Generation bidding game with potentially false attestation of flexible demand
 Yuquan Shan^{1},
 Jayaram Raghuram^{2},
 George Kesidis^{1, 2}Email author,
 David J Miller^{2},
 Anna Scaglione^{3},
 Jeff Rowe^{4} and
 Karl Levitt^{4}
https://doi.org/10.1186/s1363401502123
© Shan et al.; licensee Springer. 2015
Received: 15 December 2014
Accepted: 25 February 2015
Published: 24 March 2015
Abstract
With the onset of large numbers of energyflexible appliances, in particular plugin electric and hybridelectric vehicles, a significant portion of electricity demand will be somewhat flexible and accordingly may be responsive to changes in electricity prices. In the future, this increased degree of demand flexibility (and the onset of only shortterm predictable intermittent renewable supply) will considerably exceed present level of uncertainty in dayahead prediction of assumed inelastic demand. For such a responsive demand idealized, we consider a deregulated wholesale dayahead electricity marketplace wherein bids by generators (or energy traders) are determined through a Nash equilibrium via a common clearing price (i.e., no location marginality). This model assumes the independent system operator (ISO) helps the generators to understand how to change their bids to improve their net revenue based on a model of demandresponse. The model of demandresponse (equivalently, demandside bidding day ahead) is based on information from loadserving entities regarding their priceflexible demand. We numerically explore how collusion between generators and loads can manipulate this market. The objective is to learn how to deter such collusion, e.g., how to set penalties for significant differences between stated and actual demand, resulting in higher energy prices that benefit certain generators.
Keywords
1 Introduction
Gametheoretic approaches to the study of electricity markets, particularly under deregulation, have been explored for decades [13]. Recently, problems associated with variations of the optimum powerflow (OPF) problem [4,5] have been considered by several authors for priceelastic demand, e.g., [6,7]. Indeed, demand elasticity for electricity is motivated by the onset of potentially enormous load from flexible appliances, particularly plugin electric and hybridelectric vehicles, see, e.g., [8,9] and the references therein, where an electric vehicle represents electricity demand comparable to the rest of the household combined.
In typical models of ‘twosettlement’ (wholesale dayahead and realtime) electricity markets commonly used in practice, dayahead settlements largely account for the realtime supply. In real time, relatively minor adjustments are expected to be made to meet actual current demand. However, significant power supply may need to be quickly secured if there are failures in supply or if realtime (actual) demand is unexpectedly different than that predicted dayahead, e.g., due to weather conditions. On the other hand, excess supply may have been secured if elastic demand is deferred in real time, or (often intermittent and confidently predictable only in the shortterm) renewable supply is employed. So, in the future, actual realtime demand may be additionally difficult to predict dayahead.
To account for significant flexible demand in the dayahead market [10,11], we assume that the loadserving entities (or demand aggregators or just consumers) will inform the independent system operator (ISO) regarding their flexible demand  a kind of demandside bidding. The ISO is assumed to provide sufficient information so that the generators (suppliers) can modify their prices to improve upon their net utility, i.e., the ISO determines generation allocation sensitivities to bid prices of the OPF with demand response. In our setting, the ISO solves an OPF which minimizes the common clearing price of supply to the loads. The game is a ‘discriminatory’ sealedbid auction in that the generators earn at the price they bid but in quantity determined by the ISO [1,2]. So to simplify matters herein, we do not consider strategic bidding by the generators wherein they may infer demand and/or the bidding strategies of their competitors via a probabilistic model, nor multipart bidding to account for startup/rampup costs, secure contracts involving minimum and maximum supply per generator, and the like^{a}, nor peakpower consumption penalties [12,13]. Also, we assume each generator has a continuously differentiable and convex cost of supply [3], quadratic in particular [14]. Our model is related to noncooperative Cournot games of electricity markets reversing the direction of dayahead markets to understand how demand responds to market clearing prices in the long term [15]  here we attempt to understand demandresponse on the wholesale (generationlevel) market.
