offered to a given fraction of the population, 2. the optimized mechanism takes full advantage of personalized offers and consumption forecasts by minimizing the cost over both the discount values and the population fractions to which the discounts are offered, 3. the robust mechanism relies on personalized offers, but does not need individual consumption forecasts, 4. finally the broadcast mechanism (analogous to that in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF]) needs neither of the two features.

Interestingly, contrarily to prior studies, we find that the cost-minimization problems resulting from our DR mechanisms are not convex (even for the base mechanism). Nevertheless, simple heuristics can identify (potential) minima in a reasonable amount of time in realistic scenarios. Then, our numerical results show that the simpler robust and broadcast mechanisms achieve significantly lower cost reductions than the optimized mechanism, which is difficult to implement, but that the gap reduces when the population's flexibility increases. The paper is organized as follows. In Sec. 2 we discuss how the macroscopic models considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF][START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF][START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] hide some implicit assumptions about the user rationality or about the interactions between the provider and the user. We define our microscopic model in Sec. 3 and then describe different DR mechanisms and their corresponding optimization problems in Sec. 4. We evaluate their performance numerically in a realistic scenario in Sec. 5. Finally in Sec. 6 we discuss how our models can be tuned and which other psycological and social insights should be taken into account to explain users' decisions.

Pitfalls when Starting from Macroscopic Models

In this section, we describe in more detail the macroscopic models proposed in the literature for day-ahead price optimization. Consider a finite time horizon discretized in a set T of N timeslots and a large population S of users. The baseline aggregate energy consumption in slot j is denoted by E 0 j . The energy provider charges a flat rate B, but it can offer discount rates to incentivize the users to move some of their consumption so as to reduce the energy production cost. Due to consumption shifts, the actual aggregate consumption in time slot j is E 1 j . Observe that a usual assumption in the literature (including the papers mentioned above) is that the introduction of a DR scheme neither reduces nor increases users' demand; it merely rearranges users' consumption in a more cost effective way, so that

N j=1 E 0 j = N j=1 E 1 j . (1)
We denote the amount of consumption shifted from slot j to slot i = j as E j→i , and the amount of consumption the users refuse to shift away from j as E j→j . Then we have

E 1 i = E 0 i + N z=1 E z→i - N k=1 E i→k .
We now start to further detail the model considering some specific assumptions made in previous works. In [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF], the electricity provider offers an energy price discount R i ≥ 0 in each slot i. The users are assumed to react to these incentives by shifting a fraction of their baseline consumption from slot j to slot i (|j -i| slots away) according to the following formula:

E j→i = E 0 j S j (R i , |j -i|). (2)
S j (R i , |j -i|) is called the aggregate sensitivity function and is increasing in the discount R i and decreasing in the temporal shift |j -i|, in order to take into account the user discomfort. The provider selects the vector of discounts R in order to minimize its total cost, equal to the sum of the electricity generation costs and the loss of revenues due to the discounts. In particular the optimization problem considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] is the following:

min R i j =i R i E j→i + i c i E 1 i (3) s.t. 0 ≤ R i ≤ B ∀i = 1, . . . N, (4)
where c i (•) is the cost of electricity production in slot i. Eq. ( 4) guarantees that discounts R are non negative and smaller than the flat rate B, so that the money stream goes toward the provider.

As it often happens, the devil is hidden in the details, and in this case in Eqs. ( 2) and (3). Our first remark is that the cost of lost revenues i j =i R i E j→i in Eq. ( 3) implicitly assumes the possibility to reward only the consumption actually shifted from j to i, i.e., E j→i , but this quantity cannot be directly measured. The actual consumption E 1 i can be measured, and then E j→i can be quantified provided that we have good estimates of the sensitivity function S j (R i , |j -i|) and of the baseline consumption E 0 i . Let us assume for a moment that S j (R i , |j -i|) is known from historical data and that the aggregate baseline consumption may be predicted with a reasonably high level of accuracy on a large set of users. Then it seems possible to solve the macroscopic problem in Eqs. (3) and ( 4), but we need to consider also what should happen at the microscopic scale. While the estimates for the aggregate baseline consumption can be adequately precise, finally the billing is done at the user's granularity and each user expects to receive the price discount corresponding to the energy consumption she actually moved. If the energy bill's reduction does not correspond to her forecast, the user is likely to opt out of the program (in particular if she has experienced underpayments) or to reduce her efforts and milk occasional discounts. It appears then that Eq. (3) implicitly requires very precise predictions of individual consumptions.

We now observe that the form of the sensitivity function S j (R i , |j -i|) in Eq. [START_REF] Barbato | Optimization Models and Methods for Demand-Side Management of Residential Users: A Survey[END_REF] indicates that the amount of energy shifted from j to i depends on the discount R i but not on the other discounts. We can then ask ourselves which individual decisions may lead to this aggregate behavior, an issue ignored both in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and [START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF]. As long as a rational individual is offered two different discounts R i and R k , it seems natural that her decision to move some consumption from j to i or from j to k or to keep it in j will take into account both the discounts. To stress the point, consider a case when both S j (R i , |j -i|) and S j (R k , |j -k|) are positive, but moving the consumption from i to k is both less inconvenient (i.e., |j -k| < |j -i|) and more rewarding (i.e., R k > R i ). There is then no reason why the user would move consumption to i. The conclusion is the same for all the users and then we should have E j→i = 0 at the aggregate level, in contradiction with Eq. ( 2). We can then conclude that the expression of the sensitivity function in Eq. ( 2) is not suited to model the situation when a user is offered two or more rewards, but it can capture the case when the user decides between moving from j to i in exchange of a discount R i or staying in j. If every user is offered a single discount to move to a given slot, but different users can receive different offers, then Eq. (2) can reasonably describe the macroscopic effect of such personalized offers. The details are described in Sec. 4.1, here we only highlight that Eq. (2) requires then that the electricity provider i) calculates an offer for each user, ii) communicates individually to the user, iii) considers the individual offer when billing the user. This is clearly more demanding than simply advertising to the whole population the same set of discounts.

We observe that the equivalent sensitivity function considered in [START_REF] Subramanian | Dynamic Price Optimization Models for Managing Time-of-Day Electricity Usage[END_REF] poses similar problems. Using our notation, we have

E j→i = vi k∈T v k E 0 j
, where v i is the net utility a user gets by consuming electricity in slot i and can be a function of the timeslot itself and of the discount R i . This formula tries to capture the effect of the whole set of discounts, but it is not clear again what is the underlying user's model: if slot i has a larger utility than slot k (v i > v k ), why should the user consume in k?

Starting from a Microscopic Model

In the previous section we made the point that, while aggregate population models may be convenient, it is necessary to explicitly consider the microscopic level: how the user takes the decisions and how the provider and the user are supposed to interact. In this paper we follow the opposite path in comparison to the existing works mentioned: we move from the microscopic level to the macroscopic one. In particular, in this section, we start from a clear model of rationality for the single user and then move to describe how aggregate quantities can be derived.

