Analysis of Coordinated Multilateral Trades - CiteSeerX
Copyright 2000 IEEE. Published in the Proceedings of the Hawai'i International Conference On System Sciences, January 4-7, 2000, Maui, Hawaii.
Analysis of Coordinated Multilateral Trades Pierre-Franc¸ois D. Quet [email protected]
Jose B. Cruz, Jr. [email protected]
Ali Keyhani [email protected]
Department of Electrical Engineering, The Ohio-State University 2015 Neil Avenue, Columbus OH 43210
Abstract A new operating paradigm is still needed for the power industry in order to achieve a workable deregulated market. The two main proposals, namely the Bilateral and the Poolco models, even though their original intents were to promote efficiency in a deregulated industry, in order to be workable need a powerful centralized organization. They consequently end up operating in the same way as the regulated industry. The coordinated multilateral trading model lets the suppliers and consumers make a profit on their own while the Power System Operator only intervenes if the security of the power network is threatened. The PSO gives sufficient information about the status of the power network so that the brokers can arrange trades without endangering the security of the network. Moreover the economic data of the participants is private, thus encouraging economic competition in a free market. It is shown that the process of building such trades leads to an efficient use of the power network, i.e. maximize the social welfare. A three bus example is demonstrated.
1. Introduction Beginning in 1978 with the Public Utility Regulatory Policy Act (PURPA) the electric industry has moved toward a less regulated commerce. The intent is to encourage competition in generation and customer choice. But the debate is still open over how the transmission system should be designed in the new deregulated industry in order to achieve the goal of open transmission access which would permit an efficient energy market. The problem is that not every set of power injections at the nodes of the network are possible, because of the thermal limits of the transmission lines and the need to have balance of power across the network. Due to the nature of electricity, the quantity of energy flowing through a line is governed by all the participants at the same time, thus making this problem non-trivial to solve.
Several proposals have been made which are now opposing the Bilateral Model and the Poolco Model. But these two models use the concept of sharing the resources of the transmission network through a centralized operator. They consequently end up functioning the same way the conventional operating paradigm was working, which is a regulated monopoly. A description of the Bilateral Model can be found in [2] and a critique of both of the models is in [6]. A new operating paradigm is still needed to meet the goals of an efficient electric industry. In [5], F. Wu and P. Varaiya introduce the concept of Coordinated Multilateral Trades in which the Power System Operator intervenes only if the security of the network is threatened and gives the participants sufficient information based on which suppliers and consumers can arrange feasible trades (in the sense that the trade will not overload any line on the network). The main point of this operating paradigm is that the suppliers and the consumers arrange trades among themselves keeping their economic data private, which prevent other parties to be able to “game” based on this knowledge. Wu and Varaiya proved that such trades lead to the maximization of the social welfare (i.e. minimize the total cost for the suppliers minus the total benefit for the consumers), which is also the solution of the Optimal Power Flow problem presented next. They derived a mechanism to realize such trades considering real power only. This paper extends the concept of Coordinated Multilateral Trades to the trading of both real and reactive power production and consumption. In order to do so, a different trading mechanism is needed, which will be described in the next sections.
2. Optimal Power Flow Solution In the conventional paradigm the Power System Operator (PSO) maintains the economic efficiency of the power system operations by conducting an Optimal Power Flow (OPF) study. This analysis is based on the knowledge of the producing cost of each supplier and the benefit to all the consumers. Consider a bus power network. Let:
be the real and reactive power supply
3. Coordinated Multilateral Trading Model
at each bus be the real and reactive power demand at each bus number of suppliers and numbers of consumers ns supplier' s costs (convex and strictly increasing functions) nc consumer' s benefits (concave and strictly increasing functions) power flow equations with
where is the voltage at bus and element of the admittance matrix defined as:
is the
sum of all admittances connected at bus i (admittance connected between bus i and bus j) Some more information about power system concepts can be found in [1]. An efficient method to solve the power flow equation is carried out in [3]. MVA limit on line 1 .. . MVA limit on line L where
is the number of lines in the network and with and being the real and reactive power flowing through line respectively. The Optimal Power Flow is the maximization of the social welfare, i.e. the minimization of the total generation cost minus the total consumer benefit, given the constraints of the network:
such that:
The Optimal Power Flow problem is the one of a centralized utility which possesses the knowledge of the cost function of the generators and the benefit function of the consumers, and can dictate the level of generation for everyone. It will be shown next that the same result can be achieve using the concept of Coordinated Multilateral Trades, which gives the Power System Operator much less power and lets the suppliers and the consumers carry out their economic decisions on their own.