How an energy trader (or loadserving entity colluding with one or more generators) can manipulate dayahead electricity prices by placing uneconomic demandside bids is studied in [16,17], not considering flexible demand. In this paper, we are interested in how the wholesale dayahead market can be manipulated by colluding generators and loadserving entities, through misrepresentation of flexible demand by the latter.
This paper is organized as follows. We first specify our model of wholesale electricity market under demand response as a noncooperative game  an argument for the existence of its symmetric Nash equilibrium is given in the Appendix. We then use this model to numerically consider the effects of demand misrepresentation for the example of the benchmark IEEE 14bus power system. The paper concludes with a summary and discussion of how such computations can inform penalties of significant discrepancies between attested and actual demands, causing the benefit of certain generators or energy traders.
2 Problem setup
where D _{min} represents inflexible demand. Note that this is the simplest model of aggregatedemand flexibility and is commonly used in the literature. Although other more sophisticated model may be developed to fit cases in reality, such as plugin electric and hybridelectric vehicles, capturing flexibility of realworld aggregate loads is not a contribution of this paper.
i.e., different generators having different a _{ g } parameters.
For noncooperative generator duopoly (twoplayers), discriminatory, singlepart game with no distribution losses or constraints, and assuming that p _{1}≠p _{2} near the interior Nash equilibria, we can find surprisingly complex plurality of Nash equilibria in closed form [18].
We assume that supply allocations are the result of the optimization of a supply network by a linear program. In electricity markets, the retailer (ISO) is sometimes also the distribution system.

Let G _{ b }⊂G be the set of generators on bus b, each having generated power S _{ g }, price per unit supply p _{ g }, and minimum and maximum supply \(S_{g}^{(min)}\) and \(S_{g}^{(max)}\), respectively.

Let L _{ b }⊂B be the set of loads on bus b, each having a demand D _{ l } that depends on the clearing price P.

For each bus b∈B, θ _{ b } is its voltage angle.

Finally, let r _{ i,j } be the branch connecting bus b _{ i } and bus b _{ j }, with x _{ i,j }, the reactance of the branch, P _{ i,j }, the power ‘flowing’ from b _{ j } to b _{ i } (if we neglect power loss on the transmission line, we get P _{ i,j }=−P _{ j,i }), and c _{ i,j }, the maximum tolerable power on the branch.
2.1 Optimal power flow problem formulation
provided the generator price satisfies the condition \(p_{g} \geq 2\sqrt {a_{g} u_{g}^{(\min)}}\). If \(u_{g}^{\min } = 0\), then we observe that as a _{ g } is made larger, i.e., as the cost of supply allocation increases, the maximum supply allocation (or capacity) decreases, and vice versa.
In practice for powertransmission circuits, thermal losses may determine edge (transmission line) capacities and costs, the latter typically in a powerflow dependent fashion, e.g., ‘ I ^{2} R’ losses (Section 3.1 of [4]). In order to focus on the bidding behavior among the generators, we neglect the power loss on the transmission lines; hence, the cost in power transmission is also neglected, as in the DC approximation.
2.2 Setup of generators’ iterative game on a platform of demand response
 1.
The ISO sets an initial mean price of supply (clearing price charged to all consumers), P, say just as the mean of the initial generator prices, p _{ g }, ∀g∈G.
 2.
Determine the pricedependent loads \(\underline {D}(P)\), where D _{ l }(P)=α _{ l } D(P), ∀l∈L.
 3.
ISO solves the optimal power flow allocation problem \(\underline {S}(\underline {D}(P)), \underline {p})\) given fixed demands \(\underline {D}\) and generation/supply costs \(\underline {p}\).
 4.
ISO computes a new mean (clearing) price of supply, \(P ~=~ \frac {\sum _{g \in G} S_{g} \,p_{g}}{\sum _{g \in G} S_{g}}\).
 5.