Each user u ∈ S has a baseline energy consumption

{e 0u j } j=1,...N , (5)
that leads to the aggregate baseline consumption E 0 j = u∈S e 0u j for j = 1, . . . N . This baseline consumption corresponds to the consumptions without any incentive mechanism and is completely arbitrary. We assume in what follows that users are homogeneous, i.e., ∀ u, e 0u j = e 0

j j = 1, . . . N. (6)
In section 5.1 we show how the DR mechanisms perform when this assumption does not hold. User u is characterized by a private type D u = {d u j→i } j,i=1,...N where d u j→i indicates the discomfort due to shifting one unit of consumption from timeslot j to timeslot i. We assume that discomforts are expressed in monetary units; and that, ∀u ∈ S,

d u j→j = 0 and d u j→i > 0, ∀j, i = j, (7)
i.e., there is a strictly positive discomfort if and only if consumption is shifted from its original timeslot. The provider does not know the private type D u of each user u: from its point of view, each discomfort d u j→i is drawn from a known, continuous distribution F j→i on [0, α j,i ] (where possibly α j,i = +∞). Discomforts of distinct users are independent but note that we do not assume that, for a given user, the discomforts {d u j→i } j,i=1,...N are mutually independent.

Rational Users

We assume that a user simply chooses the option that maximizes her utility. In particular let T u j be a set of timeslots the user could move the baseline consumption e 0u j to in exchange for different discounts R u j = {R u j→k ≥ 0, k ∈ T u j }. The set pair (T u j , R u j ) defines the offer user u receives for timeslot j. The set of options includes the possibility to keep the consumption in j, i.e., j ∈ T u j . A rational user maximizes her utility by scheduling her consumption e 0u j to a timeslot i * ∈ argmax

k∈T u j R u j→k -d u j→k . (8)
We assume that if two or more timeslots are equally palatable, the whole consumption is shifted to only one of them, picked at random with equal probability.

Aggregation

We observe that the quantities d u j→k in Eq. ( 8) are random variables, then two different users could take different decisions while confronted with the same offers. The aggregate consumption E 1 i , for i ∈ T , would then be a random variable. Here we assume (as it is implicit in the other works) that we always work with large sets of the population so that the variability can be neglected by approximating actual random quantities with their expected values. In particular, if a subset Q containing a fraction q of the population receives an offer (T j , R j ), the corresponding consumption shifted from j to a time slot i, denoted as E Q j→i will be

E Q j→i = qE 0 j Prob i ∈ argmax k∈Tj R u j→k -d u j→k , (9)
if the probability that a user has two or more equally palatable timeslots is zero. When discomforts are continuous random variables (as we consider in this paper), this is always the case if each user receives only one offer (the first three mechanisms introduced below) or if the discomforts {d u j→i } j,i=1,...N are mutually independent. In Sec. 4.4, we discuss how Eq. ( 9) should be modified if this probability is not zero. We denote Prob i ∈ argmax k∈Tj R j→k -d u j→k simply as P j→i (R u j , T j ).

DR mechanisms

Under different assumptions on the provider's capabilities, we introduce different demand response mechanisms based on the microscopic model above, which are therefore practically implementable. We introduce and study the corresponding optimization problems.

We start by the base mechanism that leads to the same aggregate optimization problem considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF][START_REF] Pan | Pricing Strategy for Device-Level Demand Response in a Microgrid[END_REF].

Base mechanism

This mechanism requires that the energy provider can manage personalized offers to its customers and moreover that it has perfect knowledge (or very precise estimates) of the baseline consumption of each user.

The population is segmented into N 2 disjoint subsets Q j→i , for j, i ∈ T , respectively including a fixed fraction q j→i of the population. Each user in Q j→i is simply offered to move her baseline consumption in slot j (e 0 j ) to slot i in exchange for a price discount R i .

The total consumption that is shifted from j to i is then

E j→i = q j→i E 0 j Prob R i -d u j→i > 0
as it can be obtained from Eq. ( 9), taking into account that in this case T j = {j, i} and R j -d u j→j = 0. We observe that the probability appearing on the right-hand side only depends on the reward R i and on the random variable d u j→i . If the discomfort is only a function of the temporal distance |j -i|, then the sensitivity function (the ratio of people who move from j to i) has the same properties than in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF], in particular:

S j (R i , |i -j|) = q j→i P j→i (R i ),
where for P j→i (•) we have made explicit the only variable it depends from. As we discussed in Sec. 2, because the provider knows exactly the consumption shifted from each user, it can formulate the optimization problem [START_REF] Gyamfi | Residential peak electricity demand response-Highlights of some behavioural issues[END_REF][START_REF] Gyamfi | Price, environment and security: Exploring multi-modal motivation in voluntary residential peak demand response[END_REF]. In [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] it is stated that the problem is convex if i) the productions costs c j (•) are continuous piecewise linear and increasing and ii) the discomfort distributions F j→i (•) are continuous and concave. We show in Appendix A that this is not the case by providing a counterexample. Stronger hypotheses are required for the problem to be concave, as for example the linearity of the discomfort functions.

In particular in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] the numerical evaluation considers

S j (R i , |i -j|) = 1 N k=1 1 (|k-j|+1) R i B • (|i -j| + 1)
, that leads us to consider

q j→i = 1 |i-j|+1 N k=0 1 |k-j|+1 , P j→i (R i ) = R i B . (10)
This particular expression for P j→i can be obtained if d u j→i is a uniform random variable with support in [0, B]. The numerical results for the base mechanism in Sec. 5 are obtained considering the same expression for the fractions q j→i . Due to the non-convexity of the optimization problem (3-4) we cannot use one of the classic algorithms for convex optimization. For the results shown in section 5 we have adopted instead a multi start approach: we have generated random starting points uniformly distributed in the problem domain and we have run per each point a descendent algorithm which converged on a local minimum; the optimal offers are therefore those returning the smallest cost among these minimizers. This approach does not guarantee convergence to the global optimum but its reliability can be improved by increasing the number of starting points.

Optimized Mechanism

We have now understood which DR mechanism can lead to the optimization problem [START_REF] Gyamfi | Residential peak electricity demand response-Highlights of some behavioural issues[END_REF], but now that we look at its implementation at the microscopic level and the need for personalized offers, some specifics of the base mechanism look arbitrary and unjustified. For example, given that discounts are not broadcast but each user receives an individual offer, why should the discounts offered to the two disjoint sets of users Q j→i and Q k→i be equal to the same value R i ? It is clear that the energy provider can further reduce the cost if it can independently choose R j→i and R k→i . Moreover, there is no reason to think that the size of the sets {Q j→i } should be fixed, the fractions {q j→i } can also be optimization variables.

We allow the provider to take advantage of these additional degrees of freedom that-we repeat-do not impose any additional requirement to the system. We call this new DR mechanism optimized. The load E j→i (R j→i , q j→i ) rescheduled from j to i is now

E j→i (R j→i , q j→i ) = q j→i P j→i (R j→i )E 0 j (11)
and the cost minimization problem becomes:

min R,q cost opt. (R, q) = i z =i R z→i E z→i + i c i E 1 i (12) s.t. 0 ≤ R z→i ≤ B ∀z, i = 1, . . . N (13)
0 ≤ q z→i ≤ 1, z, i = 1, . . . N (14)
i q z→i ≤ 1, ∀z = 1 . . . N. (15)
Eq. ( 13) guarantees that discounts R are non negative and smaller than the flat rate B, Eq. ( 15) is a consequence of the fact that each user receives at most one offer for its baseline consumption in a given slot.