In the Coordinated Multilateral Trading Model [5], the Power System Operator (PSO) only intervenes if the reliability of the power network is threatened. The suppliers and consumers arrange trades and look for profit on their own. The PSO gives information to the participants over how to build secure trades, i.e. trades which will not overload transmission lines. The responsibility of maintaining security is shared by all participants. There is no need for a central authority to have the knowledge of the production cost of the suppliers or the benefit function of the consumers. The economic decisions are all private, which prevent other parties to “game” based on this knowledge. Moreover, each trade will account for its share of electric losses across the power network. Let the party which arranges a trade be called the broker. The broker can be a supplier or a consumer participating in the trade or a third party. The broker's goal is to build profitable trades, i.e. trades where the total consumer benefit minus the total cost of generating the electricity is positive. The trades are carried out neglecting the electric losses they incur, their share of losses is compensated once the trades are made. This is reasonable since the cost of compensating for the losses is small compared to the trade itself. A multilateral trade will involve two or more participants and is in fact a generalization of the Bilateral Model. But as it can be seen later that it may be necessary to have three or more participants in order to remove a congestion in the power network. The process of building multilateral trades goes as follows: 1. Brokers arrange initial trades and give their intent to the PSO. 2. If these trades would overload some lines, the PSO would then curtail them in order to make the trades feasible. 3. The PSO then computes the loading vectors for each line and announces the information to all participants. The loading vectors show the influence of a power injection at each node of the network on the loading of the lines. Based on this information, the brokers can build additional trades which will be feasible, or close to feasible. 4. If the trades are not feasible, then the PSO curtails them and process 3 is repeated until no further benefit can be obtained by carrying out additional trades. 5. The set of trades is now feasible and the participants do not want to carry out any further trades. The process of compensating for the losses is carried out next. The PSO computes the loss vectors based on which the losses are allocated. These loss vectors show the influence of a power injection at each node on the losses on each line. We will show that this process, without the loss allocation, converges to the Optimal Power Flow solution. The
explicit construction of the loading vector and the loss vector will be shown in the next sections. The transmission companies should be compensated for the service they offer. This study is not carried out here, but one can envision that the benefit made by the broker while arranging a trade can be used to compensate for the service offered by the transmission companies.
A multilateral trade
is a feasible-direction trade at
if, for
,
3.1. Feasible trades Consider the same bus power network as previously, with the same definitions for the cost and benefit functions, the power balance equations and the limits on the lines.
where for
Definition 1: A multilateral trade is represented by
such that: The set of all multilateral trades is:
for
number of trades
Lemma 1:
3.2. Profitable trades
Suppose a set of multilateral trades is not feasible. There exists a curtailment schedule
such that
is on the boundary of the feasible set. Proof: The set of feasible trades , denoted by , is a convex set in a (4n+4)-dimensional space. We denote by the boundary of . is in the feasible set. For , and for , . is a convex set, therefore there is a such that . Definition 2: Suppose
.
with and being the real and reactive power flowing through the line respectively. The operation is the defined as: . , , and are the loading vectors for the line and they represent the sensitivities on the real and reactive power on the lines due to some real and reactive power injections at each node. For example, the element of is equal to the additional amount of MW flowing on the line if 1 MW is injected into the node of the network, the same quantity being taken out from the slack bus. Thus a trade is a feasible direction trade if it does not overload the lines.
Lemma 2: Let and , being the solution of the Optimal Power Flow (see previous section). There exists a Feasible Direction trade that reduces the total cost:
Lemma 3: For a Feasible Direction trade at that reduces the total cost, there is a profit to be made in arranging such a trade [5]. Definition 3: is called a profitable multilateral trade at reduces the total cost.
if it
The suppliers and the consumers make profitable trades, the social welfare (difference between the benefit for the consumers and the cost of producing) increasing at each time, until the optimal solution is attained. But a profitable
trade may not be feasible, in that case the PSO may have to curtail it. Lemma 4: A profitable trade at in the feasible direction can always be made feasible by curtailment. Proof: Assume the constraint is violated, then using the same concept as in Lemma 1 there exists a curtailment schedule such that . Lemma 5: After curtailment, a profitable trade is still profitable. Proof: The cost functions for the suppliers are convex and the benefit functions for the consumers are concave, ns nc then is convex. This convexity guarantees that:
The trade
is then still profitable.
3.3. Trading arrangement The objective of the broker is to maximize the total cost reduction subject to the constraints that: the amount of generation and load balances out for each trade to be implemented, the trade is feasible. Let and be the sets of suppliers and consumers engaged in the trade respectively. The objective of the broker can be written as: min subject to:
and for
The loading vectors in Definition 2.
