strategies that, both in theory and in practice, can cause significant problems when adopted by many agents. In this section we discuss some possible strategies.
In general, the power balancing mechanism results in an incomplete in- formation game. This means that each agent knows how the mechanism works and the type of functions that it allows, but not necessarily the num- ber of other agents or their types. Some information, for example, a distri- bution over the types of agents that might be present, will be needed to make sensible strategic decisions. In general, only knowing how the mechanism works is not enough.
For example, consider the task of developing a pure strategy for an agent that has a fixed power that is non-zero at the very first time step of the horizon. The agent must be honest about the first time step, but they can lie in future ones. We find that for any strategy that involves lying about what the agent can achieve in a future time step, we can always construct another agent that will actually increase the costs for the fixed power agent relative to their truthful strategy (see lemma 3 in appendix B).
As agents gain information about the other agents in the market, they may start to form mixed or pure strategies that can reliably manipulate the mechanism. They may also decide to form coalitions of agents that work together to have a larger impact. The exact formation of such coalitions and any equilibria in the mechanism will be strongly dependent on what agents are present, and the nature and amount of information that is known. We leave it to future work to investigate such Bayes-Nash equilibria in incom- plete information cases.
Instead, in this chapter we focus on a simple concrete case where a single “greedy” agent has complete information about all other agents.
5.6
Greedy Agent Strategy
The “greedy” agent we develop has complete information about all other agents (which act truthfully) and optimises its costs by manipulating the mechanism. This provides it with the best possible chance of manipulating the mechanism.
This section describes a general algorithm for calculating an optimal strategy for such a greedy agent. This is an artificial scenario designed to establish the worst case harm that a single agent (or coalition of such agents working together) can cause. Later on in the chapter we will demonstrate techniques for identifying untruthful agents such as this greedy one.
The approach relies on solving a bilevel program, where the greedy agent at the top level chooses its optimal preferences given that it knows how the mechanism will respond at the lower level [Migdalas et al., 1998]. In general,
even linear bilevel programming is NP-hard [Migdalas et al., 1998, chapter 6]. This will dramatically limit the size and type of problems that can be solved optimally. We will use bounds and solve simple instances in order to gain insights despite the complexity.
The algorithm is formulated for the TRH problem. The greedy agent has index i = 1 and it knows the preferences of all other agents who act truthfully. The greedy agent can lie about their preferences by fixing their power consumptions to fixed values during each horizon. There is no un- certainty: for all h let fh,i =fi and gh,i,j =gi,j. We assume that there are
market cap prices ¯
λ, ¯λand that the greedy agent will avoid a solution which violates these because of the risk of being shed from the network.
We make the optimistic bilevel assumption, which allows the greedy agent to choose between lower level solutions if more than one exists for a given upper level decision. This is the same as allowing the greedy agent to choose between KKT points of the lower level mechanism. This allows the problem to be immediately flattened into a single level problem:
min Ph,i,λh f1(PT ,1) +λTTPT ,1 (5.27) s.t. g1,j(PT ,1)≤0 ∀j ∈ {1, . . . , N1} (5.28) Ph,i,t =Ph−1,i,t ∀i∈ {1, . . . , A}, h∈ {2, . . . , T}, t < h (5.29) λh,t =λh−1,t ∀h∈ {2, . . . , T}, t < h (5.30) Ph,1∈[ ¯ P1,P¯1]T ∀h∈ {1, . . . , T} (5.31) λh ∈[ ¯ λ,λ¯]T ∀h∈ {1, . . . , T} (5.32) (Ph,1, Ph,2, . . . , Ph,A, λh)∈KKTh ∀h∈ {1, . . . , T} (5.33)
The optimisation problem is over the powers for all agents and the prices in all horizons. The objective is to minimise the actual costs of the greedy agent (i= 1) at the end of the TRH problem. This is given by the agent’s true cost functionf1 and monetary payments. The powers and prices in the last
horizon (h=T) are used for these calculations, because at this point they are fully finalised (in earlier horizons some of these are still hypothetical).
The constraints for the agent are enforced on the final true powers for the greedy agent with inequality (5.28). Equations (5.29) and (5.30) tie together powers and prices between horizons (those that have been finalised), in accordance with the TRH formulation. Constraints (5.31) and (5.32) limit the greedy agent’s power values and the prices according to the limits discussed in section 5.4.4.