The optimization problem (12-15) can be solved with the same heuristic proposed for problem (3-4).

Robust Mechanism

The optimization problems [START_REF] Gyamfi | Residential peak electricity demand response-Highlights of some behavioural issues[END_REF][START_REF] Gyamfi | Price, environment and security: Exploring multi-modal motivation in voluntary residential peak demand response[END_REF] and [START_REF] Goulden | Smart grids, smart users? The role of the user in demand side management[END_REF][START_REF] Mouri | Log-normal distribution from a process that is not multiplicative but is additive[END_REF][START_REF] Mesarić | Exploring the Potential of Energy Consumers in Smart Grid Using Focus Group Methodology[END_REF][START_REF] Ieso Ontario | Market data[END_REF] assume that the provider has perfect knowledge of each user's baseline consumption, so that it can correctly identify the consumption shifted and reduce accordingly the energy bill. This assumption is probably unrealistic. If the provider does not have such capability, then it can offer the user a discount for all the consumption in a given timeslot i and not just for the consumption moved to i. The population is then divided into N subsets Q i , each containing a fraction q i of the users. All users in Q i receive one and only one offer: they are encouraged to shift their consumption from any timeslot in the time horizon to timeslot i and they get the discount R i for all the electricity consumed in i, including the one originally in i.

We call this scheme robust, because it does not rely on estimates of individual consumption. It is clearly simpler than the previous two, because the provider needs only to measure the amount of consumption in i for the users who got the offer and to bill them accordingly.

The load E j→i (R i , q i ) shifted from j to i is E j→i (R i , q i ) = q i P j→i (R i )E 0 j . Note that users in Q i have no interest to move their baseline consumption away from i, then E i→i = q i E 0 i . The robust mechanisms leads to the following optimization problem:

min R,q cost rob. (R, q) = i,z R i E z→i + i c i E 1 i (16) s.t. 0 ≤ R i ≤ B i = 1, . . . , N (17)
0 ≤ q i ≤ 1, i = 1, . . . , N (18)
N i=1 q i ≤ 1. (19)
Note that in Eq. ( 16) the first sum includes also E i→i because all the final consumption in i from the users in Q i is paid at a discounted price. The term does not appear in Eq. ( 3) and Eq. ( 12).

The optimization problem [START_REF] Xu | Optimization Models and Algorithms for Demand Response in Smart Grid[END_REF][START_REF] Wilson | Models of Decision Making and Residential Energy Use[END_REF](18)(19)) can be solved with the same heuristic proposed for problem (3-4).

Broadcast Mechanism

In the three mechanisms introduced above, the provider makes personalized offers to users in selected fractions of the population. This may not always be possible (due to the complexity it introduces for instance in billing) or desirable (for perceived fairness issues). Our last mechanism, which is the simplest (in its definition), does not assume personalized offers. The provider selects a single vector R of discounts for every time slot and broadcasts these discounts to all users (hence the name broadcast mechanism). Users then re-arrange their demand and pay the discounted price for their demand in each slot (hence this mechanism also does not rely on the need to estimate shifted demand).

As explained in Sec. 3.1, each individual user moves her demand from slot j to a slot (potentially j itself) that maximizes her net utility (discount minus discomfort). Recall that if several slots give equal net utility, the user chooses one of them randomly.

Until now, we have not made any assumption on the possible correlation of a given user's discomforts. This is because, in the previous three mechanisms, each user was receiving only one offer. In the broadcast mechanism, each user has several offers to compare to decide on his new demand schedule, we therefore need to describe the discomfort correlations.

Let us consider now the particular case when two slots, say h and k may appear equally attractive to a user, i.e., R h -d j→h = R k -d j→k . If we assumed that, for each user, the discomforts {d j→i } {i,j=1,••• ,N } were mutually independent, this event would have probability zero according to our assumption on F i→j , and therefore it would not appear at the aggregate level. As a result, the aggregate demand moved from j to i would be

E j→i (R) = P j→i (R)E 0 j , (20)
where

P j→i (R) = Pr R i -d j→i ≥ max k =i {R k -d j→k } . (21)
However, rather than making the above independence assumption, we prefer to assume that the discomforts have the form d j→i = β j |i -j| tj , where t j is a constant independent of the user and β j is a random variable with concave Cumulative Distribution Function (CDF) F j (•). This model describes a symmetric delay sensitivity of users (users are indifferent between moving two hours earlier or two hours later) while keeping the flexibility of users having a different flexibility of demand of different times (since β and t are indexed by the origin timeslot j); but it also introduces correlations between the discomforts of a user. As a result, the fraction of demand shifting from j to i is

P j→i (R) = Pr (R i -d j→i ≥ max k =i {R k -d j→k }) 1 + 1 R2j-i=Ri , (22)
rather than (21). The denominator in (22) accounts for cases when a slot other than i (which has to be 2j -i) gives equal net utility for all users. The broadcast mechanism then leads to the following optimization problem:

min R cost brd. (R) = z i R i E z→i + i c i E 1 i (23) s.t. 0 ≤ R i ≤ B ∀i = 1, . . . N. (24)
Unfortunately, due to indicator function in Eq. ( 22), the cost function (23) of the broadcast mechanism is not continuous, even in very simple scenarios with continuous production costs; see Appendix B for details. Discontinuity arises also in the macroscopic model in [START_REF] Yang | Quantifying the benefits to consumers for demand response with a statistical elasticity model[END_REF], but it seems to have been ignored.

In practice, we solve problem (23-24) using the same heuristic proposed for the previous problems, but we work on a continuous and smooth approximation of the cost function.

Ranking DR mechanisms

Under certain assumptions, the DR mechanisms introduced above can be partially ranked in terms of the cost savings they generate. Proposition 4.1. Assume that the discomforts have the form d j→i = β j |i-j| tj , where t j is a constant independent of the user and β j is a random variable with concave CDF F j (•); and assume that the production costs c j (•) are piecewise linear, continuous and increasing. Then, for a given initial demand E 0 , the final cost generated by the optimized scheme is smaller than the cost of the robust and base mechanism.

The proof of Proposition 4.1 is provided in Appendix C. As we show in Appendix C, the ranking cannot be extended further. In particular, the broadcast and robust mechanisms cannot be compared (one or the other is more efficient depending on the scenario).