Consider the case with one transmission congestion. A trilateral trade at least may be necessary in order to remove the congestion. The more participants are involve in a trade the better a feasible and profitable solution can be found.
3.4. Coordinated Multilateral Trading Process The trading process proceeds as follows [5]: 1. The brokers arrange initial trades . 2. If these trades would overload some lines, the PSO curtail them in order to make the trades feasible. 3. The PSO broadcasts the Loading Vectors of all lines of the network. 4. The brokers arrange additional trades following the method of section 3.3. If no feasible and profitable trades are possible any more, then stop. 5. If the set of trades overload some lines, the PSO curtails them and go to step 3. Theorem 1: It is assumed that a broker will always arrange an additional trade if it is a profitable one and that the different parties will be willing to carry it out. Under this assumption the Coordinated Multilateral Trading Process converges to the Optimal Power Flow solution [5].
3.5. Compensation of the losses We would like each individual trade to take care of the losses it incurs. Their exact evaluation is not easy since the losses caused by a trade is a function of the other trades. Yet it is believed that they can be approximated using the Loss Vectors calculation. The Loss Vectors characterize the influence of a power injection at each node on the losses on each line. More specifically: for
,
,
and
and for
are defined
Notes: If the broker arranges a trade involving all the participants, the solution is the Optimal Power Flow solution.
with and being the real and reactive power losses of the line respectively.
The losses caused by the trade 1
3
2
on the line would then be approximated by: Figure 1. 3 bus power network The data of the lines are: where and are the approximated real and reactive losses respectively caused by the trade on the line . By computing the power flow solution for all the trades the total losses across the network can be determined, being equal to the power discrepancies at the slack bus. Approximation factors can be defined as: AppP
total real losses D L
AppQ
total reactive losses D L
with D and L being the number of trades and the number of lines respectively. In order to allocate all the losses of the network, each individual trade [k] will be assigned to compensate for the losses:
r 0.06 0.0315 0.04
x 0.4 0.25 0.2
The per unit system is: 1 pu = 100 MW. The thermal limit on the line 2-3 is chosen to be 5 MVA. In order to simplify this example, the limits on the other lines are not considered. The cost functions for the suppliers and the benefit functions for the consumers are of the type: for the suppliers:
is a convex and strictly increasing function for the consumers:
with
AppP AppQ where and caused by the trade
are the real and reactive losses respectively.
The participants of the trades can then compensate for the losses by either increasing the generation or decreasing the consumption. It can be argued that once the losses are compensated it results in different trades and consequently additional losses. But in practice the loss discrepancy at the slack bus after the loss compensation process described above is negligible.
4 3-bus example The data for this example is taken in part from [5]. Consider the three bus power network shown in fig.1 with two suppliers (at bus 1 and 2) and one consumer (at bus 3).
and
In this example it is considered that the consumer wants to keep a constant power factor , then:
The function is a sum of concave functions, thus it is concave itself. The difference Total cost of generation Total consumer benefit is then a convex function. This model for the benefit function of the reactive power consumption is taken from [4]. The reactive power really serves as a service which enables the consumption of real
power. Using this point of view, the benefit of the reactive power is considered to be the avoidance of moving the reactive power from some desired level for a given power consumption. It is assumed that each consumer possesses a power factor correction device as shown in fig.2.
constraint on the lines (just constraint to the power balance). The resulting power injections, consumption and flows on the line 2-3 are as shown on fig.4. The brokers give the values of the trades to the Power System Operator. Trade 1: 76.1 MW + 17.84 MVar
Bus
1
dp+j dq 13 MVA
Filtering device 3
2
dp + j f(dp)
Figure 2. Consumer filtering device
Trade 1: 76.1 MW + 17.84 MVar Trade 2: 17.92 MW + 0.97 MVar
The cost of operating this device is equal to zero when it is performing no correction (i.e. when ) and increases, according to as it moves away from that point (see fig.3).
Trade 2: 17.92 MW + 0.97 MVar
Figure 4. Initial trades
4.2. Curtailment k(x) 0
x
The Power System Operator notices that 13 MVA would be flowing on the line 2-3 if those trades were applied (by solving the power flow equations, see [1] and [3]). The PSO then curtails the trade arranged between the generator 1 and the consumer such that only 5 MVA would be flowing through the line 2-3. See the curtailed trades on fig.5. The bus 3 is chosen to be the slack bus in this example.
Figure 3. Function Trade 1: 41.4 MW + 9.71 MVar
The numerical data for the cost functions are: 1
5 MVA
Note that the consumer would like a power factor of 0.95 after his filtering device.
