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
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.
This approach doubles the number of variables in the optimization problem (since it separates the positive and negative part of the variables).
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Acknowledgments
This work was supported by the Hungarian National Fund (OTKA NF104706) and by the Hungarian Academy of Sciences under its Momentum Programme (LP004/2010) and by grant KAP1.214/001 of the Pázmány Péter Catholic University. The author thanks Gábor Szederkényi for the helpful discussions on optimization issues.
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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:
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:
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:
Without the loss of generality, let us assume \(V_i\equiv 1\). In this case
which means \(n1\) 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:
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
where \(A\in {\mathbb {R}}^{n \times m}\) is the Nodebranch 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
where \(B(Y) \in {\mathbb {R}}^{n \times n}\) denotes the susceptance matrix whose elements are \(B_{kl}=Y_{kl}\) for the offdiagonal terms and
(the column sum of offdiagonals) 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
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
where \(B^{+}\) is the MoorePenrose pseudoinverse of \(B\). Constraint 14 becomes
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. 1–5. 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)

LBFGSB 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
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.
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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/s1010001503876
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DOI: https://doi.org/10.1007/s1010001503876