Numerical Results

In this section we evaluate the performance of the different DR mechanisms in the realistic scenario considered in [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] and based on energy data about the Ontario province in Canada. We extracted from [START_REF] Joe-Wong | Optimized Day-Ahead Pricing for Smart Grids with Device-Specific Scheduling Flexibility[END_REF] the baseline consumption E 0 , reported in Fig. 1, the flat rate B = 110$/MWh and the timeslot-independent cost function c(•) is piecewise linear with derivative: where

c (E) =    $10 E ≤ C 1 , $72.46 E ∈ (C 1 , C 2 ), $91 E ≥ C 2 , (25)
C 1 = 16.3 GWh, C 2 = 17.9 GWh, ( 26
)
represent respectively the base to intermediate load capacity and intermediate to peak load capacity. Baseline consumption E 0 and the cost functions are estimated from the IESO energy portfolio [START_REF] Ieso Ontario | Market data[END_REF], consisting of nuclear plans, hydro gas powered stations and renewable and from typical costs associated to these energy sources. We assume that discomforts take the form:

d j→i = β j |i -j|, (27)
where β j is an exponential random variable with CDF

F j (β) = 1 -e -β µ , ( 28
)
where µ is a parameter representing the population's flexibility. The larger it is, the smaller (in a stochastic order sense) the discomfort of the users to shift their consumption. In Figs. 2 and3 we respectively show final load distributions E 1 and optimal discounts and segment sizes of the four DR mechanisms, for the flexibility parameter µ = 1 10 . Since production costs c(•) are constant across the horizon, provider's costs are cut by flattening the load; this is mostly clear in the optimized scheme where the peak of 19 GWh in E 0 is reduced to 17.9 GWh which is exactly the intermediate to peak capacity C 2 , as can be seen in Fig. 2.

In Fig. 4, we report the cost savings of the DR schemes, normalized to the initial cost, for four different values of the µ parameters: 1 10 , 1 6 , 1 3 , 1. The dashed line represents the saving which could be achieved if users' demand could be rearranged at the provider's will without providing any discount (we indicate it as the dictatorial solution). Consistently with Proposition 4.1, the optimized mechanism returns larger savings than the robust and the base ones. Interestingly, the robust mechanism performs consistently better than the base one despite the fact that it does not require the ability to estimate the demand shifted and it therefore "wastes" some discount by giving it to demand that was already scheduled in a given timeslot in the baseline demand. Moreover, as the population flexibility increases, the savings gap between the optimized scheme and the robust mechanism reduces, the latter being effectively close to exploiting all the population's flexibility. In Fig. 5, we focus on the case µ = 1 3 and analyze the components of the cost for each DR mechanism. Fig. 5 confirms that the optimized scheme provides the largest savings as it can minimize the production cost while paying the smallest amount of discounts. We indicate with wasted discounts the amount of discounts paid to consumption that would in any case have been scheduled in that timeslot. The base and optimized mechanisms do not waste any discount, while the robust mechanism and the broadcast scheme do, as they provide the discount R i to all the electricity consumed in i, including the part of E 0 i that remains in i.

The effect of noise

In this section we evaluate how sensitive the DR mechanisms are to the following two different sources of randomness: 1) the number of users who accept an offer is a random variable and 2) baseline consumption forecasts can be more or less accurate.

Let U be the total number of users. Under the base mechanism, if q j→i U users (or better q j→i U ) are offered the discount R i to move their consumption from i to j, Eq. ( 9) assumes that q j→i U P j→i (R i ) of them accept. In reality the number of those who accept is a Binomial random variable U a i→j Bin( q j→i U , P j→i (R i )). Similar considerations hold for all the mechanisms. About the second aspect, we consider that the energy utility solves any of the specific optimization problems introduced above, starting from the forecast E 0 i of the aggregate baseline consumption for i = 1, • • • N . Let e 0 i = E 0 i /U be the forecast of the individual baseline consumption. We assume that the actual baseline consumptions (ẽ 0 i ) are i.i.d. random variables with expected value E[ẽ u i ] = e 0 i (i.e. the energy utility forecasts are unbiased) and coefficient of variation δ. The actual aggregate baseline consumption is then a random variable Ẽ0 i = u=1 ẽ0 i with expected value the forecast E 0 i and coefficient of variation δ/ √ U . This coefficient captures the uncertainty of aggregate consumption forecasts.

We have then evaluated how the performance metrics change if these two sources of randomness are present, but are ignored by the energy utility in the optimization phase. For the base and optimized mechanisms we maintain that the energy utility can estimate the exact amount of energy shifted between two slots, even if this value is different from what predicted the day ahead.

We considered U = 13.6 millions users (equal to the current estimate of Ontario's population). Simulating the behaviour of each user would have been very time-demanding because we would have had to generate N U random consumptions and roughly N U random choices for the offers. Instead we have approximated by matching the first two moments: i) the Binomial random variables U j→i as gaussian ones and ii) the aggregate consumption of a group of U users as a lognormal random variable as suggested in [START_REF] Mouri | Log-normal distribution from a process that is not multiplicative but is additive[END_REF]. We need then to generate only O(N 2 ) random variables.

We report in Figs. 6 and 7 respectively the final total costs and the cost savings achieved by the DR mechanisms for µ = 1/3 and different values of the aggregate relative forecast uncertainty δ √ U . Usually the consumption forecast errors over large populations are evaluated to be of a few percents. Results are averaged over 10 5 realizations of the set of random variables.

Fig. 6 indicates that costs increase with uncertainty for both the DR mechanism and the idealized schemes. This feature is explained by the Jensen's inequality as the cost functions ( 16), (3), ( 12) and ( 23) are convex in the aggregated forecast E 0 . The figures confirm the relative ranking of the DR mechanisms shown in Fig. 4, but for the base mechanism that appears to be the most sensitive to the effect of noise and becomes worse than the broadcast mechanism already for values of δ/ √ U as small as 0.2%.

Discussion

The models in this paper are based on the microeconomic theory of utility maximization and consumer rationality: users weigh expected costs and benefits and choose the most beneficial actions. This assumption is quite common in the field (see for example the literature overview in [START_REF] Barbato | Optimization Models and Methods for Demand-Side Management of Residential Users: A Survey[END_REF][START_REF] Xu | Optimization Models and Algorithms for Demand Response in Smart Grid[END_REF]). In this framework, the possibility to draw quantitative conclusions depends on the availability of accurate aggregate sensitivity functions characterizing how users react to given economic incentives. Experiments have been carried on to this purpose in many countries, like in U.S.A. (Carolina and California), Germany, Switzerland, India [START_REF] Gyamfi | Residential peak electricity demand response-Highlights of some behavioural issues[END_REF]. In particular, some of these studies measure the elasticity of substitution, that has indeed been introduced to quantify load shifting. It is defined as the relative change of the ratio of the peak to off-peak demand, divided by the relative change of the peak to off-peak price. Elasticity varies significantly from one region to another and from one season to another, but in most of the cases the measured values suggest the possibility of major cost reductions, achievable through demand response mechanisms. These empirical studies can be used to start tuning more accurate models like those we consider in this paper. Moreover, once similar pricing schemes are deployed, the system could learn over time better users' models.