4.1. Initial trades The first broker arranges a trade between the supplier at bus 1 and the consumer at bus 3. A second broker arranges another trade between the supplier at bus 2 and the consumer at bus 3. The values of the trades should be made available to the participants for discussion. It is assumed that the participants make reasonable economic decisions, i.e. that they carry out a trade if it is profitable. The outcomes will result in the same solutions as the constraint minimization described in section 3.3 without the inequality
3
2
Trade 1: 41.4 MW + 9.71 MVar Trade 2: 17.92 MW + 0.97 MVar
Trade 2: 17.92 MW + 0.97 MVar
Figure 5. Curtailed trades
4.3. Additional trades The PSO computes the loading vectors using Definition 2 and broadcasts them. The loading vectors for the line 2-3
Trade 1: 41.4 MW + 9.71 MVar Trade 3: 19.67 MW + 7.8 MVar Trade 4: 0.03 MW + 0.1 MVar
are:
1
Based on this information, a third broker arranges an additional trade among all the participants following the process described in section 3.3. The PSO checks that this additional trade would not overload the line 2-3 and broadcasts a new set of loading vectors. The broker arranges another trade based on the new information. Since there is no more benefit to be made in arranging additional trades the process stops here. All the trades are shown in fig.6. Trade 1: 41.4 MW + 9.71 MVar Trade 3: 19.67 MW + 7.80 MVar Trade 4: 0.03 MW + 0.10 MVar
5 MVA
3
2
Trade 1: 40.56 MW + 5.12 MVar Trade 2: 17.67 MW + 0.68 MVar Trade 3: 34.43 MW + 10.76 MVar Trade 4: 0.04 MW + 0.18 MVar
Figure 7. After loss compensation CMT : 61.09 MW + 17.61 MVar OPF : 61.88 MW + 12.16 MVar
1
1
5 MVA
3
Trade 2: 17.92 MW + 0.97 MVar Trade 3: 15.4 MW + 6.7 MVar Trade 4: 0.02 MW + 0.08 MVar
5 MVA
2
3 Trade 1: 41.4 MW + 9.71 MVar Trade 2: 17.92 MW + 0.97 MVar Trade 3: 35.07 MW + 14.50 MVar Trade 4: 0.05 MW + 0.19 MVar
Trade 2: 17.92 MW + 0.97 MVar Trade 3: 15.4 MW + 6.70 MVar Trade 4: 0.02 MW + 0.08 MVar
CMT : 94.42 MW + 25.36 MVar OPF : 94.24 MW + 25.29 MVar
2
CMT : 33.34 MW + 7.75 MVar OPF : 32.36 MW + 13.13 MVar
Figure 6. With additional trades Figure 8. CMT: Coordinated Multilateral Trades OPF: Optimal Power Flow solution
4.4. Loss allocation The Power System Operator computes the loss vectors following the process described in section 3.5. The loss allocation results in the following:
Real losses Reactive losses
1 0.0084 0.0458
Trades 2 3 0.0025 0.0064 0.0165 0.0374
4 0.0000 0.0001
The losses can be compensated by decreasing level of consumption. The final trades are then shown in fig.7.
It can be seen that the solutions are very close to each other for the real power injections and for the reactive power consumption. The difference in the reactive power production is explained by the low cost of producing the reactive power. This leads to equivalent solutions of constraint minimization, yet with different values. In order to check whether the solutions are equivalent, the benefit of implementing them can be computed (difference between total generation cost and total benefit for the consumers). The following results are obtained:
4.5. Comparison with the OPF solution Fig.8 shows the resulting power injections at each node of the power network for the OPF solution and for the sum of all the coordinated multilateral trades before compensating for the losses.
Coordinated Multilateral Trades benefit Optimal Power Flow benefit The values of the benefits are almost equal, thus it can be concluded that the solutions are equivalent.
References [1] J. J. Grainger and J. William D. Stevenson. Power System Analysis. McGraw-Hill, Inc, New York, 1994. [2] W. Hogan. Contract networks for electric power transmission. J. Regulatory Economics, 4:211–242, 1992. [3] A. Keyhani. Study of fast decoupled load flow algorithms with substantially reduced memory requirements. Electric Power Systems Research, 9, 1985. [4] J. Weber, T. Overbye, P. Sauer, and C. DeMarco. A simulation based approach to pricing reactive power. Hawaii International Conference on Systems Sciences (HICSS), Kona, HI, January , 1998. [5] F. Wu and P. Varaiya. Coordinated multilateral trades for electric power networks: Theory and implementation. http://www.eecs.berkeley.edu/ varaiya, 1995. [6] F. Wu, P. Varaiya, P. Spiller, and S. Oren. Folk theorems on transmission open access: proofs and counterexamples. POWER report PWP-23, University of California Energy Institute, Oct 1994.
