The Research on Equilibrium Principle in Spot Power Market

Wang Chengmin and Jiang Chuanwen
Department of Electrical Engineering
Shanghai Jiaotong University
Shanghai, Huashan Road, No. 1954
China

Abstract: - Equilibrium is the optimum operational condition for the power market by economics rule. Whereas the spot power market cannot generally achieve the equilibrium due to existing network losses and congestion. The impact of the network losses and congestion on spot power market is analyzed in order to newly establish the equilibrium model under considering the network topology. At the end, the OPF is solved by means of the equal price principle. The case study is made on IEEE-30 nodes system in order to prove the results proposed in this paper.

Key-Words: - Deregulation; Spot power market; Equilibrium; Network losses; Congestion; Power system;

1   Introduction
With the development of the research on power market about its electric utilities restructure and deregulation, power system industry has entered into an increasingly competitive era in which it becomes more realistic to improve economics and reliability of power system by enlisting market forces. Now, the essential theory for the power market, which combined the operational rule in the power system with the market principles in economics, becomes more important than ever.
     Since the deregulation of electric industry, tons of models had proposed to simulate the power market. They could be classified into two groups: 1) The auction design based on microeconomic theory, whose primary idea is to conduct auction relied on the intersection point of supply and demand curves [1]~[4] including the equal price method. 2) Introducing the traditional economic dispatching arithmetic into the power market transactions [5][6] in order to determine optimal unit commitment and marginal price.
     The simple and transparent auction process is the advantage of the first model but it is difficult to treat with constrains which are necessary to ensure that power system is operated in security. On the contrary, the second model is able to solve the optimization problem considering various constrains. But it is more complicated and lack of transparence. Therefore, it is the key issue to integrate the market principle in economics into practical operation in the power system. Literature [7] has tried to find the equilibrium point for the power pool auction by using Lagrangian relaxation approach.
     The losses and congestion are two main properties of the electric network related to the spot power transactions, therefore the responsibilities for ISO and PX are to operate the power transactions on the premise of the security of the power system. Utility engineers have produced a large number of literatures in network losses allocation and congestion dispatching[8]~[10], which similarly overlook the application of the market principles.
     The equilibrium model for the spot power market is described to indicate that equilibrium is the optimum condition for the power market, then the impact is analyzed and the equilibrium model designed for the power market is established considering the network losses and congestion. Therefore, the final OPF problem is solved by using equal price method in this paper. The following assumptions are proposed: 1) The ancillary services pricing problem is left out of account when the active power transactions are procured; 2) The supply offers are an increasing curve with the decreasing curve for the demand bids under pool mode. 3) The bidding curves offered by participants is continuously differentiable, that is logical because final arithmetic not related with the differentiability about the bidding curves whereas they are not differentiable at some points. 4) The unit commitment considered as obligation of the generation company is not included here.
     The paper is organized as follows. Section 2 introduces the equilibrium model for the power market and its properties. The impact of the network losses to the power market equilibrium is analyzed and the new equilibrium model is established in section 3. Section 4 introduces the power market equilibrium model under congestion. The final power market equilibrium mathematic model is expressed as an OPF form solved by equal price method in section 5. Section 6 is a case study made at IEEE-30 nodes system in order to prove the results proposed in this paper.

2   The advantages of market equilibrium
In the general commodity market, equilibrium means the optimum condition at which supply and demand achieve balance with maximum social benefits and equitable benefits allocation. The Walras equilibrium for a single commodity market can be described as follows.
                                     (1)
                                    (2)
where  is the total commodity demand denoted by the difference between supply and demand and  is  the price of the commodity. 
     The equilibrium of supply and demand may also be realized in the power market showed in figure 1, where line 1 denotes the generation company’s offers; line 2 denotes the consumer’s biddings;  is the equilibrium price; is the equilibrium power quantity. Thus O is the equilibrium point. At this point, the market benefits achieved maximum and the market benefit allocation is equity. The shadow area above equilibrium price denotes the generation benefits while nether part denotes the consumer’s benefits. It is important to simulate the power market because the power market can’t automatically achieve the equilibrium condition, which is the difference between power market and general commodity market. 