At the same time, experimental work has shown that individuals often do not act as the rational users envisaged by microeconomic theory. In particular many cognitive biases affect human decisions and many choices are not consistent with utility maximization forecasts [START_REF] Kahneman | Thinking, Fast and Slow[END_REF]. These observation lead to the rise of the field of behavioral economics. As regards energy use, the authors of [START_REF] Wilson | Models of Decision Making and Residential Energy Use[END_REF] show how users' choices exhibit time inconsistency, framing and reference dependence, bounded rationality and the (unconscious) adoption of different decision heuristics. In general the influence of psychological factors should be considered in the design of new energy policies and pricing schemes. For example, [START_REF] Goulden | Smart grids, smart users? The role of the user in demand side management[END_REF] advocates that decentralised energy generation and control can transform the users from simple energy consumers to energy citizens and simultaneously make them more responsive to demand response mechanisms. These conclusions are based on data from focus groups in Great Britain, and they are also supported by a similar investigation in Croatia [START_REF] Mesarić | Exploring the Potential of Energy Consumers in Smart Grid Using Focus Group Methodology[END_REF]. Another survey in New Zeland [START_REF] Gyamfi | Price, environment and security: Exploring multi-modal motivation in voluntary residential peak demand response[END_REF] confirms that users are not only price driven: their response is equally sensitive to supply security considerations (power grid congestion can increase the blackout risk) and slightly less to CO 2 emissions (from diesel generators activated to face peak demand). Overall, we see that this line of works suggests solutions that are complementary to ours: once users are sensitised to safety and envi-ronmental issues and are made aware of the large picture and involved as much as possible in all the phases of the energy cycle-as suggested by [START_REF] Gyamfi | Price, environment and security: Exploring multi-modal motivation in voluntary residential peak demand response[END_REF][START_REF] Goulden | Smart grids, smart users? The role of the user in demand side management[END_REF][START_REF] Mesarić | Exploring the Potential of Energy Consumers in Smart Grid Using Focus Group Methodology[END_REF], economic incentives have still an important role and our work can contribute to correctly design them.

Conclusions

In this paper, we have shown that macroscopic descriptions of DR mechanisms can hide important assumptions that can jeopardize the mechanisms' implementation. For this reason, our proposal moved from a microscopic description that explicitly models each user's decision. We have then introduced four DR mechanisms with various assumptions on the provider's capabilities. Interestingly, contrarily to previous studies, we find that the optimization problems that result from our mechanisms are complex and can be solved numerically only through a heuristic. Moreover, our results show that the performance of DR mechanisms under reasonable assumptions on the provider's capabilities are significantly lower than those suggested by previous studies, but that the gap reduces when the population's flexibility increases.

The derivative versus

R 1 of the cost in R = 0 is ∂cost base ∂R 1 R=0 =R 1 E 0 2 S 2 (R 1 , t) + E 0 2 S 2 (R 1 , t) + R 1 E 0 3 S 3 (R 1 , t) + E 0 2 S 3 (R 1 , t) + β 1 (E 0 2 S 2 (R 1 , t) + E 0 3 S 3 (R 1 , t)) -β 2 E 0 2 S 2 (R 1 , t) -β 3 E 0 3 S 3 (R 1 , t) = = β 1 E 0 2 e -R1/α2 α 2 |2 -1| + β 1 E 0 3 e -R1/α3 α 3 |3 -1| -β 2 E 0 2 e -R1/α2 α 2 |2 -1| -β 3 E 0 3 e -R1/α3 α 3 |3 -1| = = β 1 α 2 + β 1 α 3 - β 2 α 2 - β 3 α 3 = -5 < 0,
then it is convenient for the energy provide to increase the reward R 1 .

The second derivative (in R = 0) is timeslot-dependent costs c j (•)

∂ 2 cost base ∂R 2 1 R=0 =E 0 2 S 2 (R 1 , t) + R 1 E 0 2 S 2 (R 1 , t) + E 0 2 S 2 (R 1 , t) + E 0 3 S 3 (R 1 , t) + R 1 E 0 3 S 3 (R 1 , t) + E 0 3 S 3 (R 1 , t) + β 1 E 0 2 S 2 (R 1 , t) + β 1 E 0 3 S 3 (R 1 , t) -β 2 E 0 2 S 2 (R 1 , t) -β 3 E 0 3 S 3 (R 1 , t) = = 2 e -R1/α2 α 2 + 2 e -R1/α3 α 2 -β 1 e -R1/α2 α 2 2 -β 1 e -R1/α3 α 2 3 + β 2 e -R1/α2 α 2 2 + β 3 e -R1/α3 α 2
c j (E) =    100E j = 1 10E j = 2 1E j = 3
and a maximum discount rate B = 20; finally assume discomforts are formulated as in (27

) with P r(β j ≤ R) = 1 -e -R 6 (C.1)
Electricity provider can reduce the cost it carries by shifting consumption in timeslots 2 and 3 where there are zero demands and lower marginal production costs. We report in figures C.8 and C.9 respectively the rearranged consumptions and the minimizers of the optimized and broadcast schemes. We observe that the optimized mechanism offer to all the users to move in to slot 2, while the broadcast mechanism is able to have some of them moving to the more convenient slot 3. The final cost of the optimized scheme (12-15) is 311 and is larger than the final cost of the broadcast one (23-24), 286. The broadcast scheme is able to outperform the optimized scheme, because users can choose between two different offers and then the more flexible ones move to slot 3 and the less flexible ones to slot 2. On the contrary, in order to move some consumption to slot 3, the optimized mechanism should offer to a fraction q 1→3 to move, but those who do not accept the offer would stay in 1 with corresponding elevated costs. Observe that this example has been built in such a way that the broadcast mechanism does not "waste" rewards paying for consumption that does not move.

Note that, due to Proposition 4.1, in this scenario the broadcast scheme allows larger savings than the robust mechanism and the base scheme.

Example Appendix C.2. We show here that the robust mechanism [START_REF] Xu | Optimization Models and Algorithms for Demand Response in Smart Grid[END_REF][START_REF] Wilson | Models of Decision Making and Residential Energy Use[END_REF](18)(19) can perform better than the broadcast scheme (23-24). Consider the following three timeslots scenario: The broadcast scheme assigns only a reward R 1 to the first timeslot. Flexible enough consumption will be moved from timeslots 2 and 3. Notably, the fraction of consumption shifted from timeslot 2 is twice larger than the fraction of consumption shifted from 3 because the discomfort is linear in the distance (see Eq. ( 27)). The broadcast scheme cannot alter this proportion, though it could be beneficial as in timeslot 3 there is a larger demand. It then achieves an unbalanced consumption in slot 1 and 2

On the contrary the robust mechanism can equalize consumption in slots 1 and 2 by offering to a small fraction of the population to move to slot 2.

Observe that, because of Proposition 4.1, in this scenario also the optimized mechanism returns larger savings than the broadcast one.

Example Appendix C.3. In this final example, we show that the base scheme can perform better than the robust mechanism and broadcast scheme. Consider a two timeslots scenario with initial consumption In this scenario all the schemes the robust and broadcast mechanisms are equivalent: they both shift the same amount of consumption to slot 2 and pay the same amount of rewards. The base scheme performs better than the other two since it does not waste any discount, i.e. it does not award discount to the demand E 0 2 originally scheduled in timeslot 2. The base and broadcast schemes do

Figure 1 :

 1 Figure 1: Initial consumption E 0 extracted from [6].The red lines represent the capacities C 1 and C 2 .

Figure 2 :

 2 Figure 2: Final consumption in GWh for the four DR mechanisms (µ = 1 10 ). The black lines represents respectively C 1 and C 2 .

