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Lying generators: manipulability of centralized payoff mechanisms in electrical energy trade

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Abstract

Optimal power flow (OPF) problems are focussing on the question how a power transmission network can be operated in the most economic way. The general aim in such scenarios is to optimize generator scheduling in order to meet consumption requirements, transmission constraints and to minimize the overall generation cost and transmission losses. We use a simple lossless DC load flow model for the description of the transmission network, and assume linearly decreasing marginal cost of generators with different parameters for each generator. We consider a scenario in which the generation values regarding the OPF are calculated by a central authority who is aware of the network parameters and production characteristics. Furthermore, we assume that a central mechanism is applied for the determination of generator payoffs in order to cover their generation costs and assign them with some profit. We analyze the situation when generators may provide false information about their production parameters and thus manipulate the OPF computation in order to potentially increase their resulting profit. We consider two central payoff mechanisms and compare their vulnerability for such manipulations and analyze their effect on the total social cost.

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Notes

  1. Although it is possible that near the production capacity limits the validity of the decreasing marginal cost assumption is questionable, to keep the model as simple as possible, we restrict our analysis to the case where the linearly decreasing marginal cost assumption is valid. The model can be easily extended with more complex production characteristics.

  2. This approach doubles the number of variables in the optimization problem (since it separates the positive and negative part of the variables).

References

  • Abido M (2002) Optimal power flow using particle swarm optimization. Int J Electr Power Energy Syst 24(7):563–571

    Article  Google Scholar 

  • Achterberg T (2009) Scip: solving constraint integer programs. Math Program Comput 1(1):1–41

    Article  Google Scholar 

  • Bakirtzis AG, Biskas PN, Zoumas CE, Petridis V (2002) Optimal power flow by enhanced genetic algorithm. Power Syst IEEE Trans 17(2):229–236

    Article  Google Scholar 

  • Conejo AJ, Aguado JA (1998) Multi-area coordinated decentralized dc optimal power flow. Power Syst IEEE Trans 13(4):1272–1278

    Article  Google Scholar 

  • Contreras J (1997) A cooperative game theory approach to transmission planning in power systems. PhD thesis. University of California, Berkeley

  • Dommel HW, Tinney WF (1968) Optimal power flow solutions. power apparatus and systems. IEEE Trans 10:1866–1876

    Google Scholar 

  • Fisher EB, O’Neill RP, Ferris MC (2008) Optimal transmission switching. Power Syst IEEE Trans 23(3):1346–1355

    Article  Google Scholar 

  • Fletcher R, Leyffer S, Toint PL (2002) On the global convergence of a filter-sqp algorithm. SIAM J Optim 13(1):44–59

    Article  Google Scholar 

  • Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95–99

    Article  Google Scholar 

  • Hedman KW, O’Neill RP, Fisher EB, Oren SS (2008) Optimal transmission switching-sensitivity analysis and extensions. Power Syst IEEE Trans 23(3):1469–1479

    Article  Google Scholar 

  • Hedman KW, O’Neill RP, Fisher EB, Oren SS (2009) Optimal transmission switching with contingency analysis. Power Syst IEEE Trans 24(3):1577–1586

    Article  Google Scholar 

  • Hedman KW, Ferris MC, O’Neill RP, Fisher EB, Oren SS (2010) Co-optimization of generation unit commitment and transmission switching with n-1 reliability. Power Syst IEEE Trans 25(2):1052–1063

    Article  Google Scholar 

  • Hingorani NG (1993) Flexible AC transmission. Spectr IEEE 30(4):40–45

    Article  Google Scholar 

  • Hingorani NG, Gyugyi L, El-Hawary M (2000) Understanding FACTS: concepts and technology of flexible AC transmission systems, vol 1. IEEE press, New York

    Google Scholar 

  • Huneault M, Galiana F (1991) A survey of the optimal power flow literature. Power Syst IEEE Trans 6(2):762–770

    Article  Google Scholar 

  • Johnson SG (2010) The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/nlopt

  • Kaltenbach J, Hajdu L (1971) Optimal corrective rescheduling for power system security. IEEE Trans Power Appar Syst 90:843–851

    Article  Google Scholar 

  • Kirschen D, Strbac G (2004) Fundamentals of power system economics. Wiley, Chichester. doi:10.1002/0470020598

    Book  Google Scholar 

  • Laarhoven PJ (1987) Simulated annealing: theory and applications. In: Mathematics and its applications. Reidel, Dordrecht

  • Le Digabel S (2011) Algorithm 909: nomad: nonlinear optimization with the mads algorithm. ACM Trans Math Softw (TOMS) 37(4):44

    Article  Google Scholar 

  • Momoh JA, Adapa R, El-Hawary M (1999) A review of selected optimal power flow literature to 1993. i. nonlinear and quadratic programming approaches. Power Syst IEEE Trans 14(1):96–104

    Article  Google Scholar 

  • Momoh JA, El-Hawary M, Adapa R (1999) A review of selected optimal power flow literature to 1993. ii. newton, linear programming and interior point methods. Power Syst IEEE Trans 14(1):105–111