Analysis of Coordinated Multilateral Trades Pierre-Franc¸ois D. Quet [email protected]
Jose B. Cruz, Jr. [email protected]
Ali Keyhani [email protected]
Department of Electrical Engineering, The Ohio-State University 2015 Neil Avenue, Columbus OH 43210
Abstract A new operating paradigm is still needed for the power industry in order to achieve a workable deregulated market. The two main proposals, namely the Bilateral and the Poolco models, even though their original intents were to promote efficiency in a deregulated industry, in order to be workable need a powerful centralized organization. They consequently end up operating in the same way as the regulated industry. The coordinated multilateral trading model lets the suppliers and consumers make a profit on their own while the Power System Operator only intervenes if the security of the power network is threatened. The PSO gives sufficient information about the status of the power network so that the brokers can arrange trades without endangering the security of the network. Moreover the economic data of the participants is private, thus encouraging economic competition in a free market. It is shown that the process of building such trades leads to an efficient use of the power network, i.e. maximize the social welfare. A three bus example is demonstrated.
1. Introduction Beginning in 1978 with the Public Utility Regulatory Policy Act (PURPA) the electric industry has moved toward a less regulated commerce. The intent is to encourage competition in generation and customer choice. But the debate is still open over how the transmission system should be designed in the new deregulated industry in order to achieve the goal of open transmission access which would permit an efficient energy market. The problem is that not every set of power injections at the nodes of the network are possible, because of the thermal limits of the transmission lines and the need to have balance of power across the network. Due to the nature of electricity, the quantity of energy flowing through a line is governed by all the participants at the same time, thus making this problem non-trivial to solve.
Several proposals have been made which are now opposing the Bilateral Model and the Poolco Model. But these two models use the concept of sharing the resources of the transmission network through a centralized operator. They consequently end up functioning the same way the conventional operating paradigm was working, which is a regulated monopoly. A description of the Bilateral Model can be found in [2] and a critique of both of the models is in [6]. A new operating paradigm is still needed to meet the goals of an efficient electric industry. In [5], F. Wu and P. Varaiya introduce the concept of Coordinated Multilateral Trades in which the Power System Operator intervenes only if the security of the network is threatened and gives the participants sufficient information based on which suppliers and consumers can arrange feasible trades (in the sense that the trade will not overload any line on the network). The main point of this operating paradigm is that the suppliers and the consumers arrange trades among themselves keeping their economic data private, which prevent other parties to be able to “game” based on this knowledge. Wu and Varaiya proved that such trades lead to the maximization of the social welfare (i.e. minimize the total cost for the suppliers minus the total benefit for the consumers), which is also the solution of the Optimal Power Flow problem presented next. They derived a mechanism to realize such trades considering real power only. This paper extends the concept of Coordinated Multilateral Trades to the trading of both real and reactive power production and consumption. In order to do so, a different trading mechanism is needed, which will be described in the next sections.
2. Optimal Power Flow Solution In the conventional paradigm the Power System Operator (PSO) maintains the economic efficiency of the power system operations by conducting an Optimal Power Flow (OPF) study. This analysis is based on the knowledge of the producing cost of each supplier and the benefit to all the consumers. Consider a bus power network. Let:
be the real and reactive power supply
3. Coordinated Multilateral Trading Model
at each bus be the real and reactive power demand at each bus number of suppliers and numbers of consumers ns supplier' s costs (convex and strictly increasing functions) nc consumer' s benefits (concave and strictly increasing functions) power flow equations with
where is the voltage at bus and element of the admittance matrix defined as:
is the
sum of all admittances connected at bus i (admittance connected between bus i and bus j) Some more information about power system concepts can be found in [1]. An efficient method to solve the power flow equation is carried out in [3]. MVA limit on line 1 .. . MVA limit on line L where
is the number of lines in the network and with and being the real and reactive power flowing through line respectively. The Optimal Power Flow is the maximization of the social welfare, i.e. the minimization of the total generation cost minus the total consumer benefit, given the constraints of the network:
such that:
The Optimal Power Flow problem is the one of a centralized utility which possesses the knowledge of the cost function of the generators and the benefit function of the consumers, and can dictate the level of generation for everyone. It will be shown next that the same result can be achieve using the concept of Coordinated Multilateral Trades, which gives the Power System Operator much less power and lets the suppliers and the consumers carry out their economic decisions on their own.