Figure 3 :

 3 Figure 3: Minimizers for the four DR mechanisms (µ = 1 10 ).

Figure 4 :

 4 Figure 4: Cost savings normalized to the initial cost, for various flexibility parameters µ.

Figure 5 :

 5 Figure 5: Analysis of the components of the cost savings. All the quantities are normalized to the initial cost.

Figure 6 :

 6 Figure 6: Global cost versus relative uncertainty δ √ U (%).

Figure 7 :

 7 Figure 7: Savings versus relative uncertainty δ √ U (%). Savings are normalized by the initial cost.

Figure C. 8 :

 8 Figure C.8: Initial and final consumptions in example Appendix C.1.

Figure C. 9 :

 9 Figure C.9: Optimized and broadcast schemes minimizers in example Appendix C.1.

Figure C. 10 :

 10 Figure C.10: Initial and final consumptions in example Appendix C.2.

9 E

 9 ∈ [9 , 18] 36 E ∈ [18 , 27] 78 E ≥ 27 and a maximum discount B = 10. Discomforts as formulated in (27) with P r(β j ≤ R) = R B We report in figures C.10 and C.11 respectively the rearranged consumptions and the minimizers of the robust and broadcast schemes. The final cost of the robust mechanism (16-19), 580.75 is smaller than the final cost of the broadcast scheme (23-24), 594.

Figure C. 11 :

 11 Figure C.11: Robust mechanism and broadcast scheme minimizers in example Appendix C.2.

2 a

 2 timeslot independent cost c(•), such thatc(E) = 10E E ≤ 7 15E -35 E ≥ 7a maximum discount rate B = 10 and discomforts formulated as in (27) withP r(β j ≤ R) = R BIn this scenario the robust mechanism and the broadcast one achieve a cost equal to 154.75, larger than the cost of the base mechanism, 152.92. We report in figures C.12 and C.13 respectively the rearranged consumptions and the minimizers of the three mechanisms.

Figure C. 12 :

 12 Figure C.12: Initial and final consumptions in example Appendix C.3.

Figure C. 13 :

 13 Figure C.13: Robust mechanism, optimized and broadcast schemes minimizers in example Appendix C.3.

Acknowledgements

This work was partly funded by the French Government (National Research Agency, ANR) through the "Investments for the Future" Program reference #ANR-11-LABX-0031-01 and ANR-15-IDEX-02. We acknowledge funding from the Alexander von Humboldt Foundation.

waste some discount and therefore offer smaller discounts than the base scheme, rearrange smaller amount of demand and return a larger cost.

Appendix A. Problem (3-4) is not convex

We prove the problem is not convex by providing the following counterexample:

• N = 3 slots.

• Initial consumption in each slot:

• Sensitivity functions to rewards:

then the function is not convex. Moreover, this non-convexity is at a point where the derivative of the total cost is negative.

Appendix B. Proof of discontinuity of Broadcast mechanism cost function

Suppose the provider broadcasts discount rates R such that

The consumption shifted out of j will then be equally divided between the slots j + k and j -k. In formulas, from (22) follows:

where is a positive. Then all the consumption shifted out of j will be moved to j + k. From (22) follows

A positive arbitrary small generate a finite change in probabilties P j→i . Hence P j→i (R) are discontinuous in R which is why the final cost ( 23) is discontinuous.

Appendix C. Mechanisms ranking

We start proving proposition 4.1, i.e. that the optimized mechanism achieves a lower cost than the robust and the base mechanisms.

We first observe that the base mechanism solves the same problem solved by the optimized mechanism, but with the additional constraint that fractions q j→i have to be constant as indicated in Eq. [START_REF] Song | Demand Side Management in Smart Grids Using a Repeated Game Framework[END_REF]. The feasibility set of the base mechanism's optimization problem is then a subset of the feasibility set of the optimized mechanism's optimization problem. It follows that the optimized mechanism achieves a lower cost.

We now consider the robust mechanism, and assume that the pair (R rob. , q rob. ) is an optimizer of (( 16)-( 19)). Choose the pair (R opt , q opt ) such that:

which implies that ∀(j, i) s.t. j = i, P j→i (R i )q i = P j→i (R j→i )q j→i and set (R j→j , q j→j ) = (0, 0) ∀ j = 1, . . . , N

The two configurations (R rob. , q rob. ) and (R opt , q opt ) generate the same final distribution E 1 . However while the robust mechanism awards with a discount R i a fraction q i of the consumption E 0 i originally scheduled in i, the optimized scheme does not. So the optimized scheme can achieve the same distribution provided by the robust mechanism paying less rewards. It follows that the total cost achieved by the optimized mechanism is never larger than that of the robust mechanism. This last point completes the proof.

The ranking cannot be extended further as we show through the following examples.

Example Appendix C.1. We show first a case where the broadcast scheme (23-24) provides larger savings than the optimized one [START_REF] Goulden | Smart grids, smart users? The role of the user in demand side management[END_REF][START_REF] Mouri | Log-normal distribution from a process that is not multiplicative but is additive[END_REF][START_REF] Mesarić | Exploring the Potential of Energy Consumers in Smart Grid Using Focus Group Methodology[END_REF][START_REF] Ieso Ontario | Market data[END_REF]. Consider a three timeslots scenario with initial consumption E 0 = 10 j = 1 0 j = 2, 3


Marc Calvini-Lefebvre

Pacifists, Patriots and the Vote: The Erosion of Democratic Suffragism in Britain During the First World War

come

Review of Pacifists, Patriots and the Vote: The Erosion of Democratic Suffragism in Britain During the First World War, by Jo Vellacott

Vellacott is particularly concerned with the second of these questions. One answer, to which she previously subscribed, was that the movement had no choice: in the context of war, no better deal could be achieved. But on closer inspection, she argues here that a better deal was, in fact, possible. The political conditions were favourable: cross-party agreement that reform was necessary, a pro-suffrage House of Commons, the end of militancy, widespread admiration for women's war work, and a government keen to see the Reform Bill passed. Furthermore, Sir John Simon, a Liberal member of the Speaker's Conference, tasked by the government to advise on electoral reform, urged suffragists to take the Speaker's recommendation of an age limit of 30 or 35 for women as a baseline from which to push for more. Vellacott, while clear that this is suggestive, not conclusive evidence, maintains that '[h]ad there been a drive for a wider women's franchise, it is hard to imagine that the Bill would have been abandoned to keep the women out' (p. 164). So why was such a drive nowhere to be seen? The principle reason, she contends, was the Great War's impact on the 'nature of the feminism that prevailed' in the NUWSS (p. 1).

Drawing on her extensive knowledge of the NUWSS's minutes, methods and members, Vellacott identifies two types of suffragists in its midst. Traditional suffragists, animated by a feeling of entitlement to rule over their social inferiors and pride in the empire, tended to support the Conservative Party, and were predominantly present in the south of the country, not least in their principal power base, the London Society for Women's Suffrage (LSWS). By contrast, democratic suffragists treated their social inferiors as equals (or at the very least believed they had a responsibility to them), were critical of the empire, supported the Liberal or Labour Parties, and were predominantly present in the north. Where traditionalists only wanted the vote on the same terms as men, democrats championed adult suffrage. On the eve of war, the latter had been in charge of the NUWSS for four years, after the adoption of a new constitution had loosened the LSWS's grip on the executive and brought in new women from the periphery. Under their leadership the NUWSS grew stronger by establishing links with the Labour Party and working-class women and men. Despite their success, however, by July 1915 they had not only lost power to the traditionalists, but had left the NUWSS altogether. Why and how this happened is the subject of chapters three to eight, while the consequences for the prospects of a broader women's franchise provide the material for chapters 10 to 12.

