Unbiased economic dispatch in control areas with conventional and renewable generation sources

Electric Power Systems Research 119 (2015) 313–321

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Unbiased economic dispatch in control areas with conventional and renewable generation sources Siby Jose Plathottam ∗ , Hossein Salehfar Department of Electrical Engineering, University of North Dakota, Grand Forks, ND, USA

a r t i c l e

i n f o

Article history: Received 3 June 2014 Received in revised form 16 September 2014 Accepted 19 September 2014 Keywords: Economic dispatch Wind power LCOE Simulated annealing PSO Multiobjective

a b s t r a c t This paper describes an economic dispatch formulation for a power system control area with high penetration of renewable generation and energy storage. We develop a comprehensive objective function that is unbiased to both the conventional and renewable generation sources. Operating constraints of conventional sources are considered, wind turbine generation is treated as a dispatchable source, and solar photovoltaic generation as negative load. A simulated annealing and particle swarm optimization algorithm are separately applied to minimize the objective function. The results obtained by the proposed economic dispatch formulation show that even a variable energy resource like wind can be dispatched in a manner that benefits the control area as a whole. © 2014 Elsevier B.V. All rights reserved.

Economic dispatch (ED) algorithms have been traditionally used to minimize generation cost while maintaining load demand in a power system control area. Contemporary power systems consist of a mix of conventional and renewable energy sources as well as large energy storage systems [1,2]. The renewable sources can generally be sub-divided on the basis of their geographic location and whether they are dispatchable. The non-homogenous nature of the generation sources presents a variety of challenges as well as opportunities to the independent system operators when scheduling generation to meet the load demand. This paper describes an intuitively simple strategy for ED that can effectively reduce the cost of meeting the load demand while enabling a more sustainable and equitable operation of the grid. The paper involves the development of a comprehensive cost function, constraints and solving the optimization algorithm using two heuristic methods.

maximum limit on the energy that can be captured at a given time, modern renewable generation technologies allow a great degree of controllability over the active and reactive power generation from individual sources [3]. Additionally renewable generation sources entail a considerable initial investment and the operation of the grid has to facilitate a recoupment of the investment costs. Also the utilization of large scale energy storage in the system to make use of excess energy by optimizing its discharge and recharge cycles is a challenge. Another critical aspect is the operating and economic constraints associated with conventional dispatchable sources. For example the heat rates of coal and gas plants will show a drastic increase when they are forced to operate at low loads. Also incessant ramp ups/ramp downs and start ups/shut downs to match renewable energy production will cause excessive thermal stresses to plant equipment. Hence any meaningful ED algorithm in a modern power system will have to take into account the constraints and costs associated with both conventional and renewable sources.

2. Overview of opportunities and challenges

3. Recent work in economic dispatch

The value of non-dispatchable renewable energy sources like wind and solar can be increased by reducing the stochastic nature of their production [1]. Although the local weather conditions put a

There have been a number of contributions recently in the area of coordinating thermal-hydel generation sources with wind power. The problem of dispatching wind turbine generation with varying degree of active and reactive power controllability to comply with system operating demands as well as maximize the revenue for the wind farms has been described in [3]. However the mechanism by which the system operator arrives at limits

1. Introduction

∗ Corresponding author. Tel.: +1 7012138536. E-mail address: [email protected] (S.J. Plathottam). http://dx.doi.org/10.1016/j.epsr.2014.09.025 0378-7796/© 2014 Elsevier B.V. All rights reserved.