    Article  Google Scholar 

  • O’Neill RP, Hedman KW, Krall EA, Papavasiliou A, Oren SS (2010) Economic analysis of the n-1 reliable unit commitment and transmission switching problem using duality concepts. Energy Syst 1(2):165–195

    Article  Google Scholar 

  • Oren S, Spiller P, Varaiya P, Wu F (1995) Folk theorems on transmission access: proofs and counter examples. Working papers series of the Program on Workable Energy Regulation (POWER) PWP-023, University of California Energy Institute 2539 Channing Way Berkeley, California 94720–5180, www.ucei.berkeley.edu/ucei

  • Sauma EE, Oren SS (2007) Economic criteria for planning transmission investment in restructured electricity markets. Power Syst IEEE Trans 22(4):1394–1405

    Article  Google Scholar 

  • Song YH, Johns AT (1999) Flexible AC transmission systems (FACTS), vol 30. Institution of Electrical Engineers, Stevenage

  • Sun DI, Ashley B, Brewer B, Hughes A, Tinney WF (1984) Optimal power flow by newton approach. Power Appar Syst IEEE Trans 10:2864–2880

    Article  Google Scholar 

  • Tseng CL, Oren SS, Cheng CS, Li CA, Svoboda AJ, Johnson RB (1999) A transmission-constrained unit commitment method in power system scheduling. Decis Support Syst 24(3):297–310

    Article  Google Scholar 

  • Vaz AIF, Vicente LN (2007) A particle swarm pattern search method for bound constrained global optimization. J Glob Optim 39(2):197–219

    Article  Google Scholar 

  • Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57

    Article  Google Scholar 

  • Yao J, Oren SS, Adler I (2004) Computing cournot equilibria in two settlement electricity markets with transmission constraint. In: System Sciences. Proceedings of the 37th Annual Hawaii International Conference on IEEE, p 9

  • Yuryevich J, Wong KP (1999) Evolutionary programming based optimal power flow algorithm. Power Syst IEEE Trans 14(4):1245–1250

    Article  Google Scholar 

  • Zhu C, Byrd RH, Lu P, Nocedal J (1997) Algorithm 778: L-bfgs-b: Fortran subroutines for large-scale bound-constrained optimization. ACM Trans Math Softw (TOMS) 23(4):550–560

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the Hungarian National Fund (OTKA NF-104706) and by the Hungarian Academy of Sciences under its Momentum Programme (LP-004/2010) and by grant KAP-1.2-14/001 of the Pázmány Péter Catholic University. The author thanks Gábor Szederkényi for the helpful discussions on optimization issues.

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Correspondence to Dávid Csercsik.

Appendices

Appendix 1: DC load flow model

In the proposed model generators can be characterized by the quantity of actual and maximal generated (or supplied) power, while consumers are described by their power consumption (constant for each consumer node). We assume that a transmission line is characterized by its admittance value, denoted by \(Y_{ij}\) (which will be equal to susceptance in this case, for we neglect the real part of impedance values), and maximum transmission capacity (or branch power flow limit) \(\bar{q}_{ij}\).

According to our modelling considerations, we describe the voltage at node \(i\) with sinusoidal waveform:

$$\begin{aligned} v_i(t)=V_i\sin (\omega t+\theta _i) \end{aligned}$$
(6)

where \(V_i\) stands for the magnitude, \(\omega =2\pi f\) denotes the frequency in rad/s and \(\theta _i\) is the phase angle.

If we assume that the nodes \(i\) and \(j\) are connected by a transmission line with admittance \(Y_{ij}=Y_{ji}\), the (real) power flow from \(i\) to \(j\) can be described with:

$$\begin{aligned} q_{ij}=V_iV_jY_{ij}\sin (\theta _i-\theta _j) \end{aligned}$$
(7)

By definition \(q_{ij}>0\) if the power flows from \(i\) to \(j\). This implies \(q_{ij}=-q_{ji}\) for flows of opposite direction. We can formalize the energy conservation for each node as follows. The net power \(p_i\) injected into (or drawn from) the network at bus \(i\) addition to the total inflow is equal to the total outflow:

$$\begin{aligned} p_i=\sum _{j=1}^n q_{ij} \end{aligned}$$
(8)

Without the loss of generality, let us assume \(V_i\equiv 1\). In this case

$$\begin{aligned} p_i=\sum _{j=1}^n Y_{ij}\sin (\theta _i-\theta _j) \end{aligned}$$
(9)

which means \(n-1\) independent equations (as \(p_1+\cdots +p_n=0\)). Let us choose \(\theta _n \doteq 0\). In this case the individual line flows can be expressed as:

$$\begin{aligned} q_{ij}=Y_{ij}\sin (\theta _i-\theta _j) \end{aligned}$$
(10)

Assuming that \((\theta _i-\theta _j)\) is small, \(\sin (x)\) may be approximated with \(x\). This leads to the so called “DC load flow model”, which exhibits the following uniqueness property: Given power injections and power consumptions at each node, the phase angles \(\theta _i\) are determined by solving a system of linear equations. From the phase angle differences, the line flows can be uniquely determined.