Fig. 1. Supply and demand equilibrium figure in the power market

     In figure 1, it can be assumed that there are  participants among them and  denote the supply number and demand number respectively, then the following mathematic model can be used to express the equilibrium of supply and demand.
                                 (3)
                                  (4)
                                (5)
where  is the k-th participant’s output power which is negative for the consumer’s corresponding price . It is easy to prove that . Then equation (5) could also be omitted.
     In fact, more objective functions can be adopted as minimal generation cost or minimal consumer payment because constrains (4) and (5) are the key issues corresponding the Valras equilibrium form regardless to network properties. Otherwise, the maximal social benefits denoted by formula (3) is the decisive objective due to the new equilibrium condition can be achieved when the old one is broken.
The total social benefits consist of two parts: the first part is  denoting generation company’s benefit; the second part is denoting the consumer’s benefit. The explanations for them are as follows: in a perfect competitive market, the minimum  which denotes the generation production cost indicates maximal generation benefit because  reflects the participant’s marginal production cost. For the same reason, the maximum  indicates that consumer benefit achieves maximum because the injected power is negative.
     According to the discussion above, the characters for the power market equilibrium are concluded as follows: 1) the uniform marginal price is performed; 2) the balance of supply and demand is achieved and 3) the balance of income and expenses is achieved.

3   The impact of network losses on power market equilibrium
The accurate expression of the active power balance is as following instead of formula (4) taken into account of network losses:
                                (6)
where  is the network losses. The previous equilibrium is broken because the present network losses  which is needed to be fairly allocated among participants. The related pricing for network losses based on marginal cost theory is improper because it leads to unequal between the practical and allocated network losses. Therefore, the pricing based on embedded cost is necessary. The quadric function can be used to denote the network losses accurately.
                              (7)
where  is the vector of injected power at nodes, 、 are matrixes varying according to the state of the system while  is a constant, viz.:
                             (8)
where  is a row vector which makes (generally  if ignored) tenable. The scalar form of formula (8) is:
                 (9)
Definition:
                        (10)
as the embedded cost factor of network losses only varies according to the state if the system. So rewrite the formula (6) as the following:
                               (11)
The Lagrangian function combined formula (3) with the formula (11) as the following:
                       (12)
The optimal solutions are obtained from the following equations:
    (13)
                              (14)
Assumption:
                                    (15)
as the uniform marginal price:
                  (16)
Introducing the above formula into (3):
                      (17)
     The new power market equilibrium model taking the network losses into account can be composed of formulas (17) and (11) which is equivalent to formulas (13) and (14), at the same time the following constrain is satisfied:
                          (18)
The figure 2 shows the new equilibrium with the power  and loss  at k-th node.

Fig. 2. Market equilibrium figure included network loss

4   Power market equilibrium under congestion
The above equilibrium will be impacted when the network congestion is considered. So, an additional constrain is added to equilibrium model (17) and (11):
                      (19)
where  denote the number of line. The objective function value  called as congestion charge is greater than 0 or equal to 0 according to the optimization theory. The congestion charge is needed to be fairly allocated among participants, which relates the pricing for the network congestion and the marginal cost pricing theory. Therefore, it is considered logical because of the congestion that is a quite different concept from the network loss. It can’t consume or produce the active power but only impact the power transactions. The Lagrangian function by introducing the formula (11) and (19) into the formula (17) and defining  can be written as:
        (20)
Its optimum condition is:
 (21)
viz.:
   (22)
Definition:
                      (23)
as the marginal cost factor of network congestion reflects the impact of the injected nodal power to network congestion. Therefore, the objective function (17) can be rewritten by introducing the equation (22) into (17) as:
               (24)
and subject to:
                                     (25)
and the constrain:
                     (26)
is satisfied automatically. The equilibrium under the condition of congestion shown in figure 3 is composed of formulas (24)~(26) with the marginal price  and injected power  at node k. The congestion only occurs on one line here.
     The benefits whether for generation or the consumer decrease due to the network congestion compared with the case shown in figure 2. The final equilibrium condition is realized at point .

5   Calculations
The equal price method based on economic principle is widely used in many countries. In recent year, the development based on the equal price method enables workers to map out the generation schedule considering the network losses and congestion
     The equal price means the direction in finding optimal objective value in the equal price principle,  is equal to 0 when the congestion does not appear in the system, otherwise  is determined by formula (22) where  indicates the restriction of the limit at transmission line to the optimal direction and  denotes the shadow price correspondingly.  can be determined as the following: 
               (27)
     The calculating steps for the equal price method are listed as follows:
1、To give an initial value of the marginal price  and an increment ; 
2、To determine the participants’ power by offered and bided data;
3、To solve power flow problem and determine system states and turn to step 5 if line flow do not exceeds limit, otherwise continue;
4、The calculating process is finished if formula (25) is satisfied, otherwise return to step 2 by a added increment  to ;
5、To calculate the shadow price  and  and , then the nodal price is corrected by formula  and return to step 2.