If the change in clearing price P is significant (larger than some threshold), then go back to Step 2; else continue to Step 6.
 6.
For the current set of generator prices, consistent supply allocations, loads, and clearing price have been found. Now each generator sets a new price of supply such that there is an increase in its utility function using one of the following two approaches:
(i) Bestresponse play action: Each generator g sets a new price of supply based on (an estimate of)^{e}$$\begin{array}{@{}rcl@{}} \arg\max_{p_{g}} ~p_{g} \,S_{g}\left(p_{g}; \underline{p}_{g}\right) ~~ c_{g}\left(S_{g}\left(p_{g}; \underline{p}_{g}\right)\right), \end{array} $$(4)where c _{ g }(x) is the cost of supply (assumed =a _{ g } x ^{2} above).
(ii) Betterresponse play action: Each generator g obtains approximate left and right partial derivatives of its utility function with respect to its price p _{ g }, i.e.,$$\begin{array}{@{}rcl@{}} &&\Delta u_{g}^{+} ~=~ \frac{u_{g}\left(p_{g} + \epsilon,\, \underline{p}_{g}\right)  u_{g}\left(p_{g},\, \underline{p}_{g}\right)}{\epsilon} \\ &&\Delta u_{g}^{} ~=~ \frac{u_{g}\left(p_{g},\, \underline{p}_{g}\right)  u_{g}\left(p_{g}  \epsilon,\, \underline{p}_{g}\right)}{\epsilon}, \end{array} $$where ε↘0^{f}. If the left and right derivatives have different signs (a nondifferentiable point), then there are two possibilities. If \(\Delta u_{g}^{} > 0\) and \(\Delta u_{g}^{+} < 0\), the current price p _{ g } is a local maximum and there is no need to change p _{ g }. If \(\Delta u_{g}^{} < 0\) and \(\Delta u_{g}^{+} > 0\), the current price is a local minimum. In this case, we increase p _{ g } by a small value ζ if \(\Delta u_{g}^{+} > \Delta u_{g}^{}\); otherwise, we decrease p _{ g } by ζ. In case the derivatives have the same sign (may still be a nondifferentiable point), we increase p _{ g } by ζ if both derivatives are positive and decrease p _{ g } by ζ if both derivatives are negative. The step ζ should increase the price by a small value such that there is an increase in the value of the utility function. It should not make large changes to the price like the bestresponse play action^{g}.
 7.
Exit if there is no change in the generator prices (i.e., if an equilibrium set of prices is obtained); Else go back to Step 1.
We implicitly assume in Step 6 that the ISO helps to compute estimates of supply sensitivities to bid for each generator, ∂ S _{ g }/∂ p _{ g }. This can be done numerically through difference quotients. The result is a Jacobi iteration representing a ‘better response’ game [19,20]. Alternatively, the ISO could iteratively compute the ‘best response’ price for each generator, but this would require knowledge of each generator’s net utility.
3 Numerical study  benchmark IEEE 14bus power system
For this 14bus power system, each of the three generators can produce maximum power \(\underline {S}_{g}^{(\max)}\,=\,150\,\)MW, and minimum power of 0 MW (again, we do not consider rampup/down issues herein). The constants in D(P), our model for the total load, were chosen as the possible maximum/minimum power provided by the generators, that is, D _{max}=450 MW, D _{min}=0 MW. The total load D(P) is assumed to be proportionally divided among the individual loads, e.g., \(\alpha _{l} = \frac {1}{L}, ~\forall l \in L\). The maximum clearing price is set to P _{max}=5; for clearing prices P>P _{max}, the flexible demand is 0. Other data for this power system can be found in [23]. The constants (a _{ g }) in the utility function (2) of generators are set to 0.02, 0.025, and 0.03, respectively.
4 Effects of false demand attestations
We also numerically explored how false demand attestations may impact the revenues of generators. It is possible that demand aggregators and generators may collude, or noneconomic demandside bids may be placed by energy brokers, to cause some generator(s) to receive more revenue while possibly reducing the revenue of others.