In the Coordinated Multilateral Trading Model [5], the Power System Operator (PSO) only intervenes if the reliability of the power network is threatened. The suppliers and consumers arrange trades and look for profit on their own. The PSO gives information to the participants over how to build secure trades, i.e. trades which will not overload transmission lines. The responsibility of maintaining security is shared by all participants. There is no need for a central authority to have the knowledge of the production cost of the suppliers or the benefit function of the consumers. The economic decisions are all private, which prevent other parties to “game” based on this knowledge. Moreover, each trade will account for its share of electric losses across the power network. Let the party which arranges a trade be called the broker. The broker can be a supplier or a consumer participating in the trade or a third party. The broker's goal is to build profitable trades, i.e. trades where the total consumer benefit minus the total cost of generating the electricity is positive. The trades are carried out neglecting the electric losses they incur, their share of losses is compensated once the trades are made. This is reasonable since the cost of compensating for the losses is small compared to the trade itself. A multilateral trade will involve two or more participants and is in fact a generalization of the Bilateral Model. But as it can be seen later that it may be necessary to have three or more participants in order to remove a congestion in the power network. The process of building multilateral trades goes as follows: 1. Brokers arrange initial trades and give their intent to the PSO. 2. If these trades would overload some lines, the PSO would then curtail them in order to make the trades feasible. 3. The PSO then computes the loading vectors for each line and announces the information to all participants. The loading vectors show the influence of a power injection at each node of the network on the loading of the lines. Based on this information, the brokers can build additional trades which will be feasible, or close to feasible. 4. If the trades are not feasible, then the PSO curtails them and process 3 is repeated until no further benefit can be obtained by carrying out additional trades. 5. The set of trades is now feasible and the participants do not want to carry out any further trades. The process of compensating for the losses is carried out next. The PSO computes the loss vectors based on which the losses are allocated. These loss vectors show the influence of a power injection at each node on the losses on each line. We will show that this process, without the loss allocation, converges to the Optimal Power Flow solution. The
explicit construction of the loading vector and the loss vector will be shown in the next sections. The transmission companies should be compensated for the service they offer. This study is not carried out here, but one can envision that the benefit made by the broker while arranging a trade can be used to compensate for the service offered by the transmission companies.
A multilateral trade
is a feasible-direction trade at
if, for
,
3.1. Feasible trades Consider the same bus power network as previously, with the same definitions for the cost and benefit functions, the power balance equations and the limits on the lines.
where for
Definition 1: A multilateral trade is represented by
such that: The set of all multilateral trades is:
for
number of trades
Lemma 1:
3.2. Profitable trades
Suppose a set of multilateral trades is not feasible. There exists a curtailment schedule
such that
is on the boundary of the feasible set. Proof: The set of feasible trades , denoted by , is a convex set in a (4n+4)-dimensional space. We denote by the boundary of . is in the feasible set. For , and for , . is a convex set, therefore there is a such that . Definition 2: Suppose
.
with and being the real and reactive power flowing through the line respectively. The operation is the defined as: . , , and are the loading vectors for the line and they represent the sensitivities on the real and reactive power on the lines due to some real and reactive power injections at each node. For example, the element of is equal to the additional amount of MW flowing on the line if 1 MW is injected into the node of the network, the same quantity being taken out from the slack bus. Thus a trade is a feasible direction trade if it does not overload the lines.
Lemma 2: Let and , being the solution of the Optimal Power Flow (see previous section). There exists a Feasible Direction trade that reduces the total cost:
Lemma 3: For a Feasible Direction trade at that reduces the total cost, there is a profit to be made in arranging such a trade [5]. Definition 3: is called a profitable multilateral trade at reduces the total cost.
if it
The suppliers and the consumers make profitable trades, the social welfare (difference between the benefit for the consumers and the cost of producing) increasing at each time, until the optimal solution is attained. But a profitable
trade may not be feasible, in that case the PSO may have to curtail it. Lemma 4: A profitable trade at in the feasible direction can always be made feasible by curtailment. Proof: Assume the constraint is violated, then using the same concept as in Lemma 1 there exists a curtailment schedule such that . Lemma 5: After curtailment, a profitable trade is still profitable. Proof: The cost functions for the suppliers are convex and the benefit functions for the consumers are concave, ns nc then is convex. This convexity guarantees that:
The trade
is then still profitable.
3.3. Trading arrangement The objective of the broker is to maximize the total cost reduction subject to the constraints that: the amount of generation and load balances out for each trade to be implemented, the trade is feasible. Let and be the sets of suppliers and consumers engaged in the trade respectively. The objective of the broker can be written as: min subject to:
and for
The loading vectors in Definition 2.
Consider the case with one transmission congestion. A trilateral trade at least may be necessary in order to remove the congestion. The more participants are involve in a trade the better a feasible and profitable solution can be found.