Fig. 3. Market equilibrium figure under congestion

     In upper iteration process, the slack node is not included which can be led to hard convergence in power flow problem. So the development in the equal price method is necessary. 
     The active power at rest N-1 nodes is determined when the node N (generator node) is assumed as slack node for a given marginal price, then the active power  at slack node can be obtained by the power flow calculation, therefore formula (25) can be rewritten as:
                      (28)
viz.:
                      (29)
because:
                          (30)
so formula (30) can be rewritten as:
                   (31)
Therefore, the symbol used to determine the end of iteration in the calculating process of equal price method is that practical power obtained by power flow calculation is equal to the scheduling value obtained by the offered data at slack node N because of .

6   Case study
The case study is made on IEEE-30 nodes system with the nodal data shown in table 1 and the rest data can be found in many other literatures (for the sake of concision, they are omitted) The explanations for the data listed in table 1 are as follows: 1) the curves offered by generation company or bided by consumer are assumed as a line viz. ; 2) the node type 1 denotes generator node and type 2 denotes load node with slack node type 3; 3) the limit at transmission line is equal to 1.15 rated current. There is an assumption that only a generation company (or consumer) exists at every node.
     The  is less than a given positive number as the result of 14 times of iterations starting from a marginal initial value 280 yuan/MW. The calculating results are listed in table 2.

Table 1 Node Data for IEEE-30 Buses System
Node	Voltage 
Magnitude	Voltage
Angle	Coefficient ‘a’
for Generator	Coefficient ‘b’
for Generator	Coefficient ‘a’
for Consumer	Coefficient ‘b’
for Consumer	Node
Type	Cap.	
1	1.0500	0.0000	128.00	160.00	0.0000	0.0000	1	0	
2	1.0338	2.7374	195.00	171.00	0.0000	0.0000	1	0	
3	1.0309	4.6722	0.0000	0.0000	-752.00	382.00	2	0	
4	1.0258	5.5963	0.0000	0.0000	-761.00	388.00	2	0	
5	1.0058	9.0005	0.0000	0.0000	-760.00	388.00	2	0	
6	1.0214	6.4821	0.0000	0.0000	0.0000	0.0000	2	0	
7	1.0073	8.0435	0.0000	0.0000	-753.00	380.00	2	0	
8	1.0230	6.4864	206.80	189.00	0.0000	0.0000	3	0	
9	1.0583	8.1508	0.0000	0.0000	0.0000	0.0000	2	0	
10	1.0527	10.008	0.0000	0.0000	-734.00	385.00	2	0.19	
11	1.0913	6.3003	195.00	186.00	0.0000	0.0000	1	0	
12	1.0564	9.2015	0.0000	0.0000	-508.00	388.00	2	0	
13	1.0883	8.0216	185.00	178.00	0.0000	0.0000	1	0	
14	1.0428	10.098	0.0000	0.0000	-740.00	384.10	2	0	
15	1.0393	10.221	0.0000	0.0000	-888.00	385.11	2	0	
16	1.0476	9.8207	0.0000	0.0000	-762.00	384.00	2	0	
17	1.0459	10.159	0.0000	0.0000	-750.00	386.00	2	0	
18	1.0319	10.836	0.0000	0.0000	-762.00	383.00	2	0	
19	1.0307	11.010	0.0000	0.0000	-761.00	386.00	2	0	
20	1.0354	10.817	0.0000	0.0000	-750.00	383.00	2	0	
21	1.0404	10.466	0.0000	0.0000	-763.00	387.00	2	0	
22	1.0409	10.459	0.0000	0.0000	0.0000	0.0000	2	0	
23	1.0314	10.666	0.0000	0.0000	-791.32	388.22	2	0	
24	1.0292	10.915	0.0000	0.0000	-893.00	380.00	2	0.04	
25	1.0298	10.803	0.0000	0.0000	0.0000	0.0000	2	0	
26	1.0124	11.211	0.0000	0.0000	-750.00	383.00	2	0	
27	1.0388	10.476	0.0000	0.0000	0.0000	0.0000	2	0	
28	1.0177	6.8955	0.0000	0.0000	0.0000	0.0000	2	0	
29	1.0192	11.668	0.0000	0.0000	-762.21	383.23	2	0	
30	1.0080	12.524	0.0000	0.0000	-752.32	386.34	2	0	