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for wind power is not mentioned and generation cost of conventional sources is not optimized. The cost of wind energy has been incorporated as part of the standard ED objective function in [4]. A penalty cost for intermittency of the wind is also determined by using a Weibull probability density function. However this approach while beneficial to the transmission system operators in minimizing reserve requirements does not take into account the economic impact on the wind farm when wind variability is high. The same approach has been used to solve an optimal power flow problem in [5,6]. In [7], the investment costs of wind and solar have been used to compute the total generation cost when used in conjunction with combined heat and power (CHP) plants operating based on ED. Forecasts of wind and solar power have been used but they are considered as a negative load and are not part of the ED cost function. Various strategies using power flow sensitivity factors (PFSF) to coordinate generation in systems having multiple distributed generation sources have been described and compared in [8]. In [8], a financial evaluation has been done using net present value analysis. While this approach is feasible for real time control, the dependence of the PFSF matrix on operating condition limits its use for long term scheduling. Wind and thermal scheduling taking into account the constraints of both and generation cost of thermal power has been described in [9]. Spinning reserve requirement for meeting shortfalls in wind is also considered. In [9], the wind generation acts like a negative load with constraints imposed by the wind power profile but the long term economic state of the wind farm is not considered. In [10], the concept of balancing the risk that is incurred in using wind power and minimizing the operating cost of the thermal generation was introduced with risk being modeled by a fuzzy membership function. A recent work that proposes a means to manage the operational uncertainties in a wind–thermal system is [11] where the optimal generation schedule for a wind farm is allowed to vary within an interval instead of being limited to a fixed value. The optimization problem in [11] considers the worst case scenarios for spinning reserve and transmission line flows, but the economic condition of the wind farm is not considered. In [12], the dispatch of hydel units in a hydel-thermal power system so as to minimize the water consumption has been proposed. The water consumption in [12] is modeled as a quadratic cost function, but the relationship between water consumption and the reservoir head is not considered. The cost functions in [3–8] quantify the generation costs of wind power in one way or another but do not take into account the systems reserve availability that can offset the risk of using wind power. The problems developed in [9–11], on the other hand, quantify the risk but do not consider the economic impact of curtailing the wind power. The present work by the authors aims to develop the concept of combining the economics and the risk of using wind power in the control areas with a high degree of wind penetration as well as optimizing the generation from conventional plants. To this end the LCOE and the available system reserve will be utilized. Moreover, in the case of conventional plants, the aim of the present work is to obtain a generation schedule that optimizes certain critical performance indicators like heat rate and specific water consumption instead of minimizing the gross generation costs. Additionally, the authors have explored a few reasons that discourage the dispatching of solar PV power and justify considering it as a negative load.

4. Generation cost functions and constraints Fig. 1 is the simplified version of a modern power system control area with high degree of renewable penetration that was presented in [2]. The same is proposed to be used as the basis of this paper. The system considered has 20% wind generation and 10% distributed PV compared to the overall generation capacity. In the following

Fig. 1. Block diagram of control area.

sections of the paper, the cost functions and constraints associated with each component of Fig. 1 will be developed. In order to save space the list of symbols, units, and notations is tabulated in Appendix. 4.1. Constraints for conventional generation The conventional power plants in the problem have a common set of constraints for ramp rates (1) and operating levels (2). The subscript gen refers to either thermal or hydel generation while the superscript online refers to the maximum generation capacity available in a particular time block. min-ramp

Pgen

min-gen

Pgen

max-ramp

≤ |Pgen,i+1 − Pgen,i | ≤ Pgen

online ≤ Pgen,i ≤ Pgen,i

(1) (2)

4.2. Conventional thermal generation A power plant’s heat rate is the amount of thermal energy used by an electrical generator or power plant to generate 1 kWh of electrical energy. An ED of the thermal power plant should enable it to operate below a specific maximum heat rate for economic and environmental reasons. The heat rate of a plant is directly proportional to its fuel cost as long as the coal calorific value and cost do not change. Traditionally a polynomial cost function has been found to be sufficient to estimate the fuel cost incurred when a specific amount of thermal power is scheduled [13] and the same is used in this paper. The equations for these are not included to conserve space and prevent redundancy. We estimate the performance of the thermal power plant for N time blocks by calculating the day average of the heat rate (kJ/kWh) using (3). Here Ptherm,i is the day ahead power scheduled from the thermal plant in the ith time block. Ctherm is the fuel cost incurred for N time blocks in dollars. CoalCV and Coalcost are the calorific value of the coal (kJ/kg) and cost of the coal ($/kg) respectively. HRtherm =

N

Ctherm CoalCV Coalcost 24 N

(3)

