Resolución del MANOVA.

For a given power balancing problem, we assume that there exists at least one KKT point for each horizon (section 5.4.4 discusses what happens when a point cannot be found). Under certain regularity conditions (constraint qualifications) on the problem functions, a KKT point becomes necessary for optimality [Kuhn and Tucker, 1951, Karush, 1939]. Under other conditions such as convexity, a KKT point is sufficient for optimality [Martin, 1985].

When the functions are continuously differentiable, the KKT conditions for an agenti in horizonh are:

∇fh,i(Ph,i) + Ni X

j=1

µh,i,j∇gh,i,j(Ph,i) +λh = 0 (5.14)

gh,i,j(Ph,i)≤0 ∀j∈ {1, . . . , Ni} (5.15)

µh,i,jgh,i,j(Ph,i) = 0 ∀j∈ {1, . . . , Ni} (5.16)

µh,i,j ≥0 ∀j ∈ {1, . . . , Ni} (5.17)

where µh,i,j ∈ R are the KKT multipliers for the agent constraints and λh ∈ RT is the KKT multipliers for the power conservation constraint.

Particular powersPh,i and pricesλh satisfy the KKT conditions for agent i

if there exists someµh,i,j where this is true.

The KKT conditions for the overall problem in horizonh is the combin- ation of KKT conditions for all agents along with the power conservation constraint:

A

X

i=1

Ph,i = 0 (5.18)

When dealing with convex functions, which might be non-differentiable, we allow a subgradient to be used in place of the gradient. Therefore, if f

is a convex function, then we define ∇f(P) ∈ ∂f(P), where ∂f(P) is the subdifferential (set of subgradients) of f at P. We will explicitly mention when this overloading of the differential operator leads to any ambiguities.

Using this notation the KKT stationarity condition (5.14) remains the same, except we have the implicit assumption that ∇f(P) can be used to represent a subgradient for any convex cost or constraint functions.

5.4. POWER BALANCING MECHANISM 91

5.4.2 Payments

Agents are paid (charged) for their power production (consumption) at the marginal prices λh,t ∈ R. As discussed, these are the KKT multipliers for

the power conservation constraints:

∀t∈ {1, . . . , T}:

A

X

i=1

Ph,i,t= 0 (λh,t) (5.19)

The expected total cost ch,i for an agent i in horizon h is the combination

of their cost function and their expected payments over the horizon:

ch,i:=fh,i(Ph,i) +λThPh,i (5.20)

This is only what is expected in horizon h, because the actual costs will change as solutions are allowed to change in later horizons. Defining the costs like this assumes that agents have quasilinear utilities and are risk neutral (as discussed in section 5.2.2). The power conservation constraint ensures that our mechanism is budget balanced, as all payments between consumers and produces sum to zero for each time step.

In the receding horizon problem we assume that agents must consume or produce the power negotiated in the first time step of each horizon, which (recalling the discussion from section 5.3.2) is wheret=h. The powers and prices beyond the first time step in each horizon are just hypothetical at that point in time.

5.4.3 Convexity

Chapters 3 and 4 discussed how power consumption decisions can be discrete or non-convex in nature, which raises doubt about the applicability of the KKT conditions. However, there are some strong arguments for why in practice convex relaxations or approximations of prosumer device models will produce high quality results:

• Chapter 4 found that the most common source of discrete decisions for houses (shiftable loads) are well approximated by their convex re- laxation when the number of participants reaches realistic levels. • Many new household devices are becoming more continuously variable.

Variable speed compressors used in devices such as air conditioners can provide a more efficient alternative to discrete on/off switched compressors.

• More intelligent control can be used to apply a duty cycle to discrete components to mimic more continuous power levels.

• Household batteries will be able to compensate for the discrete power levels of other devices.

As a result, convex functions can be used with only a small hit to quality. We require, as a condition of participation, that agents reformulate any non-convex cost and constraint functions into a convex approximations or relaxations. Any increase in cost will be more than offset by the benefits of being able to efficiently solve large systems in a distributed manner.