We can summarize the equations in the following matrix formalism (Contreras 1997):

The relation between the total inlet/outlet power and power flows can be described by

$$\begin{aligned} AQ=P \end{aligned}$$
(11)

where \(A\in {\mathbb {R}}^{n \times m}\) is the Node-branch incidence matrix of the network, \(Q \in {\mathbb {R}}^{m} \) denotes the power flow vector, and \(P\in {\mathbb {R}}^n\) is the power injection vector (composed of \([p_1, p_2,\ldots ,p_n]\)).

If we substitute the individual power flows in Eq. 11 with the linearized expressions from Eq. 7, we can write

$$\begin{aligned} B(Y){\varTheta }=P \end{aligned}$$
(12)

where \(B(Y) \in {\mathbb {R}}^{n \times n}\) denotes the susceptance matrix whose elements are \(B_{kl}=-Y_{kl}\) for the off-diagonal terms and

$$\begin{aligned} B_{kk}=\sum _{k\ne l}^{} B_{kl} \end{aligned}$$
(13)

(the column sum of off-diagonals) for diagonal elements. \({\varTheta }\in {\mathbb {R}}^n\) is vector of nodal voltage angles.

The constraint describing the maximum line power flows can be derived as

$$\begin{aligned} |Q|=|B^DA^T{\varTheta }|<\bar{Q} \end{aligned}$$
(14)

where \(|\bar{Q}|\) is branch power flow limit vector (composed of the elements \(\bar{q}_{ij}\)), and \(B^D\) is a diagonal matrix with \(B^D_{kk}=Y_{ij}\).

As we know from Eq. 12, \(B {\varTheta }=P\). The matrix \(B\) is singular due to the column conservation property, but since in the calculation of flows only the differences of the elements of the vector \({\varTheta }\) are appearing (see Eq. 7), we may express it as

$$\begin{aligned} {\varTheta }=B^{+}P \end{aligned}$$
(15)

where \(B^{+}\) is the Moore-Penrose pseudoinverse of \(B\). Constraint 14 becomes

$$\begin{aligned} |B^DA^T{\varTheta }|=|B^DA^T B^{+}P|<\bar{Q} \end{aligned}$$
(16)

Appendix 2: optimization methods for the OPF

As a preliminary study we compared the currently freely available optimization tools in MATLAB in the context of solving Eqs. 15. The basic model was the topology of Network 1 as depicted in Fig. 1, however the admittance values, and production characteristics were randomized around the nominal values described in Sect. 2.2 as follows. \(\tilde{Y}_{ij}=Y_{ij}+{\varDelta }Y\) where \({\varDelta }Y \in (-0.5,0.5)\) , \(\tilde{a}^j=a^j + {\varDelta }a\) where \({\varDelta }a \in (-0.25,0.25)\), and \(\tilde{m}^j=m^j + {\varDelta }m\) where \({\varDelta }m \in (-0.03,0.03)\) (each \({\varDelta }\) value from uniform distribution).

The following algorithms were compared:

  • Interior point optimizer (IPOPT), see (Wächter and Biegler 2006)

  • Filter–SQP Algorithm (Fletcher et al. 2002)

  • L-BFGS-B or Algorithm 778 (Zhu et al. 1997)

  • NLOPT (Johnson 2010)

  • SCIP (Achterberg 2009)

  • The standard genetic algorithm (GA) of MATLAB (Goldberg and Holland 1988)

1000 repeated runs were completed with the randomized parameters for each algorithm \(\mathcal {A}\). The evaluation was carried out as follows. For each run, the reference value of the objective function \(F\) (the value of 1) at run \(i\) denoted by \(F_{ref}(i)\) was defined as the lowest value found by the various algorithms in the current run \(F_{ref}(i)= \min _{\mathcal {A}}F_\mathcal {A}(i)\)). The error of the algorithm \(\mathcal {A}\) at run \(i\) (\({\varUpsilon }_\mathcal {A}(i)\)) then is defined as

$$\begin{aligned} {\varUpsilon }_\mathcal {A}(i)=F_\mathcal {A}(i)-F_{ref}(i) \end{aligned}$$

The resulting error is the sum over the repetitive runs \({\varUpsilon }_\mathcal {A}=\sum _i {\varUpsilon }_\mathcal {A}(i)\). The resulting values are as follows.

As it can be seen from the results summarized in Tabel 3, the SCIP algorithm finds the best value of the objective function in almost every case, and it is undoubtedly the most well suited for this optimization problem.

Table 3 Comparison of various optimization methods regarding optimal power flow computation

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Csercsik, D. Lying generators: manipulability of centralized payoff mechanisms in electrical energy trade. Cent Eur J Oper Res 24, 923–937 (2016). https://doi.org/10.1007/s10100-015-0387-6

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