P i=1 therm,i

4.3. Conventional hydel generation A conventional hydel power plant is unarguably one of the cheapest sources of power since the working fluid is water and it is available in a concentrated form. However water is gradually becoming a scarce resource and the water stored behind a hydroelectric dam cannot exclusively be used for power generation. Hence the authors propose to use a cost function that represents the cost of using water (or net outflow from the reservoir) in generating a scheduled amount of power as shown in (4) where ah is a

S.J. Plathottam, H. Salehfar / Electric Power Systems Research 119 (2015) 313–321

non-monetary value representing the cost of using one cubic meter of water. The water outflow from the reservoir in the ith time block water ) is linked to the average reservoir head in that time block (Fout,i avg

using (5) while the average head (Hhydel,i ) is given by (6) [14]. 24  penalty water ah Fout,i + Chydel N N

Chydel =

(4)

i=1 water Fout,i =

avg Hhydel,i

Phydel,i avg gHhydel,i hydel

= Hhydel,i−1 +

water − (P Fin hydel,i /(gHi−1 hydel ))

 24 

2Ahydel

N

=

N 

pred

associated with not fully utilizing the predicted wind power Pwind,i C in the ith time block. Wwind is a weight that is a function of the day C Pwind,i on average of the capacity factor as shown in (12). Wreserve the other hand quantifies the risk associated with the scheduled C wind power. Wreserve is weight that is a function of the available backup generation in the system as shown in (13).

24 N

Cwind =

(6)

C Wwind,i

N 

pred

C LCOE C max(Wwind,i Cwind (Pwind,i − Pwind,i ), Wreserve,i Pwind,i )

(11)

i=1

dayavg

= f (CFwind )

(12)

C r min reserve Wreserve,i = f (Pwind,i , Parea,i )

(13)

dayavg

CFwind

=

N  P

wind,i

(14)

P rated i=1

wind

r min Pwind,i = kPwind,i

WH min(0, Hhydel,i − Hhigh )

(15)

pred

i=1

+ WL max(0, Hhydel,i − Hlow )

(7)

Specific water consumption (m3 /kWh) is used as the performance index for the hydel plant and calculated using (8). WRhydel =

pred

C LCOE (P function (11). The term Wwind,i Cwind − Pwind,i ) is the cost wind,i

(5)

water is the inflow into the reservoir and is taken to be constant Fin for the entire ED period. Also the hydraulic turbine efficiency typically peaks at the design head. Therefore to restrict variation in head between Hhigh and Hlow a penalty cost is imposed using (7) where WH and WL are costs imposed on deviation. penalty Chydel

315

N water F Ni=1 out,i

(8)

P i=1 hydel,i

In (11) Pwind,i is the predicted short term wind power available which may be obtained from physical and statistical models. A good overview of the latest short term wind power prediction dayavg methodologies are presented in [16]. The functions f (CFwind ) and reserve r min f (Pwind,i , Parea,i ) have been implemented as piecewise linear functions in this work but fuzzy functions could potentially be used to give better results. pred

cut-in Pwind,i ≤ Pwind,i ≤ Pwind,i

(16)

N Pwind,i Uwind = i=1 N pred

4.4. Wind generation

(17)

P

i=1 wind,i

Since there is no fuel cost associated with wind energy, the levelized cost of energy (LCOE) which is calculated using a net present value analysis may be used as an indicator of the generation cost [8,15]. LCOE Cwind =

I Cwind





1

8760n

1+m

design

rated CF Pwind wind

(1 + D)n − 1 D(1 + D)n

 (9) design

From (9) it can be seen that the design capacity factor CFwind of the wind farm heavily influences LCOE and in turn the tariff rate at which the wind farm enters into a power purchase agreement. Hence any ED for a wind farm must enable it to maintain an average capacity factor that is close to the one used to calculate the LCOE so as to be financially viable. The difference between actual and design LCOE is calculated using (10).



LCOE Cwind



1

=

CFactual wind

−



×

1+m



1 design

CFwind

I Cwind

8760N

(1 + D)n − 1 D(1 + D)n





1

rated Pwind

(10)

The cost function proposed in this paper is based on the following concepts.