     The flow of line 18 exceeds the limit at 5-th iteration with current 0.369323. So, the correction for nodal price is made with =9.30178 and results of nodal price and congestion factor are shown in table 3. It is can be seen that  have a leap at 5-th iteration due to congestion. The final calculating results for nodal powers and network losses are shown in table 4.
Table 2 The Results of Power Flow
Iteration	       	       	    	
1	0.193909	0.442253	279.0	
2	0.242971	0.437418	278.0	
3	0.292112	0.432582	277.0	
4	0.341334	0.427747	276.0	
5	-0.027467	0.421167	275.0	
6	0.0212349	0.416332	274.0	
7	0.0700157	0.411496	273.0	
8	0.118875	0.40666	272.0	
9	0.167814	0.401825	271.0	
10	0.216831	0.396989	270.0	
11	0.265928	0.392154	269.0	
12	0.315104	0.387318	268.0	
13	0.36436	0.382482	267.8	
14	0.380148	0.388303	267.4	

Table 3 Nodal Price and Congestion Factor
Node				
1	-0.244274	-2.272183	278.730	
2	-0.372135	-3.461517	279.920	
3	-1.435110	-13.34907	289.807	
4	-1.452280	-13.50878	289.967	
5	-1.450370	-13.49102	289.949	
6	0	0	276.458	
7	-1.437020	-13.36684	289.825	
8	0.038772	0.3606440	276.097	
9	0	0	276.458	
10	-1.400760	-13.02956	289.488	
11	-0.372135	-3.461517	279.920	
12	-0.969460	-9.017704	285.476	
13	-0.353052	-3.284012	279.742	
14	-1.412210	-13.13606	289.594	
15	-1.694640	-15.76316	292.221	
16	-1.454190	-13.52655	289.985	
17	-1.431290	-13.31354	289.772	
18	-1.454190	-13.52655	289.985	
19	-1.452280	-13.50878	289.967	
20	-1.431290	-13.31354	289.772	
21	-1.456100	-13.54432	290.002	
22	0	0	276.458	
23	-1.510140	-14.04699	290.505	
24	-1.704190	-15.85200	292.310	
25	0	0	276.458	
26	-1.431290	-13.31354	289.772	
27	0	0	276.458	
28	0	0	276.458	
29	-1.454590	-13.53027	289.988	
30	-1.435720	-13.35475	289.813	

Table 4 The Calculating Results
Node	Volt. Mag.	Voltage
Angle	Injected
Active Power	Injected
Reactive Power     	
1	1.0500	-2.4756	0.857267	0.122187	
2	1.0338	-3.6591	0.512408	0.219612	
3	1.0309	-6.5099	-0.134565	0.026162	
4	1.0258	-7.0423	-0.140648	0.0723951	
5	1.0058	-6.1781	-0.140857	-0.164064	
6	1.0180	-7.1397	0	0	
7	1.0073	-7.0301	-0.131707	-0.110505	
8	1.0230	-6.4864	0.413697	-0.027858	
9	1.0555	-9.5446	0	0	
10	1.0527	-13.267	-0.142388	-0.111837	
11	1.0913	-5.0369	0.435485	0.201830	
12	1.0564	-12.279	-0.219536	-0.212755	
13	1.0883	-8.7795	0.501308	0.2632930	
14	1.0428	-14.606	-0.139873	0.0198106	
15	1.0393	-14.915	-0.114740	0.0102849	
16	1.0476	-13.812	-0.135191	0.0292881	
17	1.0459	-13.815	-0.140305	-0.0388213	
18	1.0319	-16.801	-0.133879	0.0428145	
19	1.0307	-16.848	-0.138020	-0.0113463	
20	1.0354	-16.389	-0.136305	0.0484001	
21	1.0404	-13.929	-0.138922	-0.1407030	
22	1.0411	-14.051	0	0	
23	1.0314	-16.323	-0.134857	0.0364030	
24	1.0292	-15.997	-0.108276	-0.0536238	
25	1.0342	-17.319	0	0	
26	1.0124	-20.716	-0.136305	0.0399148	
27	1.0466	-16.369	0	0	
28	1.0120	-7.9276	0	0	
29	1.0192	-20.082	-0.134139	0.0367451	
30	1.0080	-20.401	-0.140269	-0.0089916	

7   Conclusion
In this article, the power market equilibrium is described in detail considering the network characters included losses and congestions because the equilibrium is the optimum operation state of power market, the following conclusions are got:
     1) The power market can’t generally achieve the equilibrium state due to that bids or offers of market participators can’t reflect the network characters;
     2) After correction for bidding curves or offering curves the power market achieves again the equilibrium state under considering the network losses and congestion; 
     3) The equal price method is still suitable under considering the network topology.

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