3 Probabilistic locational Marginal Prices 20 Therefore, according to the definition of LMP, LMP of bus i is equal to the shadow price of power flow equation of bus i (see the examples of appendices C and D). 3.2 Probabilistic Locational Marginal Prices In regulated power systems, probabilistic load flow is used for analyzing electric networks and transmission expansion planning -. In this environment technical criteria, such as the probability of violating line flow limits and bus voltage limits, are used for transmission expansion planning. Technical criteria are computed based on the PDFs of line flow powers and bus voltages. In deregulated power systems in addition the technical criteria, market based criteria must be used to achieve the objectives of transmission expansion planning in deregulated power systems. In order to define and compute market based criteria, we need to compute the PDFs of variables which show the performance of electric market. These variables should be affected by dynamics of both power system and electric market. This thesis proposes to compute PDFs of LMPs for assessing the performance of electric markets -. In this section a probabilistic tool, which is named “probabilistic optimal power flow” or “probabilistic locational marginal prices”, is presented for computing PDFs of LMPs. 3.2.1 Why Probability Density Functions of Locational Marginal Prices? According to equations (3.1)-(3.5), LMPs will be affected if: • producers change their bids, • producers change minimum or maximum of their submitted power, • consumers change their bids for load curtailment, • consumers change minimum or maximum of their submitted demands, • there is transmission constraint in the network, • transmission facilities (generator, transmission line, load,…) have forced outage, • input or output power to the study area change due to new contracts with neighboring areas, • transmitting power through the study area change due to new wheeling transactions, or • there is market power in the network. Hence, PDFs of LMPs contain much information about the power system and electric market. Therefore, performance of an electric market can be assessed by analyzing its PDFs of LMPs.
3 Probabilistic locational Marginal Prices 21 3.2.2 Probabilistic Optimal Power Flow We use Monte Carlo simulation to compute PDFs of LMPs for a specified scenario. The procedure of computing PDFs of LMPs using Monte Carlo simulation is as below: • Determining the PDF of each input which has random uncertainty (refer to 18.104.22.168). • Picking a sample from the PDF of each input (refer to 22.214.171.124). • Computing LMP of each bus by solving optimal power flow for the picked samples (refer to 126.96.36.199). • Repeating steps 2 and 3 a great number (number of repetition must be selected so that mean and variance of each output variable converges to a constant value) • Fitting a PDF to the samples of LMP of each bus (refer to 188.8.131.52). The above steps are described in detail in the rest of this subsection. Then, the algorithm of computing PDFs of LMPs is presented more precisely in subsection 3.2.3. 184.108.40.206 Determining the Probability Density Functions of Random Inputs To model the above random uncertainties PDF of each random input variable must be determined. In order to consider the simultaneity of loads and in order to consider the worst conditions, PDFs of random inputs should be determined for the peak load of planning horizon. Some random inputs depend on the other random inputs, for example power of some tie-lines may depend on the power of other tie-lines. In this case, only PDFs of independent random inputs are determined. The values of dependent random inputs are computed based on their relation with the independent random inputs in each iteration of Monte Carlo simulation. To model the random uncertainties of deregulated power systems the following PDFs must be determined for the peak load of planning horizon: • PDF of each load, • PDF of bid of each generator, • PDF of maximum accessible power of each IPP, and • PDF of power of each tie-line. PDFs of loads can be determined based on the load prediction and uncertainty analysis . This method can be used for computing PDFs of other random inputs. To model emergency outage of transmission facilities, unavailability of each transmission facility is determined. A standard uniform PDF is assigned to each transmission facility