Constraint (16) limits the dispatch of the wind farm within the limits imposed by the predicted wind power and cut-in wind power cut-in . The cut-in wind power is dependent on the type and number Pwind,i of operational turbines in each time block. Additionally to quantify the amount of predicted wind utilized in the dispatch we define a wind utilization factor Uwind given by (17). 4.5. Pumped hydel generation Pumped storage systems utilize energy generated at off peak hours to store energy in the form of a water head and return it back during peak times [1]. Currently they are the most widely used system for bulk energy storage in a power system with around 40 such systems in the United States alone [17]. The cost function for pumped hydel as given by (18) is effective for peak load manageretail correlates with load demand. ment as long as the retail price Rarea,i The average cost at which the utility will have to purchase a unit of energy in each time block for a given generation schedule is given by (19). This may be defined as the wholesale cost of energy in the control area. An average of the wholesale cost for the entire dispatch period may be used as another performance indicator. Cph =

N 

pump

gen

wholesale retail (Rarea,i (Pph,i Iph,i ) − Rarea,i (Pph,i Iph,i ))

(18)

i=1

(a) Assigns a cost for not using all the available wind power based on the long term economic status of the wind farm. (b) Assign a cost to the spinning reserves associated with meeting the shortfall when the scheduled wind power is not be available. The costs defined above are calculated separately for each time block and the greater of the two is the output of the wind power cost

wholesale Rarea,i =

Ptherm,i Rtherm +Phydel,i Rhydel +Pwind,i Rwind +Ptie,i Rtie,i Ptherm,i +Phydel,i +Pwind,i +Ptie,i

(19)

It is assumed that the thermal, hydel and wind generation sources sell power at fixed rates Rtherm , Rhydel , and Rwind , respecwholesale gives the tively, under a power purchase agreement. Rarea,i specific cost of energy within the control area in the ith time block.

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The charge and discharge cycles of the pumped hydel storage system is also subject to constraints imposed by the maximum and minimum energy storage levels as given by (20). stored-min stored stored-max Eph ≤ Eph,i ≤ Eph gen

pump

pump

Iph,i Pph,i effph

−

Iph,i Pph,i gen

effph

Closses

 24 N

24  = N N

(20)



stored stored Eph,i = Eph,i−1 +

transmission of power from wind farm to sub-station are not considered since it is not paid for by the transmission entity and there is no fuel cost associated with it.

 

loss Pgen,i

=

4.6. Solar PV generation Solar photovoltaic (PV) plants are segmented into three application categories – residential, commercial and utility. The PV systems installed on residential and commercial establishments are primarily for meeting their in house load and only excess energy generated is fed to grid. Utility scale systems on the other hand feed all of their power into the grid. Currently PV generation is treated by ISO’s as negative load and the same methodology is followed in this paper. A couple of compelling reasons to support this from an ED point of view are as follows. (1) Solar is highly distributed with kW size plants (residential and commercial) outnumbering utility scale MW plants. In aggregate capacity they are evenly split in the US while in Europe the commercial scale plants contribute to the majority of the capacity [18–20]. It is practically infeasible to dispatch and monitor generation from such a large number of generation sources. (2) The maximum capacity factor that solar PV plants can reach on a given day is in the range of 14–24% in case of fixed tilt systems and as high as 33% in the case of 2 axis tilt systems [21]. But average capacity factors are significantly lower because of cloud shading and other effects. Low capacity factors and the high initial investments necessitates that all energy generated from PV be utilized. The net power flow from a load with distributed solar PV can be estimated using 22. net rated Psolar,i = effPV fsun,i Psolar − Lself,i

(22)

fsun,i = f (L, ı, s , c , ε, d, ST )

(23)

rated is the name plate rating of solar PV array at standard test Psolar conditions. fsun,i gives fraction of the rated PV output available at a specific time interval and is a function of latitude (L), declination angle (ı), azimuth angle of the sun (ϕS ), collector angle (ϕC ), array tilt (ε), day of the year (d), and solar time (ST). effPV is the gross array efficiency which depends primarily on cell temperature, shading and inverter efficiency. A good example of the cross correlation between the actual and theoretical PV output curves can be seen in [22]. The magnitude of the theoretical solar output is found to be slightly higher because it does not take into account of losses due to shading.