3.4. Coordinated Multilateral Trading Process The trading process proceeds as follows [5]: 1. The brokers arrange initial trades . 2. If these trades would overload some lines, the PSO curtail them in order to make the trades feasible. 3. The PSO broadcasts the Loading Vectors of all lines of the network. 4. The brokers arrange additional trades following the method of section 3.3. If no feasible and profitable trades are possible any more, then stop. 5. If the set of trades overload some lines, the PSO curtails them and go to step 3. Theorem 1: It is assumed that a broker will always arrange an additional trade if it is a profitable one and that the different parties will be willing to carry it out. Under this assumption the Coordinated Multilateral Trading Process converges to the Optimal Power Flow solution [5].
3.5. Compensation of the losses We would like each individual trade to take care of the losses it incurs. Their exact evaluation is not easy since the losses caused by a trade is a function of the other trades. Yet it is believed that they can be approximated using the Loss Vectors calculation. The Loss Vectors characterize the influence of a power injection at each node on the losses on each line. More specifically: for
,
,
and
and for
are defined
Notes: If the broker arranges a trade involving all the participants, the solution is the Optimal Power Flow solution.
with and being the real and reactive power losses of the line respectively.
The losses caused by the trade 1
3
2
on the line would then be approximated by: Figure 1. 3 bus power network The data of the lines are: where and are the approximated real and reactive losses respectively caused by the trade on the line . By computing the power flow solution for all the trades the total losses across the network can be determined, being equal to the power discrepancies at the slack bus. Approximation factors can be defined as: AppP
total real losses D L
AppQ
total reactive losses D L
with D and L being the number of trades and the number of lines respectively. In order to allocate all the losses of the network, each individual trade [k] will be assigned to compensate for the losses:
r 0.06 0.0315 0.04
x 0.4 0.25 0.2
The per unit system is: 1 pu = 100 MW. The thermal limit on the line 2-3 is chosen to be 5 MVA. In order to simplify this example, the limits on the other lines are not considered. The cost functions for the suppliers and the benefit functions for the consumers are of the type: for the suppliers:
is a convex and strictly increasing function for the consumers:
with
AppP AppQ where and caused by the trade
are the real and reactive losses respectively.
The participants of the trades can then compensate for the losses by either increasing the generation or decreasing the consumption. It can be argued that once the losses are compensated it results in different trades and consequently additional losses. But in practice the loss discrepancy at the slack bus after the loss compensation process described above is negligible.
4 3-bus example The data for this example is taken in part from [5]. Consider the three bus power network shown in fig.1 with two suppliers (at bus 1 and 2) and one consumer (at bus 3).
and
In this example it is considered that the consumer wants to keep a constant power factor , then:
The function is a sum of concave functions, thus it is concave itself. The difference Total cost of generation Total consumer benefit is then a convex function. This model for the benefit function of the reactive power consumption is taken from [4]. The reactive power really serves as a service which enables the consumption of real
power. Using this point of view, the benefit of the reactive power is considered to be the avoidance of moving the reactive power from some desired level for a given power consumption. It is assumed that each consumer possesses a power factor correction device as shown in fig.2.
constraint on the lines (just constraint to the power balance). The resulting power injections, consumption and flows on the line 2-3 are as shown on fig.4. The brokers give the values of the trades to the Power System Operator. Trade 1: 76.1 MW + 17.84 MVar
Bus
1
dp+j dq 13 MVA
Filtering device 3
2
dp + j f(dp)
Figure 2. Consumer filtering device
Trade 1: 76.1 MW + 17.84 MVar Trade 2: 17.92 MW + 0.97 MVar
The cost of operating this device is equal to zero when it is performing no correction (i.e. when ) and increases, according to as it moves away from that point (see fig.3).
Trade 2: 17.92 MW + 0.97 MVar
Figure 4. Initial trades
4.2. Curtailment k(x) 0
x
The Power System Operator notices that 13 MVA would be flowing on the line 2-3 if those trades were applied (by solving the power flow equations, see [1] and [3]). The PSO then curtails the trade arranged between the generator 1 and the consumer such that only 5 MVA would be flowing through the line 2-3. See the curtailed trades on fig.5. The bus 3 is chosen to be the slack bus in this example.
Figure 3. Function Trade 1: 41.4 MW + 9.71 MVar
The numerical data for the cost functions are: 1
5 MVA
Note that the consumer would like a power factor of 0.95 after his filtering device.