5.4.4 Limits

Power Limits— In order to participate in the mechanism, each agent must

first negotiate a contract which restricts how much power they can consume and supply. This contract establishes limits such that ∀t ∈ {1, . . . , T} :

Pi,t ∈ [

¯

Pi,P¯i]. They can be set at the physical limits of the equipment

connecting the agent to the rest of the network, or to a subinterval of these physical limits if the agent does not need the extra capacity.

It is likely that agents will be provided with a financial incentive to set tighter limits as it provides the market operator with more certainty. A benefit of having these limits is that it restricts the outcomes available to strategic agents. For example, without any limits, a two-bedroom house could indicate to the mechanism that they will provide the network with the equivalent output of a large nuclear power station, which would clearly have serious consequences for the mechanism.

We assume that these limits are enforced in the private constraints of each agent. In practice this would be combined with a simple check by the utility, with adequate penalties that would deter agents from choosing tighter limits than they can meet.

Price Limits— A second set of limits is on the allowed prices λ, which are

commonly referred to as market caps in wholesale electricity markets. This can be set at a point where participants become indifferent towards being disconnected (shed) from the network. If the solver finds a KKT point with costs outside the price caps or if no KKT point is found at all, then the market operator can shed loads in an attempt to find a feasible solution.

Agents have the ability to respond to prices, so they will tend to reduce their loads on their own as the prices increase. The price caps and load shedding by the operator will likely only be needed as a last resort, for ex- ample, if agents do not have enough flexibility or if something goes seriously wrong.

The market operator could choose to shed those agents that are operating outside their normal behaviour combined with some degree of randomisa- tion. Although we do not go into specific approaches to shedding [Concordia and Fink, 1995], it is worth noting that the actions of strategic agents could cause them to be shed if they push the price outside the market caps. This

5.5. MANIPULATION 93 is how the price limits further restrict the possible outcomes for strategic agents.

5.5

Manipulation

This section defines truthfulness, consistency and receding horizon manip- ulation. It then provides some simple examples of manipulation with and without a receding horizon and discusses agent strategies. First some nota- tion is introduced.

When we focus on an arbitrary pair of consecutive horizonsh andh+ 1, we drop the horizon subscript from functions, powers and prices, and instead indicate the later horizon with a prime. For example, the cost functions for agentiin two consecutive horizons are fi:=fh,i and fi0 :=fh+1,i. To make

comparisons easier, we time shift powers and prices from the later horizon so that they line up with the values from the earlier horizon (see figure 5.2). These vectors are marked with an asterisk and are defined as:

P_i,t∗ := ( Ph,i,t if t=h Ph+1,i,t if h < t < h+T (5.21) λ∗_t := ( λh,t if t=h λh+1,t if h < t < h+T (5.22)

We define vectors for the change in the power and price as ∆Pi:=Pi∗−Pi

and ∆λ := λ∗ −λ. We also define the change in cost with respect to the earlier horizon as ∆ci :=fi(Pi∗) +λ∗TPi∗−fi(Pi)−λTPi.

1

2 3

T

2 3

T+1

Figure 5.2: For consecutive horizons, the shaded time steps represent the values used for the time shifted price and power vectors.

As discussed in chapter 3, agents, especially household prosumers, are exposed to a lot of uncertainty that influences their electricity consumption. Instead of modelling this through random parameters, we instead more ab- stractly allow agents to update their cost function and constraints for each horizon. We use a hat to indicate an agent’s best estimate of these functions for a particular horizon ˆfh,i and ˆgh,i,j.

The functions that the agent uses to interact with the mechanism are

fh,i and gh,i,j. When these are the same as their predicted functions, the

Definition 1 (Truthfulness). For a horizon h, let fh,i and gh,i,j be the

functions that agent i uses with the mechanism and ˆfh,i and ˆgh,i,j be the

functions the agent expects are most likely to be true at that moment in time. Agenti is truthful in a horizonh if fh,i = ˆfh,i and for all j :gh,i,j = ˆgh,i,j.

Otherwise agentiis untruthful.