loss Pd,i

=

ohmic Dgen REHV

L−L VEHV pf



N−N Pa,b,i

(25)

2 ohmic Da,b RHV

L−L VHV pf

a= / b

(24)

2

Pgen,i



loss wholesale + Pd,i Rarea,i

gen

i=1

(21)



loss Pgen,i Rgen

(26)

loss is the loss incurred while transmitting power from the Pgen,i loss is the conventional plants to a node in the control area while Pd,i loss incurred during power transmission between nodes. There is also a maximum load carrying capacity associated with each transmission line which is the ampacity times the design voltage level. In case of control areas having high degree of renewable energy penetration this becomes more important because the inter sub-station tie lines may not be upgraded to carry the extra power. However since the reactive power flows also contribute to the current flow in the transmission line, it would be more appropriate to consider these constraints in an OPF problem instead of a purely ED problem [5,6].

4.8. Spinning reserve Spinning reserve is the generation capacity that is on-line but unloaded and that can respond within 10 min to compensate for generation or transmission outages. Through multiarea reserve sharing an area can depend upon the neighboring area to meet part of its reserve capacity. However, the amount of reserve sharing is limited by tie line carrying capacity and costs associated with using that reserve. The cost function (27) is used to discourage the usage of the neighboring area reserve. In the same vein, there is also a price associated with meeting the load demand by buying power from neighboring control area which is given by (28). Creserve = Rreserve

N  

reserve Px,y,i

(27)

/ y i=1 x =

Ctie = Rtie,i 24 N

N  

Px,y,i

(28)

/ y i=1 x = reserve Parea,i =



reserve Pgen,i

(29)

reserve reserve-min Px,y ≥ Parea,i

(30)

gen reserve Parea,i +

 x= / y

4.7. Transmission losses

reserve-min Parea,i = max(Pgen,i )

(31)

Transmission and distribution losses play a key role in determining the cost of energy from the point of view of the transmission entity. Only I2 R losses are being included in the cost function since they are most directly influenced by the scheduled generation from different sources. The monetary costs arising out of I2 R losses between conventional sources and substations and between substations are accounted for in the cost function given by (24). It is also assumed that conventional sources are metered at source and renewable sources are metered at sub-station. Hence losses in

reserve online Pgen,i = Pgen,i − Pgen,i

(32)

Rreserve is the rate per MW for booking reserve capacity from a neighboring control area while Rtie,i is the cost for buying power. Optimizing the spinning reserve available based on probabilistic criteria as opposed to a deterministic criteria has been proposed in [23]. However in this paper the traditional formulation is used whereby the spinning reserve requirement is taken to be equal to the largest scheduled conventional generation source.

S.J. Plathottam, H. Salehfar / Electric Power Systems Research 119 (2015) 313–321 Table 1 Important parameter values.

4.9. Combined objective function The comprehensive cost function (33) to be minimized is obtained by combining costs from all generation sources. Due to difference in scale of the costs, they are each multiplied by a weight. The power balance constraint at the ith time block is given by (34). It is possible to gauge the overall effect of the ED on the utility by estimating the cost of buying energy (35) and profit margin using (36). Obj = xt Ctherm + xh Chydel + xw Cwind + xph Cph + xl Closses + xr Creserve

(33)

Parameter

Value

Unit

Time blocks in a day, N online Ptherm,i max-ramp Ptherm online Phydel,i

24 1000 5 800

NA MW MW/min MW

Phydel

7.5

MW/min

1

MW/min

max-ramp min-ramp

Ptherm

min-ramp

, Phydel

water Fin Ahydel pred Pwind,i

stored-max Eph

net Pload,i + Plosses,i = Ptherm,i + Phydel,i + Pwind,i + Pph,i + Psolar,i

+ Ptie,i

cost = Cutility

24 N

N 

(34)