4.1 Example with numerical study
4.2 Discussion: when to penalize
Recall the previous discussion of complications in estimating demand dayahead due to the presence of flexible demand, renewables, and storage devices. This additionally complicates detection of false attestations of demand. So, an ISO may need to detect persistent demandattestation ‘biases’ over time and let that trigger associated demandside penalties or reservation costs. General incompleteinformation gametheoretic frameworks include Bayesian games and hypergames, e.g., [24]. In both VCG and PSP auctions (e.g., [25,26]), issues of truthfulness in the disclosed bids are considered, i.e., reflecting actual demand response (by marginal valuation). More prosaic approaches simply interpolate and extrapolate from presumed honest bids (by (amount, price)) to obtain a complete estimate of other players’ demand response. These frameworks are applicable to iterated (sequential) adversarial (noncooperative) games with or without leaders. Generally, estimates are greatly simplified under the assumption that player strategies are time (playaction iteration) invariant.
The magnitude of such penalties would be informed by calculations such as those described above, i.e., by determining which generators benefit and by how much. By penalizing in this way, the motivation for deliberately false demandattestations are disincentivized.
5 Conclusions
In this paper, we studied a dayahead wholesale electricity market under demand response as a noncooperative game under certain assumptions, in particular no location marginality. We argued for the existence of symmetric Nash equilibria and numerically studied the benchmark IEEE 14bus power system. We then considered the effects of false demand attestation and again used the 14bus system to show how generators may (or may not) benefit by false attestation of demands. Such calculations may be used to penalize demand attestations that show persistent bias to the benefit of certain generators.
6 Endnotes
^{a} For example, in [5], an affine singlepart bid and associated makewhole “uplift” payments that are part of a joint integerprogramming unit commitment and continuouslinear OPF (economic dispatch) problem was considered. In practice, the generation unitcommitment decisions are typically made first, e.g. simply based on mean cost of supply over the capacity range of the generator including its rampup costs.
^{b} Herein, just the mean price of supply  again, we assume no location marginality.
^{c} We assume quadratic cost for tractability in the duopoly studied in [18]. An alternative cost structure could be asymptotic to a maximum, e.g., c(0)/(s−s _{max}) where c(0) is the cost of keeping the generator online even if zero supply is being delivered. In this paper, we do not consider ramp up/rampdown constraints for generators.
^{d} If the net consumer utility is collectively V(D)−P D, then for the linear demandresponse to price of (1), the utility is quadratic, concave and increasing, \(V(D) = (P_{\max }/2) (D_{\max }^{2}(D_{\max }D)^{2})/ (D_{\max }D_{\min })\) for D _{min}≤D≤D _{max}.
^{e} Since an explicit closedform solution is difficult to obtain, here we simply search for objectiveoptimizing prices by first defining an evenly partitioned priceset from 0 to 5.
^{f} We chose a value of ε=10^{−6}.
^{g} We chose ζ as follows. Starting with a small trial value of ζ=0.005 p _{ g }, if \(u_{g}\left (p_{g} + \zeta,\, \underline {p}_{g}\right) > u_{g}\left (p_{g},\, \underline {p}_{g}\right)\) we accept the value of ζ; Else ζ is decreased by a factor of 2 iteratively until \(u_{g}\left (p_{g} + \zeta,\, \underline {p}_{g}\right) > u_{g}\left (p_{g},\, \underline {p}_{g}\right)\).
7 Appendix: symmetry of Nash equilibrium points
In this Appendix, we simply and directly derive conditions on the parameters of our powersystem model (in DC approximation) for existence and symmetric Nash equilibria and nonexistence of asymmetric ones. The claims can be directly extended to nonquadratic cost functions for generation at the expense of closedform expressions for the conditions of the claims in terms of the model parameters.