An agent is consistent between consecutive horizons when the preferences they provide in the earlier horizon could have produced the result from the later horizon (for those time steps that overlap). This is more complicated than saying the functions are equivalent in both horizons, because in the RH problem the functions accept powers from different time steps. Formally:

Definition 2 (Consistency). Let the KKT points for the power balancing problem in two consecutive horizons be (Pi, λ) and (Pi0, λ0) for agent i. If

the later horizon time shifted point (P_i∗, λ∗) satisfies the KKT conditions for agent i in the earlier horizon, then agent i is consistent between the horizons. That is, the following conditions must hold:

∇fi(Pi∗) + Ni X j=1 µ∗_i,j∇gi,j(Pi∗) +λ ∗ = 0 (5.23) gi,j(Pi∗)≤0 ∀j ∈ {1, . . . , Ni} (5.24) µ∗_i,jgi,j(Pi∗) = 0 ∀j ∈ {1, . . . , Ni} (5.25) µ∗_i,j ≥0 ∀j∈ {1, . . . , Ni} (5.26)

for some multipliers µ∗_i,j. Otherwise agent i is inconsistent between the horizons.

An agent manipulates the receding horizon mechanism when it is un- truthful in order to create an inconsistent result. Formally:

Definition 3 (Receding Horizon Manipulation). If agent i is inconsistent between consecutive horizons and untruthful in the earlier horizon, then agentiismanipulatingthe receding horizon problem between these horizons.

5.5.1 Examples

This section provides two examples of how untruthful agents can manipulate the power balancing problem. The first example demonstrates how agents can manipulate the SH problem and the second example demonstrates re- ceding horizon manipulation (which is the focus of this chapter) in the TRH problem.

Consider a SH problem instance with a single time step on a network with a fixed load of ˆPL= 5 and two generators (A and B) with true generation

5.5. MANIPULATION 95 Figure 5.3 shows how the outcome of the mechanism changes as generator

A misreports its cost function. The optimal total cost (social optimal) is achieved when it reports truthfully, i.e. ψA = ˆψA = 2; however, the figure

shows that generator Acan reduce its individual costs if it falsely reports a higher cost function. This action fromAalso reduces the costs of generator

B, which is reporting truthfully in this example, with the savings coming at the expense of the load.

1

2

3

4

5

6

7

ψ

A

40

20

0

20

40

60

80

100

120

140

Cost

Gen A Cost

Gen B Cost

Load Cost

Total Cost

Figure 5.3: Change in outcome as generatorA changes its reported cost function. The social optimal occurs atψA= 2, when the generator truthfully reports its cost.

The lack of incentive compatibility in the general case can be insignificant in a particular problem instance if there is sufficiently strong competition, as in other markets [Makowski et al., 1999]. We expect there to be a reas- onable level of competition in future prosumer driven networks, especially in comparison to existing electricity markets.

For the second example that demonstrates receding horizon manipula- tion, consider a TRH problem with two time steps on a network with two deferrable loads (AandB) with ˆEA= ˆEB= 2 (to simplify assume no upper

power bounds) and a single generator with prices in the two time steps of ˆ

ψ1= 4 and ˆψ2 = 2. LoadAcan take advantage of the cheaper prices in the

second time step by discouragingB from consuming at this time.

The results in figure 5.4 show how the overall outcome changes as load

A lies during the first horizon about how much it needs to consume in the second time step P1,A,2. In each instance A reports zero consumption in

by stating truthfully how much power it still needs in the second time step (after which it is too late forB to make a correction).

1.0

1.5

2.0

2.5

3.0

3.5

4.0

P

1, A,2

30

20

10

0

10

20

30

40

50

Cost

Load A Cost

Load B Cost

Gen Cost

Total Cost

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Power

P1

, B,2

Figure 5.4: Change in outcome as deferrable load A changes its reported power consumption for the second time stepP1,A,2. The total costs are equivalent to the

social outcome whenP1,A,2= 2.

As load A increases P1,A,2, load B responds by decreasing its second

time step consumption P1,B,2, which in turn reduces the costs for A. The

outcome whereP1,A,2 = 2 is socially optimal, and is equivalent to outcome

of the mechanism withA acting truthfully.

5.5.2 Strategies

There are many reasons why it might be difficult for an agent to manipulate the mechanism in a practical setting, including the:

• computational intractability of computing a beneficial strategy in real- istically sized settings;

• limited information about the private preferences and constraints of others;

• strong competition between agents;

• need for collusion with other agents with complementary needs; and • rarity of circumstances where a worthwhile benefit can be achieved.

5.6. GREEDY AGENT STRATEGY 97

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