(Rtherm Ptherm,i + Rhydel Phydel,i + Rwind Pwind,i )

i=1

+ Ctie

317

106 m3 /h 106 m2 [300 300 250 275 350 350 300 300 250 275 350 375 400 400 350 375 350 300 300 250 250 275 350 50] MW 120 MWh

stored-min Eph

20

MWh

Rtherm Rhydel Rwind Rtie Rreserve design CFwind xt , xh , xl , xph xw xr

30 5 35 40 500 40 1 5 10−4

$/MWh $/MWh $/MWh $/MWh $/MW % NA NA NA

(35) 6. Results and discussion

profit

retail cost − C Cutility = Rarea,i Pload,i 24 − Cutility reserve N

(36)

xt , xh , xw , xph , xl and xr are the relative weights given to different components of the cost function. It is assumed that the consumer loads Pload,i are charged under a time of use (TOU) tariff structure given by Rretail,i . 5. Optimization algorithm Historically ED problems which involve conventional generation sources could be solved using Lagrange multipliers and other gradient methods because the objective function was convex and continuous. Numerous other algorithms based on gradient and direct search methods have been used to solve non-convex cost functions. However meta-heuristic search methods are generally found to give better solutions for non-convex functions. Simulated annealing (SA) and particle swarm optimization (PSO) methods have been successfully used in [9,10] for economic dispatch in a control area involving wind and conventional generation. A few other early attempts at using the SA and PSO methods to solve the unit commitment problem are given in [24–26]. The optimization problem developed in Section 4 of this work will be solved using both the SA and the PSO methods. The algorithm developed by the authors consists of two distinct steps. In the first step the generation of each source is allocated randomly so as to satisfy the operating (1), availability (2) and (16) and the power balance constraints (34) in each time block. In case of the SA method one set of generation schedule is created while in case of the PSO approach ‘n’ distinct sets of generation schedules are created. Here ‘n’ is the number of particles in the PSO swarm. In the SA, a fraction of the total generation is randomly shifted among the sources followed by an evaluation of the objective functions (4), (11), (18), (24), (27) and (33). The rest of the algorithm is similar to that used in the standard SA as discussed in [27]. In case of PSO, the particle with the best objective cost (least cost) is selected as the global best in the swarm. Then the position (generation schedule) of a particle is updated using the standard equations for the PSO inertia, memory and cooperation. This is followed by evaluating the objective cost function at the new position after checking whether the operating constraints are satisfied.

The optimization problem described in the previous sections was programmed and executed in MATLAB. Table 1 lists the important operational and economic parameters used in the program. In the simulation, a higher weight has been assigned to the wind component of the cost function to highlight the effect of the wind farm’s operating capacity factor. A very low weight is assigned to the reserve cost since its cost function return values several magnitudes larger than the other costs. The objective function values versus the successful iterations of the SA and PSO methods is plotted in Figs. 2 and 3, respectively. The behavior of these plots conforms to what is generally produced by these algorithms. However, in case of the SA the iterations are found to terminate prematurely even though the behavior of the objective function indicates a downward trend. It is also interesting to note that the cost incurred by the utility in purchasing power is also

Fig. 2. Objective function value versus number of iterations for SA.

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Fig. 3. Global and individual particle objective function value versus number of iterations for PSO.

Fig. 5. Comparison of predicted and scheduled wind power at near design (39%) capacity factor.

Fig. 4. Behavior of particles containing value of wind power dispatch at 6th time block versus iterations.

decreasing even though these costs are not implicit in the objective function. The behavior of the objective function value from PSO (Fig. 3) shows that the objective cost of the particles tends to oscillate above the global minimum which reaches a steady state. Fig. 4 shows the behavior of 6 particles for a single variable (wind power dispatch in the 6th time block). It can be found that the particles are oscillating about the globally best performing particle which itself is in a state of convergence. A similar behavior is exhibited by the other particles though the severity of oscillations and number of iterations taken to converge varies. In the above scenario the wind farm has been operating close to its design capacity factor till date with CFactual wind equal to 39%. Fig. 5 compares predicted wind power available with wind power scheduled by the ED algorithm and the reserve capacity available in the C LCOE ) control area. It can be observed that the effect of (Wwind Cwind C has been surpassed by that of Wreserve in this scenario and the wind dispatch is curtailed dramatically when reserve capacity is low. Conversely in the alternate scenario depicted in Fig. 6 where the wind farm had been operating at half its design capacity factor C LCOE (CFactual wind = 20%) the effect of (Wwind Cwind ) is predominant.