7.1 Proof of existence of symmetric Nash equilibrium points
For existence of a symmetric Nash equilibrium, we need show that A∩B≠∅, where A is the set of prices, p, in symmetric price vectors \(\underline {p}=\,[p,p,\ldots,p]\) from which decreasing any one price component (of a given generator) is not profitable (for that generator), and similarly B is the set of prices in price vectors from which increasing a price component is not profitable.
7.1.1 Decreasing a price component is not profitable, A
7.1.2 Increasing price is not profitable, B
Note that the three conditions can be expressed in terms of bestresponse price \(p^{*}_{i}\) (recall S _{ i }∘ is also known function of p).
7.1.2.1 Claim:
If generator parameters \(\left \{a_{i},S_{i}^{(\max)}\right \}_{i\in G}\) and aggregate demand parameter D _{max},P _{max} are such that A∩B≠∅, then there exists pricesymmetric Nash equilibria \(\underline {p}=\,[p,p,\ldots,p] (p\in A\cap B)\) with allocation symmetry S _{ g }=D(p)/G for all g∈G.
As an example of a set of parameters where such Nash equilibria exist, we refer to the benchmark IEEE 14bus power system in DC approximation described above.
7.2 Proof of nonexistence of asymmetric Nash equilibrium points
We set D _{min}=0 and the minimum power allocation of each generator to zero both w.l.o.g. Instead of simply proving nonexistence of asymmetric Nash equilibrium points for the IEEE 14bus power system under DC approximation, we herein give the conditions for the nonexistence of asymmetric Nash equilibrium points in more general form, which is useful in analyzing the systems with more than three generators.
7.2.1 Case of a unique generator having the lowest price
Recall the clearing price is \(P=\sum _{i} p_{i} S_{i}/\sum _{g} S_{g}\).
7.2.1.1 Claim:
If there is an asymmetric equilibrium point with the unique lowest price, say p _{ i }, then the total demand \(D(P)\leq S_{i}^{(\max)}\).
Proof.
For a proof by contradiction, assume a Nash equilibrium pricing point where \(D(P)> S_{i}^{(\max)}\) and p _{ i } is the lowest price. Generator i can increase its price by a small amount while maintaining lowestprice status and the condition \(D(P)> S_{i}^{(\max)}\). Doing so, the allocation S _{ i } of generator i will not be affected after the increase of price but the revenue of generator i will increase. Thus, this hypothetical equilibrium pricing point is not Nash, a contradiction. □
or approaches (is ‘infinitely close’ to) the second lowest price among generators (see the next case).
Since condition (11) does not hold for all i≠j (because the righthand side is always negative) for our IEEE 14bus power system under DC approximation, when \(\underline {a}=[.02,.025,.03]\)), asymmetric Nash equilibria with unique lowest prices do not exist.
7.2.2 Proof of the nonexistence of asymmetric Nash equilibrium point with >1 generators at the lowest price
Let generators i∈M⊂G have the lowest price \(\hat {p}\).
7.2.2.1 Claim:
If there is an asymmetric equilibrium point with G>M≥2, the total demand \(D(P)\leq M \min _{i\in M} \left \{S_{i}^{(\max)}\right \}\) where the clearing price P=p _{ i } for all i∈M.
Proof.
Similar to the proof for the first claim, if \(D(P)> M \min _{i\in M} \left \{S_{i}^{(\max)}\right \}\) and \(S_{i}^{(\max)} = \min _{j\in M} \left \{S_{j}^{(\max)}\right \}\), then i will increase its utility by increasing its price a small amount without reducing its allocation, which makes this price point not Nash. □
Note that (14) is necessary but not sufficient for existence of asymmetric Nash equilibrium.
Again, for our IEEE 14bus power system under DC approximation, if utility cost parameters [ a _{1},a _{2},a _{3}]=[.02,.025,.03] and M=2, then (14) is not satisfied; hence, no such asymmetric Nash equilibrium exists.
Declarations
Acknowledgements
This research is supported by grant NSF CCF1228717.
Authors’ Affiliations
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