Taking us to the heart of the NUWSS's internal politics, Vellacott meticulously retraces the unfolding of events between August 1914 and June 1915, as matters increasingly came to a head between the two feminisms. The democratic suffragists, arguing that there was a deep connection between militarism and women's oppression, pushed for the NUWSS to embark on a campaign to educate public opinion on the issue of peace. Traditional suffragists countered both that militarism abroad was more dangerous than militarism at home and that any such campaign would be, in Strachey's words, 'the ruin of suffrage' (quoted p. 86) for it would be perceived as unpatriotic. Their disagreements were resolved through two councils, in February and June 1915 respectively. Vellacott's almost thrilling accounts of the motions and amendments that were mooted, carried and defeated at these councils are to be commended. The ambiguous role of Fawcett, torn between her patriotism and her profound affection and admiration for the democratic suffragists (the patriotism won out); the paranoia of the patriots about the pacifists' intentions; the numerous tactical mistakes of the latter; and the abuse of the NUWSS's democratic procedures by the LSWS in order to guarantee victory, are all vividly portrayed.

In the process, two important secondary themes of this book emerge: that democratic processes are essential but fragile; and that behind their appearance of inevitability, most political victories depend on the accidental, on small things that tip the scales, on interpersonal chemistry, human fallibility and meanspiritedness. Thus, Vellacott convincingly argues that the patriots' rout of the pacifists was less a reflection of the balance of power across the NUWSS than of the way in which the former used and abused the organisation's internal politics to their advantage (for example, refusing to move the date of the second council despite knowing that a number of leading pacifists could not attend). The pacifists, meanwhile, committed a series of blunders (such as resigning too soon) that weakened their position. What emerges is that the pacifists had a broad following in the NUWSS and that anti-war sentiment went deeper and wider than was previously thought. The sense of loss, of missed opportunity, is well rendered.

Nevertheless, the NUWSS was left in the hands of the patriots, with three dire consequences for the prospect of a broad women's franchise. First, those most likely to fight for such a franchise were no longer in a position to do so. This is the 'erosion' of the book's subtitle. 'Dispersal' is perhaps a more telling term, for as Vellacott shows, the democratic suffragist vision was not so much undermined as dislodged from its principal power base. As a result it found its energies dispersed among a number of other, less influential organisations. This dispersal had begun before the LSWS takeover as the frustrated pacifists, energised by the Hague Congress, created a new feminist structure: the Women's International League (WIL), to which trades unionists, Quakers, Liberals and well-known writers such as Olive Schreiner rallied. Yet the WIL was only one of many organisations that sprang up to give voice to dissidents in Britain, and the attraction of these organisations caused democratic suffragists to drift imperceptibly away from one another. Furthermore, the WIL had the disadvantage of being new, weak and associated in the public mind only with peace. So when the suffrage issue reappeared on the political agenda, the pacifists worked through a different organisation, the National Council for Adult Suffrage (NCAS). The NCAS was persistently engaged in education work on the question of suffrage, paid attention to those left out of the Speaker's recommendations and listened to Simon. But it had no power base to speak of and was no match for the combined indifference to adult suffrage of the NUWSS and the Labour Party.

Indeed, the second consequence of the NUWSS split was that the best connected and most influential nonmilitant organisation in Britain was now dominated by a small, sociologically and ideologically homogenous group with no interest in adult suffrage and little regard for democratic process. Here Vellacott decisively undermines the prevalent view that the patriots' victory had kept the NUWSS 'a broad church'. To her, this conclusion is 'rendered indefensible by its failure to take into account class and geographical factors' (p. 95). Indeed, after the June council, the NUWSS was controlled by 'a small group of middle-and upper-class Londoners, ... none of whom could speak for the wide constituency of working women and men touched by the pre-war campaign' (p. 95). This group's lack of interest in a broad franchise was compounded by the toplevel advice it was given by most insiders, bar Simon (but including the Prime Minister), that pushing for anything more than the Speaker's recommendations would cause the loss of women's suffrage altogether. Thus, without going to council first or consulting with the constituencies, the executive decided to make a major change to NUWSS policy and accept an age limit for women rather than hold out for the vote on the same terms as men.

The final consequence of the patriots' victory was the quiet severing of the ties built up between organised feminism and organised Labour in the run up to the war. This left the most likely champion of adult suffrage -the Labour Party -free not to push for a better deal for women when the suffrage question reappeared. A first casualty of the new executive was the Election Fighting Fund (EFF), a policy set up in 1912 to bring financial and organisational support to Labour candidates in three-cornered contests following the party's pledge to reject franchise reforms that excluded women. Supported by a strong majority in the NUWSS executive that included Fawcett, it had nonetheless been a controversial policy in those societies with close links to the Liberal Party. It was now all but abandoned by an executive whose 11 new members came 'from or [were] connected with the relatively few societies which had never been comfortable with the EFF policy' (p. 93). Relations with the working class were further undermined by the change in policy regarding women's war work. At the outbreak of war, the democratic suffragist-dominated NUWSS embarked on relief work that sought to alleviate women's suffering caused by war, particularly unemployment. The LSWS, by contrast, sought ways to release more men for the front by putting women in their place. Its priority, Vellacott contends, was less the welfare of women workers than the needs of the war machine. Thus, under its new leadership, the organisation became more confrontational towards the trades unions, tended to reproduce the classist assumptions (i.e., the poor in manual jobs, the rich as overseers) that dominated the attribution of jobs to women, and showed some interest but little commitment to protecting women workers' rights. Let off the hook by this estrangement, the Labour Party gave in to its fear of losing manhood suffrage and to the rising sexism among its male rank and file, embittered by women's wartime encroachment into the workplace. Vellacott points out that there was something rather self-defeating about this position: after all, the women who got the vote were likely to vote Conservative, those who did not would have voted Labour. Calling for more research into this area, she suggests that the long term consequence for the Labour Party was to turn it away from 'the potential of an integrative feminism towards an exclusionary and even patriarchal emphasis on the class struggle' (p. 178).