4.1. Initial trades The first broker arranges a trade between the supplier at bus 1 and the consumer at bus 3. A second broker arranges another trade between the supplier at bus 2 and the consumer at bus 3. The values of the trades should be made available to the participants for discussion. It is assumed that the participants make reasonable economic decisions, i.e. that they carry out a trade if it is profitable. The outcomes will result in the same solutions as the constraint minimization described in section 3.3 without the inequality
3
2
Trade 1: 41.4 MW + 9.71 MVar Trade 2: 17.92 MW + 0.97 MVar
Trade 2: 17.92 MW + 0.97 MVar
Figure 5. Curtailed trades
4.3. Additional trades The PSO computes the loading vectors using Definition 2 and broadcasts them. The loading vectors for the line 2-3
Trade 1: 41.4 MW + 9.71 MVar Trade 3: 19.67 MW + 7.8 MVar Trade 4: 0.03 MW + 0.1 MVar
are:
1
Based on this information, a third broker arranges an additional trade among all the participants following the process described in section 3.3. The PSO checks that this additional trade would not overload the line 2-3 and broadcasts a new set of loading vectors. The broker arranges another trade based on the new information. Since there is no more benefit to be made in arranging additional trades the process stops here. All the trades are shown in fig.6. Trade 1: 41.4 MW + 9.71 MVar Trade 3: 19.67 MW + 7.80 MVar Trade 4: 0.03 MW + 0.10 MVar
5 MVA
3
2
Trade 1: 40.56 MW + 5.12 MVar Trade 2: 17.67 MW + 0.68 MVar Trade 3: 34.43 MW + 10.76 MVar Trade 4: 0.04 MW + 0.18 MVar
Figure 7. After loss compensation CMT : 61.09 MW + 17.61 MVar OPF : 61.88 MW + 12.16 MVar
1
1
5 MVA
3
Trade 2: 17.92 MW + 0.97 MVar Trade 3: 15.4 MW + 6.7 MVar Trade 4: 0.02 MW + 0.08 MVar
5 MVA
2
3 Trade 1: 41.4 MW + 9.71 MVar Trade 2: 17.92 MW + 0.97 MVar Trade 3: 35.07 MW + 14.50 MVar Trade 4: 0.05 MW + 0.19 MVar
Trade 2: 17.92 MW + 0.97 MVar Trade 3: 15.4 MW + 6.70 MVar Trade 4: 0.02 MW + 0.08 MVar
CMT : 94.42 MW + 25.36 MVar OPF : 94.24 MW + 25.29 MVar
2
CMT : 33.34 MW + 7.75 MVar OPF : 32.36 MW + 13.13 MVar
Figure 6. With additional trades Figure 8. CMT: Coordinated Multilateral Trades OPF: Optimal Power Flow solution
4.4. Loss allocation The Power System Operator computes the loss vectors following the process described in section 3.5. The loss allocation results in the following:
Real losses Reactive losses
1 0.0084 0.0458
Trades 2 3 0.0025 0.0064 0.0165 0.0374
4 0.0000 0.0001
The losses can be compensated by decreasing level of consumption. The final trades are then shown in fig.7.
It can be seen that the solutions are very close to each other for the real power injections and for the reactive power consumption. The difference in the reactive power production is explained by the low cost of producing the reactive power. This leads to equivalent solutions of constraint minimization, yet with different values. In order to check whether the solutions are equivalent, the benefit of implementing them can be computed (difference between total generation cost and total benefit for the consumers). The following results are obtained:
4.5. Comparison with the OPF solution Fig.8 shows the resulting power injections at each node of the power network for the OPF solution and for the sum of all the coordinated multilateral trades before compensating for the losses.
Coordinated Multilateral Trades benefit Optimal Power Flow benefit The values of the benefits are almost equal, thus it can be concluded that the solutions are equivalent.
References [1] J. J. Grainger and J. William D. Stevenson. Power System Analysis. McGraw-Hill, Inc, New York, 1994. [2] W. Hogan. Contract networks for electric power transmission. J. Regulatory Economics, 4:211–242, 1992. [3] A. Keyhani. Study of fast decoupled load flow algorithms with substantially reduced memory requirements. Electric Power Systems Research, 9, 1985. [4] J. Weber, T. Overbye, P. Sauer, and C. DeMarco. A simulation based approach to pricing reactive power. Hawaii International Conference on Systems Sciences (HICSS), Kona, HI, January , 1998. [5] F. Wu and P. Varaiya. Coordinated multilateral trades for electric power networks: Theory and implementation. http://www.eecs.berkeley.edu/ varaiya, 1995. [6] F. Wu, P. Varaiya, P. Spiller, and S. Oren. Folk theorems on transmission open access: proofs and counterexamples. POWER report PWP-23, University of California Energy Institute, Oct 1994.