Fig. 6. Comparison of predicted and scheduled wind power at 20% capacity factor.

However in both scenarios all the predicted wind power is dispatched when enough reserve is available. The above examples illustrate that the algorithm is able to respond effectively to control area reserve constraints and be influenced by the wind farm economic situation at the same time. This behavior is also reflected in the average wholesale cost of energy (Section 4.5) versus iterations plot for different capacity factors shown in Fig. 7. The average wholesale cost is tending to a higher value for the scenario with a actual low CFactual wind compared to that with a high CFwind . This is intuitive since wind power is more costly than hydel and thermal power and a higher fraction of wind power in the generation mix would entail a higher cost for utility. But this is a desirable premium that guarantees the sustainability of the wind farms. The performances indices for thermal and hydel plant described earlier are plotted in Fig. 8 for scenario 1. According to [28] the average heat rate of coal fired plants in the US as of 2011 was nearly

S.J. Plathottam, H. Salehfar / Electric Power Systems Research 119 (2015) 313–321

319

Fig. 7. Average whole sale cost of energy versus iterations.

Fig. 9. Day ahead economic dispatch for all generation sources.

Table 3 Costs obtained when wind power is treated negative load. Parameter

Run 1

MAPE Fraction of predicted wind utilized, Uwind Optimal objective value Reserve costs, Creserve cost Utility costs, Cutility Cost of losses, Closses

Fig. 8. Thermal and hydel plant performance index versus iterations. Table 2 Costs obtained for three runs. Parameter Wind farm operating capacity factor Fraction of predicted wind utilized, Uwind Optimal objective value Reserve costs, Creserve cost Utility costs, Cutility Cost of losses, Closses

Run 1

Run 2

0.2

0.2

83%

88%

Run 3 0.39 25%

7.21× 105 2.44× 106 6.98× 105

7.24× 105 2.02× 106 7.41× 105

4.83× 105 5.18× 106 6.9× 105

1.42× 104

1.37× 104

2.06× 104

9.8% 101%

Run 2 10% 95%

Run 3 9.6% 101%

2.19 × 105 1.13 × 106 6.52 × 105

2.29 × 105 1.39 × 106 6.6 × 105

2.73 × 105 2.31 × 106 7.18 × 105

1.18 × 104

1.17 × 104

1.31 × 104

correlates with reserve costs Creserve . This is to expected since higher share of wind power translates to more spare capacity available in the thermal and hydel generation within the control area. But as will be seen in the next section this does not scale. Another fact observed is that lower objective function values translate to lower utility energy costs and improve thermal heat rate, or losses. The results show the scope for significant improvement in costs by proper tuning of the weights to fit different operational and economic priorities. However the authors acknowledge that in the proposed formulation for wind power dispatch there is always a possibility that wind power which might have been available is lost for gains that are not easily quantifiable. In defense, a few results are presented that are obtained by removing the wind cost from the objective function and treating wind as a negative load. The wind generation actual is generated from P pred with a specific mean in each block Pwind,i wind,i absolute predicted error (MAPE) using (37). pred

11,000 kJ/kWh which concurs with the calculated value. It is difficult to generalize the specific water consumption rates for hydel plants because of the large differences in water head and tend to vary from 10 to 3000 m3 /MWh if some of the largest plants in the world are considered, as in [29]. The day ahead ED for all the generation sources is plotted in Fig. 9. The dispatch solution is able to meet the load in most circumstances and the ramp rate constraints are always obeyed. Table 2 tabulates the final costs obtained for three different runs of the algorithm. It can be immediately observed that Uwind

MAPE = 100

actual ) abs(Pwind,i − Pwind,i rated Pwind

(37)