There is no doubt that Vellacott's fresh take on how the vote was won enriches our understanding. In proposing her counterfactual reading, however, it is arguable that she makes too much of Simon's advice -he appears to have been a rather lonely voice among insiders. She may also be too optimistic in thinking that a broad franchise was likely before the guns began blazing in August 1914. Nevertheless, by insisting on the importance of political rapports de force, her book offers an illuminating and, I would argue, a complementary counter-point to the latest studies, which have given more weight to the war's impact on cultural understandings of gender and citizenship in explaining the winning of the vote for women. [START_REF]The Blood of Our Sons': Men, Women, and the Renegotiation of British Citizenship During the Great War[END_REF] In addition to exploring women's enfranchisement, Vellacott compares the different responses to war that came out of the women's suffrage movement and evaluates them as forms of feminism. This too is an area that her rediscovery of the anti-war suffragists had prompted her to explore in a previous, and controversial, article. If some suffragists had opposed the war on feminist grounds, she argued, then this raised awkward questions about the feminist credentials of those who had not. On her reading, what the different responses revealed was a division in the movement between those 'whose suffrage claims still rested mainly on a demand to be admitted to the existing male political system, unchanged' (4) and those who wanted the vote in order radically to change that system. The latter developed a 'sophisticated feminist view of the war' [START_REF] Vellacott | Feminist consciousness and the First World War[END_REF] and opposed it; whereas the former, from the militant Emmeline Pankhurst to the constitutionalist Fawcett, did no such thing, falling unselfconsciously into the support roles expected of women in wartime. This relationship between attitudes to suffrage and attitudes to war, she continued, should give pause to contemporary feminists. In particular, it should lead them to dislodge the militants from their privileged position in the history of feminism in favour of those who had consistently rejected violence both in pursuit of the vote and in opposition to the war. Subsequent scholarship has challenged her position and in this book she both defends and clarifies it.

The principal attack on Vellacott's argument was mounted by Susan Kent who described the actions and ideas of the pacifists as grounded in the essentialist (i.e. un-and possibly even anti-feminist) language of motherhood. To her, far from being inspirational foremothers, these women, by accepting the ubiquitous wartime dichotomy of women-as-carers/men-as-killers, had largely contributed to taking the British women's movement down the blind alley of 'difference' in which it remained terminally stuck until the Second World War and to which it must not, on any account, return.( 6) Denouncing an 'ahistorical theoretical analysis that makes no sense in the context of the time' (p. 31), Vellacott counters that 'we need to distinguish between vocabulary and action, and to situate the discourse in the context of the time' (p. 48). While I agree with her critique of Kent and her emphasis on contextualisation, I find this proposed action/rhetoric dichotomy problematic. Indeed, action and rhetoric exist not as two independent fields but in a symbiotic relationship. Those feminists launched into relief and pacifist work precisely because of their belief in women's special insights into the world provided by their biological and social roles as mothers. That they did not think motherhood was the only role for women (p. 32) is beside the point: it is the role that matters, the role from which their claim to transform the political arises. In other words, we cannot focus exclusively on what they did and treat what they said as a sign of the times that can be safely ignored. That we might want, as Vellacott justifiably does, to build a case similar to theirs in less essentialist terms is one thing, but we do a disservice to our historical understanding if we pretend that what they said and what they did were not intimately intertwined.

In addition to defending the pacifists' feminism, Vellacott offers a clarification of her thesis. Indeed, although never explicitly accusing the pro-war group of being non-or anti-feminist, her argument could be read as such. Any ambiguity is here lifted: '[i]t was of course not by definition anti-feminist to support the war effort' (p. 32). This formulation advantageously takes the heat out of the discussion of the split in the NUWSS and allows for an illuminating comparison, as we have seen, of the two feminisms' contrasting attitudes to war, Labour and suffrage. That said, I have found her reading of the pro-war suffragist line, at times, to be unduly close to that of the democratic suffragists' and wonder if a more sympathetic (if not less critical) reading could not have been given. Take, for example, Fawcett's bitter disagreement with pacifist Helena Swanwick over the NUWSS's attitude to the war. Swanwick argued that suffragists must oppose the war because they had always opposed the idea that politics, whether national or international, rested on force. Vellacott describes Fawcett's answer, grounded in 'a virulent attack on German philosophy, policies and practices', as 'irrelevant' (p. 50). I have not read the document in question but I suspect that Fawcett's argument may have been that it was precisely because suffragists were opposed to the use of physical force and in favour of the rule of law that they should oppose Germany, a country whose philosophers claimed that might is right, whose policy had been to violate international law by invading neutral Belgium, and whose war practice was to use force even against civilians. These may be poor arguments, but they were not irrelevant to the question and often recurred in the animated debates between pacifists and patriots that rocked the correspondence pages of the NUWSS's organ, Common Cause.(7) It would be unfortunate if Vellacott's important work of recovery of the pacifists led to an obscuring of our understanding of their opponents' views.

Which brings me to Vellacott's treatment of Emmeline and Christabel Pankhurst's response to the war. Within her clarified framework, their Hun-hating brand of patriotism and staging of military recruitment campaigns continue to fail to make the feminist cut, either in rhetoric or action. Thus, where Jacqueline de Vries suggested that their gendered interpretation of the war, whereby the 'feminine' values and qualities of Britain and her allies were pitted against the 'masculine' ones of Germany, was arguably feminist, Vellacott detects only a 'feminist theoretical flavour', noting that 'the representation of Germany as the male bully and Belgium as the assaulted maiden was also much used by both government and press for pro-war propaganda' (p. 21). This, however, misses the point: the government and press did not use these gendered images to go on to argue that this was a war for women's freedom, or use the deeply gendered language of war to argue for women's right to citizenship. As for their actions, Vellacott stresses their striking similarity with those of the leading anti-suffragist Mrs Humphrey Ward, and argues that they amounted to 'conforming to the role expected of women in wartime' as 'docile supporters of men's wars' (p. 48). Now, they were certainly supporters, but they were far from docile. There are chilling editorials from Christabel Pankhurst calling for the country to spare none of its male blood in this holy war. [START_REF] Pankhurst | War service for all[END_REF] Nor did they see it as the men's war. On the contrary, they continuously insisted that this was women's war above all because of the threat that Germany, a nation that privileged men and masculinity over women and femininity, posed to women's freedom. The point with political arguments is not so much what you say, as Vellacott rightly claims when defending the pacifists, but what you are trying to do in saying it. [START_REF]For the sophisticated version of this point, see Meaning and Context[END_REF] The pacifists were trying to use the widespread association of mothering with peace to make a case for women's active and immediate involvement in a transnational peace campaign. The Pankhursts were trying to use the widespread association of Germany with the oppression of women to make a case for women's involvement as active political actors in the war effort and the polity.

The cumulative effect of Vellacott's discussion is to frame the Great War as dealing a nasty blow to British feminism's historical evolution. This marks a sharp turn in her thinking. Her earlier work, written in the context of Greenham Common Women's Peace Camp, unsettled the dominant narrative of pre-war apogee and post-war decline of British feminism by claiming that 1915, the year of the Women's Peace Congress and of the publication of some of the most striking pacifist feminist writings, should now be seen and celebrated as a landmark in the annals of feminism. [START_REF]Militarism versus Feminism: Writings on Women and War / Catherine Marshall[END_REF] Writing in the context of a global 'war on terror' which feminists have criticised but not (so far) effectively countered, and no longer focusing exclusively on ideas and their legacy but taking in the history of the organisation as well, her assessment is more sombre. But it is not all despair. For if the Great War 'eroded' the democratic suffragist vision, it did not obliterate it. The book's ultimate message, then, is one of hope. Even in the darkest of hours, feminists have fought, if clumsily, for peace. Even when no one was listening, as during the inter-war period mentioned briefly in the epilogue, they have articulated a powerful alternative vision for the conduct of international relations. Calling on feminists to reclaim that vision in the 21st century, the book ends with one final familiar and important theme of her work: the use of history to re-open urgently needed avenues of political thinking.

Notes