KKTh is defined to be the set of feasible KKT points for the problem
when for all time steps wheret < hthe prices and powers are fixed to some arbitrary values, and for all time steps the powers of agent 1 are fixed to some arbitrary values. For a givenh, constraint (5.33) can be expanded into
5.6. GREEDY AGENT STRATEGY 99 the constraints: A X i=1 Ph,i,t = 0 ∀t≥h (5.34)
and for alli∈ {2, . . . , A}:
∇tfi(Ph,i) + Ni X
j=1
µh,i,j∇tgi,j(Ph,i) +λh,t = 0 ∀t≥h (5.35)
gi,j(Ph,i)≤0 ∀j∈ {1, . . . , Ni} (5.36)
µh,i,jgi,j(Ph,i) = 0 ∀j ∈ {1, . . . , Ni} (5.37)
µh,i,j ≥0 ∀j∈ {1, . . . , Ni} (5.38)
The stationarity (5.35) and complementary slackness (5.37) equalities and the multiplication of λT in the objective (5.27) are all possible sources of
non-convexity. We will now further develop our three devices so that they can be used in this formulation as one of the truthful agents.
5.6.1 Generator
The KKT conditions for a truthful generatori can be reduced to∀t≥h: 2ψi,tPh,i,t+µuh,i,t−µlh,i,t+λh,t= 0 (5.39)
Ph,i,t ≤0 (5.40) ¯ Pi−Ph,i,t≤0 (5.41) µuh,i,tPh,i,t = 0 (5.42) µlh,i,t( ¯ Pi−Ph,i,t) = 0 (5.43) µuh,i,t≥0 (5.44) µlh,i,t≥0 (5.45)
The generator has upper and lower power bounds at each time step, for which we associate the dual variablesµuh,i,tandµlh,i,t. The time index is used to indicate which time step the dual variables are for. This is reformulated as a series of mixed-integer linear constraints by introducing binary variables
zh,i,tu and zh,i,tl , and combining the KKT multipliers. The complementary slackness requires thatµuh,i,t >0 =⇒ µlh,i,t= 0 and µlh,i,t >0 =⇒ µuh,i,t = 0. This means that we can replace each pair of upper and lower bound KKT multipliers by a single multiplier,νh,i,t∈R, whereνh,i,t =µuh,i,t−µlh,i,t. Big-
M style constant bounds
¯νi,t and ¯νi,t are used for these new multipliers. The reformulation is∀t≥h:
2ψi,tPh,i,t+νh,i,t+λh,t = 0 (5.46)
νh,i,t ≤zh,i,tu ν¯i,t, Ph,i,t ≥(1−zh,i,tu )¯Pi (5.47)
The market price caps and the stationarity condition provide bounds for the multiplierνh,i,t: ¯νi,t =−¯λ (5.49) ¯ νi,t =− ¯ λ−2ψi,t ¯ Pi (5.50) 5.6.2 Deferrable Load
For a truthful deferrable load, the KKT conditions can be transformed into mixed-integer linear constraints in the same way as the generator:
γh,i+νh,i,t+λh,t = 0 ∀t≥h (5.51) T
X
t=1
Ph,i,t=Ei (5.52)
νh,i,t ≤zuh,i,tν¯i,t, Ph,i,t ≥zuh,i,tP¯i,t ∀t≥h (5.53)
νh,i,t≥zh,i,tl ¯νi,t, Ph,i,t≤(1−zlh,i,t) ¯Pi,t ∀t≥h (5.54)
A difference here is the presence of a KKT multiplier γh,i associated with
the energy consumption constraint, which makesνh,i,tunbounded in general.
However, there exist finite bounds which do not cut off any feasible KKT point if we are only interested in the values of Ph,i and λh (which are all
that matter for the overall problem).
If ( ˜Ph,i,˜λh,z˜h,iu ,z˜h,il ,γ˜h,i,ν˜h,i) is a KKT point, then so is ( ˜Ph,i,λ˜h,z˜h,iu ,˜zh,il ,
˜
γh,i+,ν˜h,i−~1), where= ˜νh,i,τ givenτ = arg mint∈{h,...,T}|ν˜h,i,t|, and~1 is
the the appropriate all-ones vector. This means that for any values (Ph,i, λh)
that satisfy the KKT conditions, we can choose the other multipliers in such a way that there exists some τ where νh,i,τ = 0. By cancelling γh,i, the
stationarity conditions require that for all t≥h:
νh,i,t+λh,t =νh,i,τ +λh,τ (5.55)
=⇒ νh,i,t+λh,t =λh,τ (5.56)
Therefore the market price caps provide the bounds:
¯νi,t=¯λ− ¯ λ (5.57) ¯ νi,t =− ¯ λ+ ¯λ (5.58)
5.6.3 Fixed Power Device
The conditions for a truthful fixed power device are trivial∀t≥h: Ph,i,t =
PF,t. When the fixed power device is greedy the problem is also nicely