MAPE takes into account both over and under prediction of actual may be higher or lower available wind power and hence Pwind,i pred

than Pwind,i . The modified optimization problem may be considered to be a crude approximation to an ED running in real time and the results are tabulated in Table 3. Among the three runs shown in Table 3 it is interesting to observe that higher wind utilization does not always result in lower utility energy and reserve costs

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and vice versa. Instead the optimal objective value of the comprehensive objective function is found to correlate with lower costs. Additionally, an example of the potential of dispatching wind can be observed by comparing run 2 in Table 2 with run 3 in Table 3. It can be seen that 88% wind utilization can give better reserve costs compared to 100% wind utilization. 7. Conclusion and further work An ED optimization problem using traditional and nontraditional cost functions was developed for conventional and renewable generation sources in a control area. The preliminary results indicate that it is possible to dispatch both conventional and renewable sources in a manner that provides equitable benefit to both generation and transmission entities. Also the fact that the problem could be solved using two different heuristic methods enhances the feasibility of the proposed cost function. Comparing results obtained by the ED problem when wind is a negative load, there is evidence to suggest that dispatching wind can provide better results even if all available wind is not utilized. However, further fine tuning of the algorithm and cost function weights are required to improve the solution. It may also be possible to improve the solution using other heuristic cost functions like Genetic algorithm. 8. Appendix

N Ptherm,i online Ptherm,i reserve Ptherm,i

Phydel,i reserve Phydel,i

Chydel penalty

number of time blocks into which a day (24 h) is divided into power scheduled from thermal generation in ith time block (MW) thermal power generation capacity available in the ith time block (MW) thermal power reserve capacity available in the ith time block (MW) power scheduled from hydel generation in ith time block (MW) thermal power reserve capacity available in the ith time block (MW) cost of hydel power generation

Chydel

cost of violating hydel reservoir level limits

water Fout,i

flow rate of water out of hydel reservoir in ith time block

(m3 /h) surface area of reservoir (m2 ) Ahydel Hhydel,i water level in hydel reservoir in ith time block (m) avg Hi effective head in hydel reservoir in ith time block (m) hydel hydraulic turbine efficiency (%) Hhigh , Hlow maximum and minimum water level to be maintained in hydel reservoir (m) WH , WL penalty for deviating from reservoir high and low limits LCOE levelized cost of wind energy ($/MWh) Cwind design

CFwind rated Pwind I Cwind

D m

n C Wwind

dayavg

CFwind r min Pwind Uwind

design capacity factor of wind generation rated (nameplate) capacity of wind generation (MW) capital investment on wind power generation ($) discount rate of wind power capital investment annual maintenance cost of wind power as a fraction of capital investment operational lifetime of wind power generation (years) penalty for low day average of the capacity factor day average of the wind farm capacity factor min conventional generation capacity required as backup for wind power (MW) fraction of predicted wind power scheduled for dispatch

Lself,i

in house load of distributed PV plants in ith time blocks (MW) Pph,i power scheduled from pumped-hydel generation in ith time block (MW) stored Eph,i energy stored in pumped-hydel system in ith time block (MWh) gen pump effph , effph generation and pumping efficiency of the pumpedhydel system stored-min , E stored-max energy storage limits of pumped-hydel sysEph ph tem (MWh) pump gen Iph,i , Iph,i pumping/generating status of pumped-hydel storage wholesale wholesale cost of energy in the control area ($/MWh) Rarea,i retail Rarea,i

retail cost of energy in the control area ($/MWh)

Rtherm Rwind Rhydel Rtie,i

cost at which utility buys thermal power ($/MWh) cost at which utility buys wind power ($/MWh) cost at which utility buys hydel power ($/MWh) cost at which utility buys power from neighboring control area ($/MWh) acceleration due to gravity (MW/min) g  density of water (kg/m3 ) Dt , Dh , Dab length of transmission line for thermal, hydel and between nodes. The values are 200 km, 400 km and 100 km respectively ohmic , Rohmic resistance per unit length of EHV and HV transmisREHV HV sion lines (0.02 /km) L−L L−L VEHV , VHV voltage level of EHV (500 kV) and HV (135 kV) transmission lines References

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